import { type PracticeQuestion } from "../../types/lesson";

export const NONLINEAR_FUNC_EASY: PracticeQuestion[] = [
  {
    id: "044c1cb7",
    type: "mcq",
    questionHtml:
      "<strong>h (x) = x² − 3</strong><br>Which table gives three values of <strong>x</strong> and their corresponding values of <strong>h (x)</strong> for the given function <strong>h</strong>?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>1</strong><br><strong>2</strong><br><strong>3</strong><br><br><strong>h (x)</strong><br><strong>4</strong><br><strong>5</strong><br><strong>6</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>1</strong><br><strong>2</strong><br><strong>3</strong><br><br><strong>h (x)</strong><br><strong>−2</strong><br><strong>1</strong><br><strong>6</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>1</strong><br><strong>2</strong><br><strong>3</strong><br><br><strong>h (x)</strong><br><strong>−1</strong><br><strong>1</strong><br><strong>3</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>1</strong><br><strong>2</strong><br><strong>3</strong><br><br><strong>h (x)</strong><br><strong>−2</strong><br><strong>1</strong><br><strong>3</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It′s given that <strong>h (x) = x² − 3</strong>. Each table gives <strong>1</strong>, <strong>2</strong>, and <strong>3</strong> as the three given values of <strong>x</strong>. Substituting <strong>1</strong> for <strong>x</strong> in the equation <strong>h (x) = x² − 3</strong> yields <strong>h (1) = (1)² − 3</strong>, or <strong>h (1) = −2</strong>. Substituting <strong>2</strong> for <strong>x</strong> in the equation <strong>h (x) = x² − 3</strong> yields <strong>h (2) = (2)² − 3</strong>, or <strong>h (2) = 1</strong>. Finally, substituting <strong>3</strong> for <strong>x</strong> in the equation <strong>h (x) = x² − 3</strong> yields <strong>h (3) = (3)² − 3</strong>, or <strong>h (3) = 6</strong>. Therefore, <strong>h (x)</strong> is <strong>−2</strong> when <strong>x</strong> is <strong>1</strong>, <strong>h (x)</strong> is <strong>1</strong> when <strong>x</strong> is <strong>2</strong>, and <strong>h (x)</strong> is <strong>6</strong> when <strong>x</strong> is <strong>3</strong>. Choice B is a table with these values of <strong>x</strong> and their corresponding values of <strong>h (x)</strong>.<br>Choice A is incorrect. This is a table of values for the function <strong>h (x) = x + 3</strong>, not <strong>h (x) = x² − 3</strong>.<br>Choice C is incorrect. This is a table of values for the function <strong>h (x) = 2 x − 3</strong>, not <strong>h (x) = x² − 3</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "09f58996",
    type: "spr",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 6 + √(x)</strong>. What is the value of <strong>f (36)</strong>?",
    choices: [],
    correctAnswer: "12",
    explanation:
      "The correct answer is <strong>12</strong>. The value of <strong>f (36)</strong> is the value of <strong>f (x)</strong> when <strong>x = 36</strong>. Substituting <strong>36</strong> for <strong>x</strong> in the given equation yields <strong>f (36) = 6 + √(36)</strong>, which is equivalent to <strong>f (36) = 6 + 6</strong>, or <strong>f (36) = 12</strong>. Thus, the value of <strong>f (36)</strong> is <strong>12</strong>.",
    hasFigure: false,
  },
  {
    id: "1ee962ec",
    type: "mcq",
    questionHtml:
      "The parabola opens upward.<br>The parabola passes through the following points:<br><br>(0 comma 0)<br>(3 comma negative 12)<br>(6 comma 0)<br><br>Scientists recorded data about the ocean water levels at a certain location over a period of <strong>6</strong> hours. The graph shown models the data, where <strong>y = 0</strong> represents sea level. Which table gives values of <strong>x</strong> and their corresponding values of <strong>y</strong> based on the model?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>−12</strong><br><br><strong>0</strong><br><strong>3</strong><br><br><strong>3</strong><br><strong>6</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>3</strong><br><strong>12</strong><br><br><strong>0</strong><br><strong>−6</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>3</strong><br><strong>−12</strong><br><br><strong>6</strong><br><strong>0</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>12</strong><br><strong>3</strong><br><br><strong>−6</strong><br><strong>0</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Each point <strong>(x, y)</strong> on the graph represents an elapsed time <strong>x</strong>, in hours, and the corresponding ocean water level <strong>y</strong>, in feet, at a certain location based on the model. The graph shown passes through the points <strong>(0, 0)</strong>, <strong>(3 −12)</strong>, and <strong>(6, 0)</strong>. Thus, the table in choice C gives the values of <strong>x</strong> and their corresponding values of <strong>y</strong> based on the model.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/1ee962ec_svg1.svg",
  },
  {
    id: "20722644",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = x³ + 9</strong>. What is the value of <strong>f (2)</strong>?",
    choices: [
      { label: "A", text: "<strong>14</strong>" },
      { label: "B", text: "<strong>15</strong>" },
      { label: "C", text: "<strong>17</strong>" },
      { label: "D", text: "<strong>18</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>f (x) = x³ + 9</strong>. Substituting <strong>2</strong> for <strong>x</strong> in this equation yields <strong>f (2) = (2)³ + 9</strong>. This is equivalent to <strong>f (2) = 8 + 9</strong>, or <strong>f (2) = 17</strong>.<br>Choice A is incorrect. This is the value of <strong>2 + 3 + 9</strong>, not <strong>2³ + 9</strong>.<br>Choice B is incorrect. This is the value of <strong>2 (3) + 9</strong>, not <strong>2³ + 9</strong>.<br>Choice D is incorrect. This is the value of <strong>3² + 9</strong>, not <strong>2³ + 9</strong>.",
    hasFigure: false,
  },
  {
    id: "2cf7f039",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 8 √(x)</strong>. For what value of <strong>x</strong> does <strong>f (x) = 48</strong>?",
    choices: [
      { label: "A", text: "<strong>6</strong>" },
      { label: "B", text: "<strong>8</strong>" },
      { label: "C", text: "<strong>36</strong>" },
      { label: "D", text: "<strong>64</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>f (x) = 8 √(x)</strong>. Substituting <strong>48</strong> for <strong>f (x)</strong> in this equation yields <strong>48 = 8 √(x)</strong>. Dividing both sides of this equation by <strong>8</strong> yields <strong>6 = √(x)</strong>. This can be rewritten as <strong>√(x) = 6</strong>. Squaring both sides of this equation yields <strong>x = 36</strong>. Therefore, the value of <strong>x</strong> for which <strong>f (x) = 48</strong> is <strong>36</strong>.<br>Choice A is incorrect. If <strong>x = 6</strong>, <strong>f (x) = 8 √(6)</strong>, not <strong>48</strong>.<br>Choice B is incorrect. If <strong>x = 8</strong>, <strong>f (x) = 8 √(8)</strong>, not <strong>48</strong>.<br>Choice D is incorrect. If <strong>x = 64</strong>, <strong>f (x) = 8 √(64)</strong>, which is equivalent to <strong>64</strong>, not <strong>48</strong>.",
    hasFigure: false,
  },
  {
    id: "3ea87153",
    type: "mcq",
    questionHtml:
      "The function <strong>g</strong> is defined by <strong>g (x) = x² + 9</strong>. For which value of <strong>x</strong> is <strong>g (x) = 25</strong>?",
    choices: [
      { label: "A", text: "<strong>4</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>9</strong>" },
      { label: "D", text: "<strong>13</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>g (x) = x² + 9</strong>. Substituting <strong>25</strong> for <strong>g (x)</strong> in this equation yields <strong>25 = x² + 9</strong>. Subtracting <strong>9</strong> from both sides of this equation yields <strong>16 = x²</strong>. Taking the square root of each side of this equation yields <strong>x = + or − 4</strong>. It follows that <strong>g (x) = 25</strong> when the value of <strong>x</strong> is <strong>4</strong> or <strong>−4</strong>. Only <strong>4</strong> is listed among the choices.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "4fbffc0a",
    type: "mcq",
    questionHtml:
      "All values are approximate.<br>The parabola opens downward.<br>The vertex of the parabola is at (0.6 comma 5.5).<br>The parabola passes through the following points:<br><br>(0.0 comma 3.8)<br>(0.6 comma 5.5)<br>(1.0 comma 4.8)<br>(1.7 comma 0.0)<br><br>The graph shows the height above ground, in meters, of a ball <strong>x</strong> seconds after the ball was launched upward from a platform. Which statement is the best interpretation of the marked point <strong>(1.0, 4.8)</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "<strong>1.0</strong> second after being launched, the ball's height above ground is <strong>4.8</strong> meters.",
      },
      {
        label: "B",
        text: "<strong>4.8</strong> seconds after being launched, the ball's height above ground is <strong>1.0</strong> meter.",
      },
      {
        label: "C",
        text: "The ball was launched from an initial height of <strong>1.0</strong> meter with an initial velocity of <strong>4.8</strong> meters per second.",
      },
      {
        label: "D",
        text: "The ball was launched from an initial height of <strong>4.8</strong> meters with an initial velocity of <strong>1.0</strong> meter per second.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the graph shows the height above ground, in meters, of a ball <strong>x</strong> seconds after the ball was launched upward from a platform. In the graph shown, the x-axis represents time, in seconds, and the y-axis represents the height of the ball above ground, in meters. It follows that for the marked point <strong>(1.0, 4.8)</strong>, <strong>1.00</strong> represents the time, in seconds, after the ball was launched upward from a platform and <strong>4.80</strong> represents the height of the ball above ground, in meters. Therefore, the best interpretation of the marked point <strong>(1.0, 4.8)</strong> is <strong>1.00</strong> second after being launched, the ball's height above ground is <strong>4.80</strong> meters.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: true,
    figureUrl: "/practice-images/4fbffc0a_svg1.svg",
  },
  {
    id: "50418728",
    type: "spr",
    questionHtml:
      "In quadrant 3, the curve rises to its maximum at point (0 comma negative 3).<br>In quadrant 4, the curve then falls.<br><br>The graph of the polynomial function <strong>f</strong>, where <strong>y = f (x)</strong>, is shown. The y-intercept of the graph is <strong>(0, y)</strong>. What is the value of <strong>y</strong>?",
    choices: [],
    correctAnswer: "-3",
    explanation:
      "The correct answer is <strong>−3</strong>. The y-intercept of the graph of a function in the xy-plane is the point where the graph crosses the y-axis. The graph of the polynomial function shown crosses the y-axis at the point <strong>(0 −3)</strong>. It's given that the y-intercept of the graph is <strong>(0, y)</strong>. Thus, the value of <strong>y</strong> is <strong>−3</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/50418728_svg1.svg",
  },
  {
    id: "5377d9cf",
    type: "mcq",
    questionHtml:
      "If <strong>f of x = the fraction with numerator x² − 6 x + 3, and denominator x − 1, end fraction</strong>, what is <strong>f of − 1</strong> ?",
    choices: [
      { label: "A", text: "–5" },
      { label: "B", text: "–2" },
      { label: "C", text: "2" },
      { label: "D", text: "5" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Substituting –1 for x in the equation that defines f gives <strong>f of − 1 = the fraction with numerator, (−1, ), ² − 6 · (−1, ) + 3, and denominator − 1 − 1, end fraction</strong>. Simplifying the expressions in the numerator and denominator yields <strong>the fraction with numerator 1 + 6 + 3, and denominator − 2</strong>, which is equal to <strong>10 over − 2</strong> or –5.Choices B, C, and D are incorrect and may result from misapplying the order of operations when substituting –1 for x.",
    hasFigure: false,
  },
  {
    id: "6abec9a8",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve passes from quadrant 3 to quadrant 4.<br>In quadrant 3, the curve trends up sharply to point (0 comma negative 5).<br>In quadrant 4, the curve trends up gradually.<br><br>As x increases, the curve approaches the line y equals negative 4.<br>The curve passes through the following points:<br><br>(negative 1 comma negative 9)<br>(0 comma negative 5)<br>(1 comma negative 4.2)<br><br>What is the <strong>y</strong>-intercept of the graph shown?",
    choices: [
      { label: "A", text: "<strong>(−1 −9)</strong>" },
      { label: "B", text: "<strong>(0 −5)</strong>" },
      { label: "C", text: "<strong>(0 −4)</strong>" },
      { label: "D", text: "<strong>(0, 0)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The y-intercept of a graph in the xy-plane is the point <strong>(x, y)</strong> on the graph where <strong>x = 0</strong>. At <strong>x = 0</strong>, the corresponding value of <strong>y</strong> is <strong>−5</strong>. Therefore, the y-intercept of the graph shown is <strong>(0 −5)</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice C is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at <strong>y = −4</strong>, not <strong>y = −5</strong>.<br>Choice D is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at <strong>y = 0</strong>, not <strong>y = −5</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/6abec9a8_svg1.svg",
  },
  {
    id: "7160cbb3",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve passes from quadrant 3 to quadrant 4.<br>In quadrant 3, the curve trends up sharply to point (0 comma negative 6).<br>In quadrant 4, the curve trends up gradually.<br><br>As x increases, the curve approaches the line y equals negative 5.<br>The curve passes through the following points:<br><br>(0 comma negative 6)<br>(1 comma negative 5 and one sixth)<br><br>What is the y-intercept of the graph shown?",
    choices: [
      { label: "A", text: "<strong>(0 −6)</strong>" },
      { label: "B", text: "<strong>(−6, 0)</strong>" },
      { label: "C", text: "<strong>(0, 0)</strong>" },
      { label: "D", text: "<strong>(−5 −5)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The y-intercept of a graph in the xy-plane is the point <strong>(x, y)</strong> on the graph where <strong>x = 0</strong>. For the graph shown, at <strong>x = 0</strong>, the corresponding value of y is <strong>−6</strong>. Therefore, the y-intercept of the graph shown is <strong>(0 −6)</strong>.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: true,
    figureUrl: "/practice-images/7160cbb3_svg1.svg",
  },
  {
    id: "72ae8a87",
    type: "mcq",
    questionHtml:
      "The function <strong>f (x) = 200, 000 (1.21)^(x)</strong> gives a company’s predicted annual revenue, in dollars, <strong>x</strong> years after the company started selling light bulbs online, where <strong>0 < x < or = 10</strong>. What is the best interpretation of the statement “<strong>f (5)</strong> is approximately equal to <strong>518, 748</strong>” in this context?",
    choices: [
      {
        label: "A",
        text: "<strong>5</strong> years after the company started selling light bulbs online, its predicted annual revenue is approximately <strong>518, 748</strong> dollars.",
      },
      {
        label: "B",
        text: "<strong>5</strong> years after the company started selling light bulbs online, its predicted annual revenue will have increased by a total of approximately <strong>518, 748</strong> dollars.",
      },
      {
        label: "C",
        text: "When the company’s predicted annual revenue is approximately <strong>518, 748</strong> dollars, it is <strong>5</strong> times the predicted annual revenue for the previous year.",
      },
      {
        label: "D",
        text: "When the company’s predicted annual revenue is approximately <strong>518, 748</strong> dollars, it is <strong>5 % sign</strong> greater than the predicted annual revenue for the previous year.",
      },
    ],
    correctAnswer: "A",
    explanation:
      'Choice A is correct. It\'s given that the function <strong>f (x) = 200, 000 (1.21)^(x)</strong> gives a company\'s predicted annual revenue, in dollars, <strong>x</strong> years after the company started selling light bulbs online. It follows that <strong>f (x)</strong> represents the company\'s predicted annual revenue, in dollars, <strong>x</strong> years after the company started selling light bulbs online. Since the value of <strong>f (5)</strong> is the value of <strong>f (x)</strong> when <strong>x = 5</strong>, it follows that "<strong>f (5)</strong> is approximately equal to <strong>518, 748</strong>" means that <strong>f (x)</strong> is approximately equal to <strong>518, 748</strong> when <strong>x = 5</strong>. Therefore, the best interpretation of the statement "<strong>f (5)</strong> is approximately equal to <strong>518, 748</strong>" in this context is <strong>5</strong> years after the company started selling light bulbs online, its predicted annual revenue is approximately <strong>518, 748</strong> dollars.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.',
    hasFigure: false,
  },
  {
    id: "75915e3c",
    type: "mcq",
    questionHtml:
      "For the function f defined above, what is the value of <strong>f of 2</strong>?",
    choices: [
      { label: "A", text: "9" },
      { label: "B", text: "12" },
      { label: "C", text: "18" },
      { label: "D", text: "36" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The value of <strong>f of 2</strong> is found by evaluating the expression <strong>2 · 3 raised to the x power</strong> when <strong>x = 2</strong>. Substituting 2 for x in the given equation yields <strong>f of 2 = 2 · 3²</strong>. Simplifying <strong>3²</strong> in the equation results in <strong>f of 2 = 2 · 9</strong>. Evaluating the right-hand side of the equation yields <strong>f of 2 = 18</strong>. Therefore, the value of <strong>f of 2</strong> is 18.Choice A is incorrect and may result from evaluating the expression as <strong>3²</strong>. Choice B is incorrect and may result from evaluating the expression as <strong>2 · (3 · 2, )</strong>. Choice D is incorrect and may result from evaluating the expression as <strong>(2 · 3, ), ²</strong>.",
    hasFigure: false,
  },
  {
    id: "788bfd56",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 4 + √(x)</strong>. What is the value of <strong>f (144)</strong>?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>16</strong>" },
      { label: "C", text: "<strong>40</strong>" },
      { label: "D", text: "<strong>76</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The value of <strong>f (144)</strong> is the value of <strong>f (x)</strong> when <strong>x = 144</strong>. It's given that the function <strong>f</strong> is defined by <strong>f (x) = 4 + √(x)</strong>. Substituting <strong>144</strong> for <strong>x</strong> in this equation yields <strong>f (144) = 4 + √(144)</strong>. Since the positive square root of <strong>144</strong> is <strong>12</strong>, it follows that this equation can be rewritten as <strong>f (144) = 4 + 12</strong>, or <strong>f (144) = 16</strong>. Therefore, the value of <strong>f (144)</strong> is <strong>16</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the value of <strong>f (1, 296)</strong>, not <strong>f (144)</strong>.<br>Choice D is incorrect. This is the value of <strong>f (5, 184)</strong>, not <strong>f (144)</strong>.",
    hasFigure: false,
  },
  {
    id: "79ba511a",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = x³ + 15</strong>. What is the value of <strong>f (2)</strong>?",
    choices: [
      { label: "A", text: "<strong>20</strong>" },
      { label: "B", text: "<strong>21</strong>" },
      { label: "C", text: "<strong>23</strong>" },
      { label: "D", text: "<strong>24</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The value of <strong>f (2)</strong> is the value of <strong>f (x)</strong> when <strong>x = 2</strong>. Substituting <strong>2</strong> for <strong>x</strong> in the given function yields <strong>f (2) = (2)³ + 15</strong>, or <strong>f (2) = 8 + 15</strong>, which is equivalent to <strong>f (2) = 23</strong>. Therefore, the value of <strong>f (2)</strong> is <strong>23</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the value of <strong>f (2)</strong> when <strong>f (x) = x (3) + 15</strong>, rather than <strong>f (x) = x³ + 15</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "837e9da7",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = (1) / (6 x)</strong>. What is the value of <strong>f (x)</strong> when <strong>x = 3</strong>?",
    choices: [
      { label: "A", text: "<strong>one third</strong>" },
      { label: "B", text: "<strong>one sixth</strong>" },
      { label: "C", text: "<strong>one ninth</strong>" },
      { label: "D", text: "<strong>one eighteenth</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that <strong>f (x) = (1) / (6 x)</strong>. Substituting <strong>3</strong> for <strong>x</strong> in this equation yields <strong>f (3) = (1) / (6 (3))</strong>, or <strong>f (3) = one eighteenth</strong>. Therefore, when <strong>x = 3</strong>, the value of <strong>f (x)</strong> is <strong>one eighteenth</strong>.<br>Choice A is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 0.5</strong>.<br>Choice B is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 1</strong>. <br>Choice C is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 1.5</strong>.",
    hasFigure: false,
  },
  {
    id: "9da41c80",
    type: "mcq",
    questionHtml:
      "A ball is dropped from an initial height of <strong>22</strong> feet and bounces off the ground repeatedly. The function <strong>h</strong> estimates that the maximum height reached after each time the ball hits the ground is <strong>85 % sign</strong> of the maximum height reached after the previous time the ball hit the ground. Which equation defines <strong>h</strong>, where <strong>h (n)</strong> is the estimated maximum height of the ball after it has hit the ground <strong>n</strong> times and <strong>n</strong> is a whole number greater than <strong>1</strong> and less than <strong>10</strong>?",
    choices: [
      { label: "A", text: "<strong>h (n) = 22 (0.22)^(n)</strong>" },
      { label: "B", text: "<strong>h (n) = 22 (0.85)^(n)</strong>" },
      { label: "C", text: "<strong>h (n) = 85 (0.22)^(n)</strong>" },
      { label: "D", text: "<strong>h (n) = 85 (0.85)^(n)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that for the function <strong>h</strong>, <strong>h (n)</strong> is the estimated maximum height, in feet, of the ball after it has hit the ground <strong>n</strong> times. It's also given that the function <strong>h</strong> estimates that the maximum height reached after each time the ball hits the ground is <strong>85 % sign</strong> of the maximum height reached after the previous time the ball hit the ground. It follows that <strong>h</strong> is a decreasing exponential function that can be written in the form <strong>h (n) = a ((p) / (100))^(n)</strong>, where <strong>a</strong> is the initial height, in feet, the ball was dropped from and the function estimates that the maximum height reached after each time the ball hits the ground is <strong>p % sign</strong> of the maximum height reached after the previous time the ball hit the ground. It's given that the ball is dropped from an initial height of <strong>22</strong> feet. Therefore, <strong>a = 22</strong>. Since the function <strong>h</strong> estimates that the maximum height reached after each time the ball hits the ground is <strong>85 % sign</strong> of the maximum height reached after the previous time the ball hit the ground, <strong>p = 85</strong>. Substituting <strong>22</strong> for <strong>a</strong> and <strong>85</strong> for <strong>p</strong> in the equation <strong>h (n) = a ((p) / (100))^(n)</strong> yields <strong>h (n) = 22 ((85) / (100))^(n)</strong>, or <strong>h (n) = 22 (0.85)^(n)</strong>.<br>Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is <strong>22 % sign</strong>, not <strong>85 % sign</strong>, of the maximum height reached after the previous time the ball hit the ground.<br>Choice C is incorrect. This function estimates that the ball is dropped from an initial height of <strong>85</strong> feet, not <strong>22</strong> feet, and that the maximum height reached after each time the ball hits the ground is <strong>22 % sign</strong>, not <strong>85 % sign</strong>, of the maximum height reached after the previous time the ball hit the ground.<br>Choice D is incorrect. This function estimates that the ball is dropped from an initial height of <strong>85</strong> feet, not <strong>22</strong> feet.",
    hasFigure: false,
  },
  {
    id: "ad376f1a",
    type: "mcq",
    questionHtml:
      "For the first curve: <br><br>Moving from left to right:<br><br>The curve passes from quadrant 2 to quadrant 1.<br>In quadrant 2, the curve trends up gradually to point (0 comma 3).<br>In quadrant 1, the curve trends up sharply.<br><br>As x increases, the curve approaches the line x equals 4.<br>As x decreases, the curve approaches the line y equals 1.<br>The curve passes through the following points:<br><br>(negative 3 comma 2)<br>(0 comma 3)<br><br>For the second curve: <br><br>Moving from left to right:<br><br>The curve is in quadrant 4.<br>In quadrant 4, the curve trends up sharply.<br><br>As x increases, the curve approaches the line y equals 1.<br>As x decreases, the curve approaches the line x equals 4.<br>The curve passes through the following point:<br><br>(6 comma negative 4)<br><br>The graph of <strong>y = f (x)</strong> is shown in the xy-plane. What is the value of <strong>f (0)</strong>?",
    choices: [
      { label: "A", text: "<strong>−3</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>three fifths</strong>" },
      { label: "D", text: "<strong>3</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Because the graph of <strong>y = f (x)</strong> is shown, the value of <strong>f (0)</strong> is the value of <strong>y</strong> on the graph that corresponds with <strong>x = 0</strong>. When <strong>x = 0</strong>, the corresponding value of <strong>y</strong> is <strong>3</strong>. Therefore, the value of <strong>f (0)</strong> is <strong>3</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: true,
    figureUrl: "/practice-images/ad376f1a_svg1.svg",
  },
  {
    id: "b39d74a0",
    type: "mcq",
    questionHtml:
      "The table shown includes some values of x and their corresponding values of y. Which of the following graphs in the xy-plane could represent the relationship between x and y ?",
    choices: [
      {
        label: "A",
        text: "<strong>The answer choice presents the graph of a function in the xy plane. The origin is labeled O, and the numbers − 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins above the x axis and to the left of the y axis, moves downward and to the right, and extends below the x axis and to the right of the y axis. From left to right, the curve moves downward, at first quickly, and passes through −1, 1. It then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly downward, then more quickly, and passes through, 1 −1. It then extends downward and to the right.</strong>",
      },
      {
        label: "B",
        text: "<strong>The answer choice presents the graph of a function in the xy plane. The origin is labeled O, and the numbers − 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, passes through −1 −1, and then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly upward, then more quickly as it passes through, 1, 1. It then extends upward and to the right.</strong>",
      },
      {
        label: "C",
        text: "<strong>The answer choice presents the graph of a function in the xy plane. The origin is labeled O, and the numbers − 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, crosses the x axis at − 1, and then moves more slowly as it crosses the y axis at 1. After crossing the y axis, the curve continues to move slowly upward, then moves more quickly as it passes through, 1, 2. It then extends upward and to the right.</strong>",
      },
      {
        label: "D",
        text: "<strong>The answer choice presents the graph of a function in the xy plane. The origin is labeled O, and the numbers − 4 and 4 are indicated on each axis. The graph of the function is a curve. It begins below the x axis and to the left of the y axis, moves upward and to the right, and extends above the x axis and to the right of the y axis. From left to right, the curve moves upward, at first quickly, passes through −1 −3, and then moves more slowly as it passes through the origin. After crossing the origin, the curve continues to move slowly upward, then moves more quickly as it passes through, 1, 3. It then extends upward and to the right.</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Each pair of values shown in the table gives the ordered pair of coordinates for a point that lies on the graph that represents the relationship between x and y in the xy-plane: <strong>0, 0</strong>, <strong>1, 1</strong>, <strong>2, 8</strong>, and <strong>3, 27</strong>. Only the graph in choice B passes through the points listed in the table that are visible in the given choices.Choices A, C, and D are incorrect. None of these graphs passes through the point <strong>with coordinates 1, 1</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/b39d74a0_img1.png",
  },
  {
    id: "b5c43226",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve passes from quadrant 2 to quadrant 1.<br>In quadrant 2, the curve trends up gradually to point (0 comma 2).<br>In quadrant 1, the curve trends up sharply.<br><br>The curve passes through the following points:<br><br>(negative 1 comma 1)<br>(0 comma 2)<br>(2 comma 8)<br><br>What is the y-intercept of the graph shown?",
    choices: [
      { label: "A", text: "<strong>(0, 0)</strong>" },
      { label: "B", text: "<strong>(0, 2)</strong>" },
      { label: "C", text: "<strong>(2, 0)</strong>" },
      { label: "D", text: "<strong>(2, 2)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point <strong>(0, 2)</strong>. Therefore, the y-intercept of the graph shown is <strong>(0, 2)</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/b5c43226_svg1.svg",
  },
  {
    id: "cc2601cb",
    type: "spr",
    questionHtml:
      "The parabola opens upward.<br>The vertex is at point (7 comma 0).<br>The parabola passes through the following points:<br><br>(6 comma 3)<br>(7 comma 0)<br>(8 comma 3)<br><br>The x-intercept of the graph shown is <strong>(x, 0)</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "7",
    explanation:
      "The correct answer is <strong>7</strong>. It’s given that the x-intercept of the graph shown is <strong>(x, 0)</strong>. The graph passes through the point <strong>(7, 0)</strong>. Therefore, the value of <strong>x</strong> is <strong>7</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/cc2601cb_svg1.svg",
  },
  {
    id: "d46da42c",
    type: "mcq",
    questionHtml:
      "The function f is defined as shown. Which of the following graphs in the xy-plane could be the graph of <strong>y = f of x</strong> ?",
    choices: [
      {
        label: "A",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 8 −4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers − 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers − 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at − 4 on the y axis. The parabola crosses the x axis at − 2 and 2.</strong>",
      },
      {
        label: "B",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 8 −4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers − 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers − 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at − 4 on the x axis. The parabola passes through − 6, 4, and − 2, 4.</strong>",
      },
      {
        label: "C",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 8 −4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers − 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers − 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at 4 on the x axis. The parabola passes through 2, 4, and 6, 4.</strong>",
      },
      {
        label: "D",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 8 −4, 4, and 8 are indicated on each axis. There are horizontal gridlines at the numbers − 8 through 8, in increments of 2, on the y axis, and there are vertical gridlines at the numbers − 8 through 8, in increments of 2, on the x axis. The figure presents an upward opening parabola, whose vertex lies at 4 on the y axis. The parabola passes through − 2, 8, and 2, 8.</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. For the quadratic function <strong>f of x = x² + 4</strong>, the vertex of the graph is <strong>0, 4</strong>, and because the <strong>x²</strong> term is positive, the vertex is the minimum of the function. Choice D is the only option that meets these conditions.Choices A, B, and C are incorrect. The vertex of each of these graphs doesn’t correspond to the minimum of the given function.",
    hasFigure: true,
    figureUrl: "/practice-images/d46da42c_img3.png",
  },
  {
    id: "de362c2f",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 5 x²</strong>. What is the value of <strong>f (8)</strong>?",
    choices: [
      { label: "A", text: "<strong>40</strong>" },
      { label: "B", text: "<strong>50</strong>" },
      { label: "C", text: "<strong>80</strong>" },
      { label: "D", text: "<strong>320</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the function <strong>f</strong> is defined by <strong>f (x) = 5 x²</strong>. Substituting <strong>8</strong> for <strong>x</strong> in <strong>f (x) = 5 x²</strong> yields <strong>f (8) = 5 (8)²</strong>, which is equivalent to <strong>f (8) = 5 (64)</strong>, or <strong>f (8) = 320</strong>. Therefore, the value of <strong>f (8)</strong> is <strong>320</strong>.<br>Choice A is incorrect. This is the value of <strong>f (8)</strong> if <strong>f (x) = 5 x</strong>.<br>Choice B is incorrect. This is the value of <strong>f (8)</strong> if <strong>f (x) = 5 (x + 2)</strong>.<br>Choice C is incorrect. This is the value of <strong>f (8)</strong> if <strong>f (x) = (5 x) (2)</strong>.",
    hasFigure: false,
  },
  {
    id: "e166aca6",
    type: "mcq",
    questionHtml:
      "The parabola opens upward.<br>The vertex is at point (4 comma 0).<br>The parabola passes through the following points:<br><br>(3 comma 5)<br>(4 comma 0)<br>(5 comma 5)<br><br>What is the x-intercept of the graph shown?",
    choices: [
      { label: "A", text: "<strong>(−5, 0)</strong>" },
      { label: "B", text: "<strong>(5, 0)</strong>" },
      { label: "C", text: "<strong>(−4, 0)</strong>" },
      { label: "D", text: "<strong>(4, 0)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The x-intercept of the graph shown is the point <strong>(x, y)</strong> on the graph where <strong>y = 0</strong>. At <strong>y = 0</strong>, the corresponding value of <strong>x</strong> is <strong>4</strong>. Therefore, the x-intercept of the graph shown is <strong>(4, 0)</strong>.<br>Choice A is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at <strong>x = −5</strong>, not <strong>x = 4</strong>.<br>Choice B is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at <strong>x = 5</strong>, not <strong>x = 4</strong>.<br>Choice C is incorrect. This is the x-intercept of a graph in the xy-plane that intersects the x-axis at <strong>x = −4</strong>, not <strong>x = 4</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/e166aca6_svg1.svg",
  },
  {
    id: "ebe4bde0",
    type: "mcq",
    questionHtml:
      "The quadratic function graphed above models a particular measure of plant diversity as a function of the elevation in a region of Switzerland. According to the model, which of the following is closest to the elevation, in meters, at which plant diversity is greatest?",
    choices: [
      { label: "A", text: "13,500" },
      { label: "B", text: "3,000" },
      { label: "C", text: "1,250" },
      { label: "D", text: "250" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Each point <strong>with coordinates x, y</strong> on the graph represents the elevation x, in meters, and the corresponding measure of plant diversity y in a region of Switzerland. Therefore, the point on the graph with the greatest y-coordinate represents the location that has the greatest measure of plant diversity in the region. The greatest y-coordinate of any point on the graph is approximately 13,500. The x-coordinate of that point is approximately 1,250. Therefore, the closest elevation at which the plant diversity is the greatest is 1,250 meters.Choice A is incorrect. This value is closest to the greatest y-coordinate of any point on the graph and therefore represents the greatest measure of plant diversity, not the elevation where the greatest measure of plant diversity occurs. Choice B is incorrect. At an elevation of 3,000 meters the measure of plant diversity is approximately 4,000. Because there are points on the graph with greater y-coordinates, 4,000 can’t be the greatest measure of plant diversity, and 3,000 meters isn’t the elevation at which the greatest measure of plant diversity occurs. Choice D is incorrect. At an elevation of 250 meters, the measure of plant diversity is approximately 11,000. Because there are points on the graph with greater y-coordinates, 11,000 can’t be the greatest measure of plant diversity and 250 meters isn’t the elevation at which the greatest measure of plant diversity occurs.",
    hasFigure: true,
    figureUrl: "/practice-images/ebe4bde0_img1.png",
  },
  {
    id: "ee05c84e",
    type: "mcq",
    questionHtml:
      "The function f is defined above. What is the value of <strong>f of 20</strong> ?",
    choices: [
      { label: "A", text: "250" },
      { label: "B", text: "500" },
      { label: "C", text: "750" },
      { label: "D", text: "2,000" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Adding the like terms x and <strong>0 . 2 5 x</strong> yields the equation <strong>f of x = 1 . 2 5 x · (50 − x, )</strong>. Substituting 20 for x yields <strong>f of 20 = (1 . 2 5 · 20, ) · (50 − 20, )</strong>. The product <strong>1 . 2 5 · 20</strong> is equal to 25, and the difference <strong>50 − 20</strong> is equal to 30. Substituting these values in the given equation gives <strong>f of 20 = 25 · 30</strong>, and multiplying 25 by 30 results in <strong>f of 20 = 750</strong>.Choices A, B, and D are incorrect and may result from conceptual or computational errors when finding the value of <strong>f of 20</strong>.",
    hasFigure: false,
  },
  {
    id: "f547a8b1",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve passes from quadrant 2 to quadrant 1.<br>In quadrant 2, the curve trends up gradually to point (0 comma 8).<br>In quadrant 1, the curve trends up sharply.<br><br>The curve passes through the following points:<br><br>(0 comma 8)<br>(1 comma 9)<br><br>What is the y-intercept of the graph shown?",
    choices: [
      { label: "A", text: "<strong>(−8, 0)</strong>" },
      { label: "B", text: "<strong>(−6, 0)</strong>" },
      { label: "C", text: "<strong>(0, 6)</strong>" },
      { label: "D", text: "<strong>(0, 8)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point <strong>(0, 8)</strong>. Therefore, the y-intercept of the graph shown is <strong>(0, 8)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/f547a8b1_svg1.svg",
  },
];

export const NONLINEAR_FUNC_MEDIUM: PracticeQuestion[] = [
  {
    id: "0aaef7aa",
    type: "mcq",
    questionHtml:
      "The function <strong>p</strong> is defined by <strong>p (n) = 7 n³</strong>. What is the value of <strong>n</strong> when <strong>p (n)</strong> is equal to <strong>56</strong>?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>eight thirds</strong>" },
      { label: "C", text: "<strong>7</strong>" },
      { label: "D", text: "<strong>8</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>p (n) = 7 n³</strong>. Substituting <strong>56</strong> for <strong>p (n)</strong> in this equation yields <strong>56 = 7 n³</strong>. Dividing each side of this equation by <strong>7</strong> yields <strong>8 = n³</strong>. Taking the cube root of each side of this equation yields <strong>2 = n</strong>. Therefore, when <strong>p (n)</strong> is equal to <strong>56</strong>, the value of <strong>n</strong> is <strong>2</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "100030d9",
    type: "spr",
    questionHtml:
      "A rubber ball bounces upward one-half the height that it falls each time it hits the ground. If the ball was originally dropped from a distance of 20.0 feet above the ground, what was its maximum height above the ground, in feet, between the third and fourth time it hit the ground?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 2.5. After hitting the ground once, the ball bounces to <strong>20 . zero ÷ 2 = 10 . zero</strong> feet. After hitting the ground a second time, the ball bounces to <strong>10 . zero ÷ 2 = 5 . zero</strong> feet. After hitting the ground for the third time, the ball bounces to <strong>5 . zero ÷ 2 = 2 . 5</strong> feet. Note that 2.5 and 5/2 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "203774bc",
    type: "mcq",
    questionHtml:
      "The product of two positive integers is <strong>546</strong>. If the first integer is <strong>11</strong> greater than twice the second integer, what is the smaller of the two integers?",
    choices: [
      { label: "A", text: "<strong>7</strong>" },
      { label: "B", text: "<strong>14</strong>" },
      { label: "C", text: "<strong>39</strong>" },
      { label: "D", text: "<strong>78</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Let <strong>x</strong> be the first integer and let <strong>y</strong> be the second integer. If the first integer is <strong>11</strong> greater than twice the second integer, then <strong>x = 2 y + 11</strong>. If the product of the two integers is <strong>546</strong>, then <strong>xy = 546</strong>. Substituting <strong>2 y + 11</strong> for <strong>x</strong> in this equation results in <strong>(2 y + 11) y = 546</strong>. Distributing the <strong>y</strong> to both terms in the parentheses results in <strong>2 y² + 11 y = 546</strong>. Subtracting <strong>546</strong> from both sides of this equation results in <strong>2 y² + 11 y − 546 = 0</strong>. The left-hand side of this equation can be factored by finding two values whose product is <strong>2 (−546)</strong>, or <strong>−1, 092</strong>, and whose sum is <strong>11</strong>. The two values whose product is <strong>−1, 092</strong> and whose sum is <strong>11</strong> are <strong>39</strong> and <strong>−28</strong>. Thus, the equation <strong>2 y² + 11 y − 546 = 0</strong> can be rewritten as <strong>2 y² + 28 y − 39 y − 546 = 0</strong>, which is equivalent to <strong>2 y (y − 14) + 39 (y − 14) = 0</strong>, or <strong>(2 y + 39) (y − 14) = 0</strong>. By the zero product property, it follows that <strong>2 y + 39 = 0</strong> and <strong>y − 14 = 0</strong>. Subtracting <strong>39</strong> from both sides of the equation <strong>2 y + 39 = 0</strong> yields <strong>2 y = −39</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>y = −(39) / (2)</strong>. Since <strong>y</strong> is a positive integer, the value of <strong>y</strong> is not <strong>−(39) / (2)</strong>. Adding <strong>14</strong> to both sides of the equation <strong>y − 14 = 0</strong> yields <strong>y = 14</strong>. Substituting <strong>14</strong> for <strong>y</strong> in the equation <strong>xy = 546</strong> yields <strong>x (14) = 546</strong>. Dividing both sides of this equation by <strong>14</strong> results in <strong>x = 39</strong>. Therefore, the two integers are <strong>14</strong> and <strong>39</strong>, so the smaller of the two integers is <strong>14</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the larger of the two integers.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "252a3b3a",
    type: "mcq",
    questionHtml:
      "In quadrant 3:<br><br>The curve rises sharply to touch the x axis at point (negative 5 comma 0).<br>The curve falls gradually to a relative minimum at point (negative 2 comma negative 11).<br>The curve rises gradually to cross both axes at the origin.<br><br>In quadrant 1:<br><br>The curve rises gradually to a relative maximum at point (2.5 comma 21).<br>The curve falls sharply to cross the x axis at point (4 comma 0).<br><br>In quadrant 4 the curve falls sharply.<br><br>Which of the following could be the equation of the graph shown in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>y = −one tenth x (x − 4) (x + 5)</strong>" },
      {
        label: "B",
        text: "<strong>y = −one tenth x (x − 4) (x + 5)²</strong>",
      },
      { label: "C", text: "<strong>y = −one tenth x (x − 5) (x + 4)</strong>" },
      {
        label: "D",
        text: "<strong>y = −one tenth x (x − 5)² (x + 4)</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Each of the given choices is an equation of the form <strong>y = − one tenth x (x − a)^(m) (x + b)^(n)</strong>, where <strong>a</strong>, <strong>b</strong>, <strong>m</strong>, and <strong>n</strong> are positive constants. In the xy-plane, the graph of an equation of this form has x-intercepts at <strong>x = 0</strong>, <strong>x = a</strong>, and <strong>x = −b</strong>. The graph shown has x-intercepts at <strong>x = 0</strong>, <strong>x = 4</strong>, and <strong>x = −5</strong>. Therefore, <strong>a = 4</strong> and <strong>b = 5</strong>. Of the given choices, only choices A and B have <strong>a = 4</strong> and <strong>b = 5</strong>. For an equation in the form <strong>y = − one tenth x (x − a)^(m) (x + b)^(n)</strong>, if all values of <strong>x</strong> that are less than <strong>−b</strong> or greater than <strong>a</strong> correspond to negative y-values, then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of <strong>x</strong> less than <strong>−5</strong> or greater than <strong>4</strong> correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation <strong>y = − one tenth x (x − 4)^(m) (x + 5)^(n)</strong> must be even. For choice A, the sum of these exponents is <strong>1 + 1 + 1</strong>, or <strong>3</strong>, which is odd. For choice B, the sum of these exponents is <strong>1 + 1 + 2</strong>, or <strong>4</strong>, which is even. Therefore, <strong>y = − one tenth x (x − 4) (x + 5)²</strong> could be the equation of the graph shown.<br>Choice A is incorrect. For the graph of this equation, all values of <strong>x</strong> less than <strong>−5</strong> correspond to positive, not negative, y-values.<br>Choice C is incorrect. The graph of this equation has x-intercepts at <strong>x = −4</strong>, <strong>x = 0</strong>, and <strong>x = 5</strong>, rather than x-intercepts at <strong>x = −5</strong>, <strong>x = 0</strong>, and <strong>x = 4</strong>.<br>Choice D is incorrect. The graph of this equation has x-intercepts at <strong>x = −4</strong>, <strong>x = 0</strong>, and <strong>x = 5</strong>, rather than x-intercepts at <strong>x = −5</strong>, <strong>x = 0</strong>, and <strong>x = 4</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/252a3b3a_svg1.svg",
  },
  {
    id: "281a4f3b",
    type: "mcq",
    questionHtml:
      "A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled each year will be 2% less than the number of students enrolled the year before. Which of the following functions models the relationship between the number of students enrolled, <strong>f of x</strong>, and the number of years after 2015, x ?",
    choices: [
      {
        label: "A",
        text: "<strong>f of x = 0 . 0 2 · (3, 000, ), to the x power</strong>",
      },
      {
        label: "B",
        text: "<strong>f of x = 0 . 9 8 · (3, 000, ), to the x power</strong>",
      },
      {
        label: "C",
        text: "<strong>f of x = 3, 000 · (0 . 0 2, ), to the x power</strong>",
      },
      {
        label: "D",
        text: "<strong>f of x = 3, 000 · (0 . 9 8, ), to the x power, </strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Because the change in the number of students decreases by the same percentage each year, the relationship between the number of students and the number of years can be modeled with a decreasing exponential function in the form <strong>f of x = a · (1 − r, ), to the x power</strong>, where <strong>f of x</strong> is the number of students, a is the number of students in 2015, r is the rate of decrease each year, and x is the number of years since 2015. It’s given that 3,000 students were enrolled in 2015 and that the rate of decrease is predicted to be 2%, or 0.02. Substituting these values into the decreasing exponential function yields <strong>f of x = 3, 000 · (1 − 0 . 0 2, ), to the x power</strong>, which is equivalent to <strong>f of x = 3, 000 · (0 . 9 8, ), to the x power</strong>.Choices A, B, and C are incorrect and may result from conceptual errors when translating the given information into a decreasing exponential function.",
    hasFigure: false,
  },
  {
    id: "341ba5db",
    type: "mcq",
    questionHtml:
      "<strong>g (x) = x² + 55</strong><br>What is the minimum value of the given function?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>55</strong>" },
      { label: "C", text: "<strong>110</strong>" },
      { label: "D", text: "<strong>3, 025</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. For a quadratic function defined by an equation of the form <strong>g (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants and <strong>a > 0</strong>, the minimum value of the function is <strong>k</strong>. In the given function, <strong>a = 1</strong>, <strong>h = 0</strong>, and <strong>k = 55</strong>. Therefore, the minimum value of the given function is <strong>55</strong>.<br>Choice A is incorrect. This is the value of <strong>x</strong> for which the given function reaches its minimum value, not the minimum value of the function.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "3918e8bc",
    type: "mcq",
    questionHtml:
      "An object is kicked from a platform. The equation <strong>h = − 4.9 t² + 7 t + 9</strong> represents this situation, where <strong>h</strong> is the height of the object above the ground, in meters, <strong>t</strong> seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>4.9</strong>" },
      { label: "C", text: "<strong>7</strong>" },
      { label: "D", text: "<strong>9</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the equation <strong>h = − 4.9 t² + 7 t + 9</strong> represents this situation, where <strong>h</strong> is the height, in meters, of the object <strong>t</strong> seconds after it is kicked. It follows that the height, in meters, from which the object was kicked is the value of <strong>h</strong> when <strong>t = 0</strong>. Substituting <strong>0</strong> for <strong>t</strong> in the equation <strong>h = − 4.9 t² + 7 t + 9</strong> yields <strong>h = − 4.9 (0)² + 7 (0) + 9</strong>, or <strong>h = 9</strong>. Therefore, the object was kicked from a height of <strong>9</strong> meters.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "3c600337",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 270 (0.1)^(x)</strong>. What is the value of <strong>f (0)</strong>?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>1</strong>" },
      { label: "C", text: "<strong>27</strong>" },
      { label: "D", text: "<strong>270</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The value of <strong>f (0)</strong> is the value of <strong>f (x)</strong> when <strong>x = 0</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the given function yields <strong>f (0) = 270 (0.1)⁰</strong>, or <strong>f (0) = 270 (1)</strong>, which is equivalent to <strong>f (0) = 270</strong>. Therefore, the value of <strong>f (0)</strong> is <strong>270</strong>.<br>Choice A is incorrect. This is the value of <strong>x</strong>, not <strong>f (x)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the value of <strong>f (1)</strong>, not <strong>f (0)</strong>.",
    hasFigure: false,
  },
  {
    id: "4209aefe",
    type: "mcq",
    questionHtml:
      "The function <strong>f (x) = 206 (1.034)^(x)</strong> models the value, in dollars, of a certain bank account by the end of each year from <strong>1957</strong> through <strong>1972</strong>, where <strong>x</strong> is the number of years after <strong>1957</strong>. Which of the following is the best interpretation of “<strong>f (5)</strong> is approximately equal to <strong>243</strong>” in this context?",
    choices: [
      {
        label: "A",
        text: "The value of the bank account is estimated to be approximately <strong>5</strong> dollars greater in <strong>1962</strong> than in <strong>1957</strong>.",
      },
      {
        label: "B",
        text: "The value of the bank account is estimated to be approximately <strong>243</strong> dollars in <strong>1962</strong>.",
      },
      {
        label: "C",
        text: "The value, in dollars, of the bank account is estimated to be approximately <strong>5</strong> times greater in <strong>1962</strong> than in <strong>1957</strong>.",
      },
      {
        label: "D",
        text: "The value of the bank account is estimated to increase by approximately <strong>243</strong> dollars every <strong>5</strong> years between <strong>1957</strong> and <strong>1972</strong>.",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that the function <strong>f (x) = 206 (1.034)^(x)</strong> models the value, in dollars, of a certain bank account by the end of each year from <strong>19 57</strong> through <strong>19 72</strong>, where <strong>x</strong> is the number of years after <strong>19 57</strong>. It follows that <strong>f (x)</strong> represents the estimated value, in dollars, of the bank account <strong>x</strong> years after <strong>19 57</strong>. Since the value of <strong>f (5)</strong> is the value of <strong>f (x)</strong> when <strong>x = 5</strong>, it follows that “<strong>f (5)</strong> is approximately equal to <strong>243</strong>” means that <strong>f (x)</strong> is approximately equal to <strong>243</strong> when <strong>x = 5</strong>. In the given context, this means that the value of the bank account is estimated to be approximately <strong>243</strong> dollars <strong>5</strong> years after <strong>19 57</strong>. Therefore, the best interpretation of the statement “<strong>f (5)</strong> is approximately equal to <strong>243</strong>” in this context is the value of the bank account is estimated to be approximately <strong>243</strong> dollars in <strong>19 62</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "44076c7d",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>−1</strong><br><strong>10</strong><br><br><strong>0</strong><br><strong>14</strong><br><br><strong>1</strong><br><strong>20</strong><br><br>For the quadratic function <strong>f</strong>, the table shows three values of <strong>x</strong> and their corresponding values of <strong>f (x)</strong>. Which equation defines <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 3 x² + 3 x + 14</strong>" },
      { label: "B", text: "<strong>f (x) = 5 x² + x + 14</strong>" },
      { label: "C", text: "<strong>f (x) = 9 x² − x + 14</strong>" },
      { label: "D", text: "<strong>f (x) = x² + 5 x + 14</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The equation of a quadratic function can be written in the form <strong>f (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants. It’s given in the table that when <strong>x = −1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>10</strong>. Substituting <strong>−1</strong> for <strong>x</strong> and <strong>10</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> gives <strong>10 = a (−1 − h)² + k</strong>, which is equivalent to <strong>10 = a (1 + 2 h + h²) + k</strong>, or <strong>10 = a + 2 a h + a h² + k</strong>. It’s given in the table that when <strong>x = 0</strong>, the corresponding value of <strong>f (x)</strong> is <strong>14</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>14</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> gives <strong>14 = a (0 − h)² + k</strong>, or <strong>14 = a h² + k</strong>. It’s given in the table that when <strong>x = 1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>20</strong>. Substituting <strong>1</strong> for <strong>x</strong> and <strong>20</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> gives <strong>20 = a (1 − h)² + k</strong>, which is equivalent to <strong>20 = a (1 − 2 h + h²) + k</strong>, or <strong>20 = a − 2 a h + a h² + k</strong>. Adding <strong>20 = a − 2 a h + a h² + k</strong> to the equation <strong>10 = a + 2 a h + a h² + k</strong> gives <strong>30 = 2 a + 2 a h² + 2 k</strong>. Dividing both sides of this equation by <strong>2</strong> gives <strong>15 = a + a h² + k</strong>. Since <strong>14 = a h² + k</strong>, substituting <strong>14</strong> for <strong>a h² + k</strong>  into the equation <strong>15 = a + a h² + k</strong> gives <strong>15 = a + 14</strong>. Subtracting <strong>14</strong> from both sides of this equation gives <strong>a = 1</strong>. Substituting <strong>1</strong> for <strong>a</strong> in the equations <strong>14 = a h² + k</strong> and <strong>20 = a h² − 2 a h + a + k</strong> gives <strong>14 = h² + k</strong> and <strong>20 = 1 − 2 h + h² + k</strong>, respectively. Since <strong>14 = h² + k</strong>, substituting <strong>14</strong> for <strong>h² + k</strong> in the equation <strong>20 = 1 − 2 h + h² + k</strong> gives <strong>20 = 1 − 2 h + 14</strong>, or <strong>20 = 15 − 2 h</strong>. Subtracting <strong>15</strong> from both sides of this equation gives <strong>5 = − 2 h</strong>. Dividing both sides of this equation by <strong>−2</strong> gives <strong>−five halves = h</strong>. Substituting <strong>−five halves</strong> for <strong>h</strong> into the equation <strong>14 = h² + k</strong> gives <strong>14 = (−five halves)² + k</strong>, or <strong>14 = (25) / (4) + k</strong>. Subtracting <strong>(25) / (4)</strong> from both sides of this equation gives <strong>(31) / (4) = k</strong>. Substituting <strong>1</strong> for <strong>a</strong>, <strong>−five halves</strong> for <strong>h</strong>, and <strong>(31) / (4)</strong> for <strong>k</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> gives <strong>f (x) = (x + five halves)² + (31) / (4)</strong>, which is equivalent to <strong>f (x) = x² + 5 x + (25) / (4) + (31) / (4)</strong>, or <strong>f (x) = x² + 5 x + 14</strong>. Therefore, <strong>f (x) = x² + 5 x + 14</strong> defines <strong>f</strong>.<br>Choice A is incorrect. If <strong>f (x) = 3 x² + 3 x + 14</strong>, then when <strong>x = −1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>14</strong>, not <strong>10</strong>.<br>Choice B is incorrect. If <strong>f (x) = 5 x² + x + 14</strong>, then when <strong>x = −1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>18</strong>, not <strong>10</strong>.<br>Choice C is incorrect. If <strong>f (x) = 9 x² − x + 14</strong>, then when <strong>x = −1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>24</strong>, not <strong>10</strong>, and when <strong>x = 1</strong>, the corresponding value of <strong>f (x)</strong> is <strong>22</strong>, not <strong>20</strong>.",
    hasFigure: false,
  },
  {
    id: "45df91ee",
    type: "mcq",
    questionHtml:
      "<strong>g (x) = 11 (one twelfth)^(x)</strong><br>If the given function <strong>g</strong> is graphed in the xy-plane, where <strong>y = g (x)</strong>, what is the y-intercept of the graph?",
    choices: [
      { label: "A", text: "<strong>(0, 11)</strong>" },
      { label: "B", text: "<strong>(0, 132)</strong>" },
      { label: "C", text: "<strong>(0, 1)</strong>" },
      { label: "D", text: "<strong>(0, 12)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The x-coordinate of any y-intercept of a graph is <strong>0</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the given equation yields <strong>g (0) = 11 (one twelfth)⁰</strong>. Since any nonzero number raised to the <strong>0 th</strong> power is <strong>1</strong>, this gives <strong>g (0) = 11 dot 1</strong>, or <strong>g (0) = 11</strong>. The y-intercept of the graph is, therefore, the point <strong>(0, 11)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "4993b828",
    type: "mcq",
    questionHtml:
      "The area <strong>A</strong>, in square centimeters, of a rectangular cutting board can be represented by the expression <strong>w (w + 9)</strong>, where <strong>w</strong> is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?",
    choices: [
      { label: "A", text: "<strong>w (w + 9)</strong>" },
      { label: "B", text: "<strong>w</strong>" },
      { label: "C", text: "<strong>9</strong>" },
      { label: "D", text: "<strong>(w + 9)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the expression <strong>w (w + 9)</strong> represents the area, in square centimeters, of a rectangular cutting board, where <strong>w</strong> is the width, in centimeters, of the cutting board. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the cutting board is represented by the expression <strong>(w + 9)</strong>. <br>Choice A is incorrect. This expression represents the area, in square centimeters, of the cutting board, not its length, in centimeters.<br>Choice B is incorrect. This expression represents the width, in centimeters, of the cutting board, not its length.<br>Choice C is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the cutting board, not its length, in centimeters.",
    hasFigure: false,
  },
  {
    id: "50e40f08",
    type: "spr",
    questionHtml:
      "<strong>f (x) = (x + 6) (x − 4)</strong><br>If the given function <strong>f</strong> is graphed in the xy-plane, where <strong>y = f (x)</strong>, what is the x-coordinate of an x-intercept of the graph?",
    choices: [],
    correctAnswer: "-6, 4",
    explanation:
      "The correct answer is either <strong>−6</strong> or <strong>4</strong>. The x-intercepts of a graph in the xy-plane are the points where <strong>y = 0</strong>. Thus, for an x-intercept of the graph of <strong>y = f (x)</strong>, <strong>0 = f (x)</strong>. Substituting <strong>0</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = (x + 6) (x − 4)</strong> yields <strong>0 = (x + 6) (x − 4)</strong>. By the zero product property, <strong>x + 6 = 0</strong> and <strong>x − 4 = 0</strong>. Subtracting <strong>6</strong> from both sides of the equation <strong>x + 6 = 0</strong> yields <strong>x = −6</strong>. Adding <strong>4</strong> to both sides of the equation <strong>x − 4 = 0</strong> yields <strong>x = 4</strong>. Therefore, the x-coordinates of the x-intercepts of the graph of <strong>y = f (x)</strong> are <strong>−6</strong> and <strong>4</strong>. Note that -6 and 4 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "52b1700c",
    type: "mcq",
    questionHtml:
      "Time (years)<br>Total amount (dollars)<br><br><strong>0</strong><br><strong>604.00</strong><br><br><strong>1</strong><br><strong>606.42</strong><br><br><strong>2</strong><br><strong>608.84</strong><br><br>Rosa opened a savings account at a bank. The table shows the exponential relationship between the time <strong>t</strong>, in years, since Rosa opened the account and the total amount <strong>n</strong>, in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between <strong>t</strong> and <strong>n</strong>?",
    choices: [
      { label: "A", text: "<strong>|n = (1 + 604)^(t|)</strong>" },
      { label: "B", text: "<strong>|n = (1 + 0.004)^(t|)</strong>" },
      { label: "C", text: "<strong>|n = 604 (1 + 0.004)^(t|)</strong>" },
      { label: "D", text: "<strong>|n = 0.004 (1 + 604)^(t|)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the relationship between <strong>t</strong> and <strong>n</strong> is exponential. The table shows that the value of <strong>n</strong> increases as the value of <strong>t</strong> increases. Therefore, the relationship between <strong>t</strong> and <strong>n</strong> can be represented by an increasing exponential equation of the form <strong>n = a (1 + b)^(t)</strong>, where <strong>a</strong> and <strong>b</strong> are positive constants. The table shows that when <strong>t = 0</strong>, <strong>n = 604</strong>. Substituting <strong>0</strong> for <strong>t</strong> and <strong>604</strong> for <strong>n</strong> in the equation <strong>n = a (1 + b)^(t)</strong> yields <strong>604 = a (1 + b)⁰</strong>, which is equivalent to <strong>604 = a (1)</strong>, or <strong>604 = a</strong>. Substituting <strong>604</strong> for <strong>a</strong> in the equation <strong>n = a (1 + b)^(t)</strong> yields <strong>n = 604 (1 + b)^(t)</strong>. The table also shows that when <strong>t = 1</strong>, <strong>n = 606.42</strong>. Substituting <strong>1</strong> for <strong>t</strong> and <strong>606.42</strong> for <strong>n</strong> in the equation <strong>n = 604 (1 + b)^(t)</strong> yields <strong>606.42 = 604 (1 + b)¹</strong>, or <strong>606.42 = 604 (1 + b)</strong>. Dividing both sides of this equation by <strong>604</strong> yields approximately <strong>1.004 = 1 + b</strong>. Subtracting <strong>1</strong> from both sides of this equation yields that the value of <strong>b</strong> is approximately <strong>0.004</strong>. Substituting <strong>0.004</strong> for <strong>b</strong> in the equation <strong>n = 604 (1 + b)^(t)</strong> yields <strong>n = 604 (1 + 0.004)^(t)</strong>. Therefore, of the choices, choice C best represents the relationship between <strong>t</strong> and <strong>n</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "5bf0f84a",
    type: "mcq",
    questionHtml:
      "The function above models the height h, in feet, of an object above ground t seconds after being launched straight up in the air. What does the number 72 represent in the function?",
    choices: [
      { label: "A", text: "The initial height, in feet, of the object" },
      { label: "B", text: "The maximum height, in feet, of the object" },
      {
        label: "C",
        text: "The initial speed, in feet per second, of the object",
      },
      {
        label: "D",
        text: "The maximum speed, in feet per second, of the object",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The variable t represents the seconds after the object is launched. Since <strong>h of 0 = 72</strong>, this means that the height, in feet, at 0 seconds, or the initial height, is 72 feet.Choices B, C, and D are incorrect and may be the result of misinterpreting the function in context.",
    hasFigure: false,
  },
  {
    id: "5c00c2c1",
    type: "mcq",
    questionHtml:
      "There were no jackrabbits in Australia before 1788 when 24 jackrabbits were introduced. By 1920 the population of jackrabbits had reached 10 billion. If the population had grown exponentially, this would correspond to a 16.2% increase, on average, in the population each year. Which of the following functions best models the population <strong>p of t</strong> of jackrabbits t years after 1788?",
    choices: [
      {
        label: "A",
        text: "<strong>p of t = 1 . 1 6 2 · 24 raised to the t power</strong>",
      },
      {
        label: "B",
        text: "<strong>p of t = 24 · 2 raised to the 1 . 1 6 2, t power</strong>",
      },
      {
        label: "C",
        text: "<strong>p of t = 24 · 1 . 1 6 2 raised to the t power</strong>",
      },
      {
        label: "D",
        text: "<strong>p of t = (24 · 1 . 1 6 2, ), raised to the t power</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. This exponential growth model can be written in the form <strong>p of t = A · (1 + r, ), raised to the t power</strong>, where <strong>p of t</strong> is the population t years after 1788, A is the initial population, and r is the yearly growth rate, expressed as a decimal. Since there were 24 jackrabbits in Australia in 1788, <strong>A = 24</strong>. Since the number of jackrabbits increased by an average of 16.2% each year, <strong>r = 0 . 1 6 2</strong>. Therefore, the equation that best models this situation is <strong>p of t = 24 · 1 . 1 6 2 raised to the t power</strong>.Choices A, B, and D are incorrect and may result from misinterpreting the form of an exponential growth model.",
    hasFigure: false,
  },
  {
    id: "6676f055",
    type: "mcq",
    questionHtml:
      "An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the function above models the airflow <strong>f of theta</strong>, in cubic feet per minute, as a function of the angle of the fan <strong>theta</strong>, in degrees. According to the model, what angle, in degrees, gives the greatest airflow?",
    choices: [
      { label: "A", text: "–0.28" },
      { label: "B", text: "0.28" },
      { label: "C", text: "27" },
      { label: "D", text: "880" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The function f is quadratic, so it will have either a maximum or a minimum at the vertex of the graph. Since the coefficient of the quadratic term (–0.28) is negative, the vertex will be at a maximum. The equation f(<strong>theta</strong>) = –0.28(<strong>theta</strong> – 27)2 + 880 is given in vertex form, so the vertex is at <strong>theta</strong> = 27. Therefore, an angle of 27 degrees gives the greatest airflow.Choices A and B are incorrect and may be the result of misidentifying which value in a quadratic equation in vertex form represents the vertex. Choice D is incorrect. This choice identifies the maximum value of f(<strong>theta</strong>) rather than the value of <strong>theta</strong> for which f(<strong>theta</strong>) is maximized.",
    hasFigure: false,
  },
  {
    id: "67f4b449",
    type: "mcq",
    questionHtml:
      "The function <strong>f (w) = 6 w²</strong> gives the area of a rectangle, <strong>in square feet (ft²)</strong>, if its width is <strong>w ft</strong> and its length is <strong>6</strong> times its width. Which of the following is the best interpretation of <strong>f (14) = 1, 176</strong>?",
    choices: [
      {
        label: "A",
        text: "If the width of the rectangle is <strong>14 ft</strong>, then the area of the rectangle is <strong>1, 176 ft²</strong>.",
      },
      {
        label: "B",
        text: "If the width of the rectangle is <strong>14 ft</strong>, then the length of the rectangle is <strong>1, 176 ft</strong>.",
      },
      {
        label: "C",
        text: "If the width of the rectangle is <strong>1, 176 ft</strong>, then the length of the rectangle is <strong>14 ft</strong>.",
      },
      {
        label: "D",
        text: "If the width of the rectangle is <strong>1, 176 ft</strong>, then the area of the rectangle is <strong>14 ft²</strong>.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The function <strong>f</strong> gives the area of the rectangle, in <strong>ft²</strong>, if its width is <strong>w ft</strong>. Since the value of <strong>f (14)</strong> is the value of <strong>f (w)</strong> if <strong>w = 14</strong>, it follows that <strong>f (14) = 1, 176</strong> means that <strong>f (w)</strong> is <strong>1, 176</strong> if <strong>w = 14</strong>. In the given context, this means that if the width of the rectangle is <strong>14 ft</strong>, then the area of the rectangle is <strong>1, 176 ft²</strong>.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from interpreting <strong>f (w)</strong> as the width, in <strong>ft</strong>, of the rectangle if its area is <strong>w ft²</strong>, rather than as the area, in <strong>ft²</strong>, of the rectangle if its width is <strong>w ft</strong>.",
    hasFigure: false,
  },
  {
    id: "6ecdbcb4",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = (x + 6) (x + 5) (x − 4)</strong><br>The function <strong>f</strong> is given. Which table of values represents <strong>y = f (x) − 3</strong>?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>−6</strong><br><strong>−9</strong><br><br><strong>−5</strong><br><strong>−8</strong><br><br><strong>4</strong><br><strong>1</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>−6</strong><br><strong>−3</strong><br><br><strong>−5</strong><br><strong>−3</strong><br><br><strong>4</strong><br><strong>−3</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>−6</strong><br><strong>−3</strong><br><br><strong>−5</strong><br><strong>−2</strong><br><br><strong>4</strong><br><strong>7</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>y</strong><br><br><strong>−6</strong><br><strong>3</strong><br><br><strong>−5</strong><br><strong>3</strong><br><br><strong>4</strong><br><strong>3</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>f (x) = (x + 6) (x + 5) (x − 4)</strong> and <strong>y = f (x) − 3</strong>. Substituting <strong>(x + 6) (x + 5) (x − 4)</strong> for <strong>f (x)</strong> in the equation <strong>y = f (x) − 3</strong> yields <strong>y = (x + 6) (x + 5) (x − 4) − 3</strong>. Substituting <strong>−6</strong> for <strong>x</strong> in this equation yields <strong>y = (−6 + 6) (−6 + 5) (−6 − 4) − 3</strong>, or <strong>y = −3</strong>. Substituting <strong>−5</strong> for <strong>x</strong> in the equation <strong>y = (x + 6) (x + 5) (x − 4) − 3</strong> yields <strong>y = (−5 + 6) (−5 + 5) (−5 − 4) − 3</strong>, or <strong>y = −3</strong>. Substituting <strong>4</strong> for <strong>x</strong> in the equation <strong>y = (x + 6) (x + 5) (x − 4) − 3</strong> yields <strong>y = (4 + 6) (4 + 5) (4 − 4) − 3</strong>, or <strong>y = −3</strong>. Therefore, when <strong>x = −6</strong> then <strong>y = −3</strong>, when <strong>x = −5</strong> then <strong>y = −3</strong>, and when <strong>x = 4</strong> then <strong>y = −3</strong>. Thus, the table of values in choice B represents <strong>y = f (x) − 3</strong>.<br>Choice A is incorrect. This table represents <strong>y = x − 3</strong> rather than <strong>y = f (x) − 3</strong>.<br>Choice C is incorrect. This table represents <strong>y = x + 3</strong> rather than <strong>y = f (x) − 3</strong>.<br>Choice D is incorrect. This table represents <strong>y = f (x) + 3</strong> rather than <strong>y = f (x) − 3</strong>.",
    hasFigure: false,
  },
  {
    id: "70ebd3d0",
    type: "mcq",
    questionHtml:
      "The function N defined above can be used to model the number of species of brachiopods at various ocean depths d, where d is in hundreds of meters. Which of the following does the model predict?",
    choices: [
      {
        label: "A",
        text: "For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.",
      },
      {
        label: "B",
        text: "For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.",
      },
      {
        label: "C",
        text: "For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.",
      },
      {
        label: "D",
        text: "For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The function N is exponential, so it follows that <strong>N of d</strong> changes by a fixed percentage for each increase in d by 1. Since d is measured in hundreds of meters, it also follows that the number of brachiopod species changes by a fixed percentage for each increase in ocean depth by 100 meters. Since the base of the exponent in the model is 0.90, which is less than 1, the number of brachiopod species decreases as the ocean depth increases. Specifically, the number of brachiopod species at a depth of <strong>d + 100</strong> meters is 90% of the number of brachiopod species at a depth of d meters. This means that for each increase in ocean depth by 100 meters, the number of brachiopod species decreases by 10%.Choices A and C are incorrect. These describe situations where the number of brachiopod species are decreasing linearly rather than exponentially. Choice B is incorrect and results from interpreting the decrease in the number of brachiopod species as 10% for every 1-meter increase in ocean depth rather than for every 100-meter increase in ocean depth.",
    hasFigure: false,
  },
  {
    id: "735a0a00",
    type: "mcq",
    questionHtml:
      "<strong>y = 0.25 x² − 7.5 x + 90.25</strong><br>The equation gives the estimated stock price <strong>y</strong>, in dollars, for a certain company <strong>x</strong> days after a new product launched, where <strong>0 < or = x < or = 20</strong>. Which statement is the best interpretation of <strong>(x, y) = (1, 83)</strong>  in this context?",
    choices: [
      {
        label: "A",
        text: "The company's estimated stock price increased <strong>dollar sign 83</strong> every day after the new product launched.",
      },
      {
        label: "B",
        text: "The company's estimated stock price increased <strong>dollar sign 1</strong> every <strong>83</strong> days after the new product launched.",
      },
      {
        label: "C",
        text: "<strong>1</strong> day after the new product launched, the company's estimated stock price is <strong>dollar sign 83</strong>.",
      },
      {
        label: "D",
        text: "<strong>83</strong> days after the new product launched, the company's estimated stock price is <strong>dollar sign 1</strong>.",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. In the given equation, <strong>x</strong> represents the number of days after a new product launched, where <strong>0 < or = x < or = 20</strong>, and <strong>y</strong> represents the estimated stock price, in dollars, for a certain company. Therefore, the best interpretation of <strong>(x, y) = (1, 83)</strong> in this context is that <strong>1</strong> day after the new product launched, the company's estimated stock price is <strong>dollar sign 83</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "752055d1",
    type: "mcq",
    questionHtml:
      "A scientist initially measures <strong>12, 000</strong> bacteria in a growth medium. <strong>4</strong> hours later, the scientist measures <strong>24, 000</strong> bacteria. Assuming exponential growth, the formula <strong>P = C (2)^(r t)</strong> gives the number of bacteria in the growth medium, where <strong>r</strong> and <strong>C</strong> are constants and <strong>P</strong> is the number of bacteria <strong>t</strong> hours after the initial measurement. What is the value of <strong>r</strong>?",
    choices: [
      { label: "A", text: "<strong>(1) / (12, 000)</strong>" },
      { label: "B", text: "<strong>one fourth</strong>" },
      { label: "C", text: "<strong>4</strong>" },
      { label: "D", text: "<strong>12, 000</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that the formula <strong>P = C (2)^(r t)</strong> gives the number of bacteria in a growth medium, where <strong>r</strong> and <strong>C</strong> are constants and <strong>P</strong> is the number of bacteria <strong>t</strong> hours after the initial measurement. It’s also given that a scientist initially measures <strong>12, 000</strong> bacteria in the growth medium. Since the initial measurement is <strong>0</strong> hours after the initial measurement, it follows that when <strong>t = 0</strong>, <strong>P = 12, 000</strong>. Substituting <strong>0</strong> for <strong>t</strong> and <strong>12, 000</strong> for <strong>P</strong> in the given equation yields <strong>12, 000 = C (2)^(r (0))</strong>, or <strong>12, 000 = C (2)⁰</strong>, which is equivalent to <strong>12, 000 = C</strong>.  It’s given that <strong>4</strong> hours later, the scientist measures <strong>24, 000</strong> bacteria, or when <strong>t = 4</strong>, <strong>P = 24, 000</strong>. Substituting <strong>4</strong> for <strong>t</strong>, <strong>24, 000</strong> for <strong>P</strong>, and <strong>12, 000</strong> for <strong>C</strong> in the given equation yields <strong>24, 000 = 12, 000 (2)^(4 r)</strong>. Dividing each side of this equation by <strong>12, 000</strong> yields <strong>2 = 2^(4 r)</strong>, or <strong>2¹ = 2^(4 r)</strong>, which is equivalent to <strong>1 = 4 r</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>one fourth = r</strong>. Therefore, the value of <strong>r</strong> is <strong>one fourth</strong>.<br>Choice A is incorrect. This is the value of the reciprocal of <strong>C</strong>.<br>Choice C is incorrect. This is the value of the reciprocal of <strong>r</strong>.<br>Choice D is incorrect. This is the value of <strong>C</strong>.",
    hasFigure: false,
  },
  {
    id: "78d5f91a",
    type: "mcq",
    questionHtml:
      "For the function f defined above, what is the value of <strong>f of − 1</strong>?",
    choices: [
      { label: "A", text: "<strong>−11</strong>" },
      { label: "B", text: "<strong>−7</strong>" },
      { label: "C", text: "<strong>7</strong>" },
      { label: "D", text: "<strong>11</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Substituting <strong>−1</strong> for x in the given function f gives <strong>f of − 1 = (−1, ), ³ + 3 · (−1, ), ² − 6 · −1 − 1</strong>, which simplifies to <strong>f of − 1 = −1 + 3 · 1 − 6 · −1 − 1</strong>. This further simplifies to <strong>f of − 1 = −1 + 3 + 6 − 1</strong>, or <strong>f of − 1 = 7</strong>.Choice A is incorrect and may result from correctly substituting <strong>−1</strong> for x in the function but incorrectly simplifying the resulting expression to <strong>f of − 1 = −1 − 3 − 6 − 1</strong>, or <strong>−11</strong>. Choice B is incorrect and may result from arithmetic errors. Choice D is incorrect and may result from correctly substituting <strong>−1</strong> for x in the function but incorrectly simplifying the expression to <strong>f of − 1 = 1 + 3 + 6 + 1</strong>, or 11.",
    hasFigure: false,
  },
  {
    id: "7ba694f3",
    type: "mcq",
    questionHtml:
      "The number of bacteria in a liquid medium doubles every day. There are <strong>44, 000</strong> bacteria in the liquid medium at the start of an observation. Which represents the number of bacteria, <strong>y</strong>, in the liquid medium <strong>t</strong> days after the start of the observation?",
    choices: [
      { label: "A", text: "<strong>y = one half (44, 000)^(t)</strong>" },
      { label: "B", text: "<strong>y = 2 (44, 000)^(t)</strong>" },
      { label: "C", text: "<strong>y = 44, 000 (one half)^(t)</strong>" },
      { label: "D", text: "<strong>y = 44, 000 (2)^(t)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since the number of bacteria doubles every day, the relationship between <strong>t</strong> and <strong>y</strong> can be represented by an exponential equation of the form <strong>y = a (b)^(t)</strong>, where <strong>a</strong> is the number of bacteria at the start of the observation and the number of bacteria increases by a factor of <strong>b</strong> every day. It’s given that there are <strong>44, 000</strong> bacteria at the start of the observation. Therefore, <strong>a = 44, 000</strong>. It’s also given that the number of bacteria doubles, or increases by a factor of <strong>2</strong>, every day. Therefore, <strong>b = 2</strong>. Substituting <strong>44, 000</strong> for <strong>a</strong> and <strong>2</strong> for <strong>b</strong> in the equation <strong>y = a (b)^(t)</strong> yields <strong>y = 44, 000 (2)^(t)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This equation represents a situation where the number of bacteria is decreasing by half, not doubling, every day.",
    hasFigure: false,
  },
  {
    id: "7e5a3640",
    type: "mcq",
    questionHtml:
      "Bacteria are growing in a liquid growth medium. There were <strong>300, 000</strong> cells per milliliter during an initial observation. The number of cells per milliliter doubles every <strong>3</strong> hours. How many cells per milliliter will there be <strong>15</strong> hours after the initial observation?",
    choices: [
      { label: "A", text: "<strong>1, 500, 000</strong>" },
      { label: "B", text: "<strong>2, 400, 000</strong>" },
      { label: "C", text: "<strong>4, 500, 000</strong>" },
      { label: "D", text: "<strong>9, 600, 000</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Let <strong>y</strong> represent the number of cells per milliliter <strong>x</strong> hours after the initial observation. Since the number of cells per milliliter doubles every <strong>3</strong> hours, the relationship between <strong>x</strong> and <strong>y</strong> can be represented by an exponential equation of the form <strong>y = a (b)^((x) / (k))</strong>, where <strong>a</strong> is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of <strong>b</strong> every <strong>k</strong> hours. It’s given that there were <strong>300, 000</strong> cells per milliliter during the initial observation. Therefore, <strong>a = 300, 000</strong>. It’s also given that the number of cells per milliliter doubles, or increases by a factor of <strong>2</strong>, every <strong>3</strong> hours. Therefore, <strong>b = 2</strong> and <strong>k = 3</strong>. Substituting <strong>300, 000</strong> for <strong>a</strong>, <strong>2</strong> for <strong>b</strong>, and <strong>3</strong> for <strong>k</strong> in the equation <strong>y = a (b)^((x) / (k))</strong> yields <strong>y = 300, 000 (2)^((x) / (3))</strong>. The number of cells per milliliter there will be <strong>15</strong> hours after the initial observation is the value of <strong>y</strong> in this equation when <strong>x = 15</strong>. Substituting <strong>15</strong> for <strong>x</strong> in the equation <strong>y = 300, 000 (2)^((x) / (3))</strong> yields <strong>y = 300, 000 (2)^((15) / (3))</strong>, or <strong>y = 300, 000 (2)⁵</strong>. This is equivalent to <strong>y = 300, 000 (32)</strong>, or <strong>y = 9, 600, 000</strong>. Therefore, <strong>15</strong> hours after the initial observation, there will be <strong>9, 600, 000</strong> cells per milliliter.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "90bcaa61",
    type: "spr",
    questionHtml:
      "The function <strong>f (t) = 60, 000 (2)^((t) / (410))</strong> gives the number of bacteria in a population <strong>t</strong> minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?",
    choices: [],
    correctAnswer: "410",
    explanation:
      "The correct answer is <strong>410</strong>. It's given that <strong>t</strong> minutes after an initial observation, the number of bacteria in a population is <strong>60, 000 (2)^((t) / (410))</strong>. This expression consists of the initial number of bacteria, <strong>60, 000</strong>, multiplied by the expression <strong>2^((t) / (410))</strong>. The time it takes for the number of bacteria to double is the increase in the value of <strong>t</strong> that causes the expression <strong>2^((t) / (410))</strong> to double. Since the base of the expression <strong>2^((t) / (410))</strong> is <strong>2</strong>, the expression <strong>2^((t) / (410))</strong> will double when the exponent increases by <strong>1</strong>. Since the exponent of the expression <strong>2^((t) / (410))</strong> is <strong>(t) / (410)</strong>, the exponent will increase by <strong>1</strong> when <strong>t</strong> increases by <strong>410</strong>. Therefore the time, in minutes, it takes for the number of bacteria in the population to double is <strong>410</strong>.",
    hasFigure: false,
  },
  {
    id: "926c246b",
    type: "mcq",
    questionHtml:
      "The equation above estimates the global data traffic D, in terabytes, for the year that is t years after 2010. What is the best interpretation of the number 5,640 in this context?",
    choices: [
      {
        label: "A",
        text: "The estimated amount of increase of data traffic, in terabytes, each year",
      },
      {
        label: "B",
        text: "The estimated percent increase in the data traffic, in terabytes, each year",
      },
      {
        label: "C",
        text: "The estimated data traffic, in terabytes, for the year that is t years after 2010",
      },
      { label: "D", text: "The estimated data traffic, in terabytes, in 2010" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since t represents the number of years after 2010, the estimated data traffic, in terabytes, in 2010 can be calculated using the given equation when <strong>t = 0</strong>. Substituting 0 for t in the given equation yields <strong>D = 5, 640 · (1 . 9, ), to the 0 power</strong>, or <strong>5, 640 · 1 = 5, 640</strong>. Thus, 5,640 represents the estimated data traffic, in terabytes, in 2010.Choice A is incorrect. Since the equation is exponential, the amount of increase of data traffic each year isn’t constant. Choice B is incorrect. According to the equation, the percent increase in data traffic each year is 90%. Choice C is incorrect. The estimated data traffic, in terabytes, for the year that is t years after 2010 is represented by D, not the number 5,640.",
    hasFigure: false,
  },
  {
    id: "97158b3a",
    type: "mcq",
    questionHtml:
      "The area A, in square centimeters, of a rectangular painting can be represented by the expression <strong>w (w + 29)</strong>, where <strong>w</strong> is the width, in centimeters, of the painting.  Which expression represents the length, in centimeters, of the painting?",
    choices: [
      { label: "A", text: "<strong>w</strong>" },
      { label: "B", text: "<strong>29</strong>" },
      { label: "C", text: "<strong>(w + 29)</strong>" },
      { label: "D", text: "<strong>w (w + 29)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that the expression <strong>w (w + 29)</strong> represents the area, in square centimeters, of a rectangular painting, where <strong>w</strong> is the width, in centimeters, of the painting. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length, in centimeters, of the painting is represented by the expression <strong>(w + 29)</strong>.<br>Choice A is incorrect. This expression represents the width, in centimeters, of the painting, not its length, in centimeters.<br>Choice B is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the painting, not its length, in centimeters.<br>Choice D is incorrect. This expression represents the area, in square centimeters, of the painting, not its length, in centimeters.",
    hasFigure: false,
  },
  {
    id: "981aca65",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = (a − 19) / (x) + 5</strong><br>In the given function <strong>f</strong>, <strong>a</strong> is a constant. The graph of function <strong>f</strong> in the xy-plane, where <strong>y = f (x)</strong>, is translated <strong>3</strong> units down and <strong>4</strong> units to the right to produce the graph of <strong>y = g (x)</strong>. Which equation defines function <strong>g</strong>?",
    choices: [
      { label: "A", text: "<strong>g (x) = (a − 19) / (x + 4) + 2</strong>" },
      { label: "B", text: "<strong>g (x) = (a − 19) / (x − 4) + 2</strong>" },
      { label: "C", text: "<strong>g (x) = (a − 22) / (x + 4) + 5</strong>" },
      { label: "D", text: "<strong>g (x) = (a − 22) / (x − 4) + 5</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that the graph of <strong>y = g (x)</strong> is produced by translating the graph of <strong>y = f (x)</strong> <strong>3</strong> units down and <strong>4</strong> units to the right in the xy-plane. Therefore, function <strong>g</strong> can be defined by an equation in the form <strong>g (x) = f (x − 4) − 3</strong>. Function <strong>f</strong> is defined by the equation <strong>f (x) = (a − 19) / (x) + 5</strong>, where <strong>a</strong> is a constant. Substituting <strong>x − 4</strong> for <strong>x</strong> in the equation <strong>f (x) = (a − 19) / (x) + 5</strong> yields <strong>f (x − 4) = (a − 19) / (x − 4) + 5</strong>. Substituting <strong>(a − 19) / (x − 4) + 5</strong> for <strong>f (x − 4)</strong> in the equation <strong>g (x) = f (x − 4) − 3</strong> yields <strong>g (x) = (a − 19) / (x − 4) + 5 − 3</strong>, or <strong>g (x) = (a − 19) / (x − 4) + 2</strong>. Therefore, the equation that defines function <strong>g</strong> is <strong>g (x) = (a − 19) / (x − 4) + 2</strong>.<br>Choice A is incorrect. This equation defines a function whose graph is produced by translating the graph of <strong>y = f (x)</strong> <strong>3</strong> units down and <strong>4</strong> units to the left, not <strong>3</strong> units down and <strong>4</strong> units to the right.<br>Choice C is incorrect. This equation defines a function whose graph is produced by translating the graph of  <strong>y = f (x)</strong> <strong>4</strong> units to the left, not <strong>3</strong> units down and <strong>4</strong> units to the right.<br>Choice D is incorrect. This equation defines a function whose graph is produced by translating the graph of <strong>y = f (x)</strong> <strong>4</strong> units to the right, not <strong>3</strong> units down and <strong>4</strong> units to the right.",
    hasFigure: false,
  },
  {
    id: "a26c29f7",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 7 x³</strong>. In the xy-plane, the graph of <strong>y = g (x)</strong> is the result of shifting the graph of <strong>y = f (x)</strong> down <strong>2</strong> units. Which equation defines function <strong>g</strong>?",
    choices: [
      { label: "A", text: "<strong>g (x) = seven halves x³</strong>" },
      { label: "B", text: "<strong>g (x) = 7 x^(three halves)</strong>" },
      { label: "C", text: "<strong>g (x) = 7 x³ + 2</strong>" },
      { label: "D", text: "<strong>g (x) = 7 x³ − 2</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. If the graph of <strong>y = g (x)</strong> is the result of shifting the graph of <strong>y = f (x)</strong> down <strong>k</strong> units in the xy-plane, the function <strong>g</strong> can be defined by an equation of the form <strong>g (x) = f (x) − k</strong>. It’s given that <strong>f (x) = 7 x³</strong> and the graph of <strong>y = g (x)</strong> is the result of shifting the graph of <strong>y = f (x)</strong> down <strong>2</strong> units. Substituting <strong>7 x³</strong> for <strong>f (x)</strong> and <strong>2</strong> for <strong>k</strong> in the equation <strong>g (x) = f (x) − k</strong> yields <strong>g (x) = 7 x³ − 2</strong>. <br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect. This equation defines a function <strong>g</strong> for which the graph of <strong>y = g (x)</strong> is the result of shifting the graph of <strong>y = f (x)</strong> up, not down, <strong>2</strong> units.",
    hasFigure: false,
  },
  {
    id: "a31417d1",
    type: "mcq",
    questionHtml:
      "From 2005 through 2014, the number of music CDs sold in the United States declined each year by approximately 15% of the number sold the preceding year. In 2005, approximately 600 million CDs were sold in the United States. Of the following, which best models C, the number of millions of CDs sold in the United States, t years after 2005?",
    choices: [
      {
        label: "A",
        text: "<strong>C = 600 · (0 . 1 5, ), raised to the t power</strong>",
      },
      {
        label: "B",
        text: "<strong>C = 600 · (0 . 8 5, ), raised to the t power</strong>",
      },
      {
        label: "C",
        text: "<strong>C = 600 · (1 . 1 5, ), raised to the t power</strong>",
      },
      {
        label: "D",
        text: "<strong>C = 600 · (1 . 8 5, ), raised to the t power</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A model for a quantity C that decreases by a certain percentage per time period t is an exponential equation in the form <strong>C = I · (1 − the fraction r over 100, ), to the t power</strong>, where I is the initial value at time <strong>t = 0</strong> for r% annual decline. It’s given that C is the number of millions of CDs sold in the United States and that t is the number of years after 2005. It’s also given that 600 million CDs were sold at time <strong>t = 0</strong>, so <strong>I = 600</strong>. This number declines by 15% per year, so <strong>r = 15</strong>. Substituting these values into the equation produces <strong>C = 600 · (1 − the fraction 15 over 100, ), to the t power</strong>, or <strong>C = 600 · (0 . 8 5, ), to the t power</strong>.Choice A is incorrect and may result from errors made when representing the percent decline. Choices C and D are incorrect. These equations model exponential increases in CD sales, not exponential decreases.",
    hasFigure: false,
  },
  {
    id: "b7cd6ca6",
    type: "mcq",
    questionHtml:
      "The equation <strong>E (t) = 5 (1.8)^(t)</strong> gives the estimated number of employees at a restaurant, where <strong>t</strong> is the number of years since the restaurant opened. Which of the following is the best interpretation of the number <strong>5</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "The estimated number of employees when the restaurant opened",
      },
      {
        label: "B",
        text: "The increase in the estimated number of employees each year",
      },
      { label: "C", text: "The number of years the restaurant has been open" },
      {
        label: "D",
        text: "The percent increase in the estimated number of employees each year",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. For an exponential function of the form <strong>E (t) = a (b)^(t)</strong>, where <strong>a</strong> and <strong>b</strong> are constants, the initial value of the function—that is, the value of the function when <strong>t = 0</strong>—is <strong>a</strong> and the value of the function increases by a factor of <strong>b</strong> each time <strong>t</strong> increases by <strong>1</strong>. Since the function <strong>E (t) = 5 (1.8)^(t)</strong> gives the estimated number of employees at a restaurant and <strong>t</strong> is the number of years since the restaurant opened, the best interpretation of the number <strong>5</strong> in this context is the estimated number of employees when <strong>t = 0</strong>, or when the restaurant opened.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "be0c419e",
    type: "mcq",
    questionHtml:
      "Immanuel purchased a certain rare coin on January 1. The function <strong>f (x) = 65 (1.03)^(x)</strong>, where <strong>0 < or = x < or = 10</strong>, gives the predicted value, in dollars, of the rare coin <strong>x</strong> years after Immanuel purchased it. What is the best interpretation of the statement “<strong>f (8)</strong> is approximately equal to <strong>82</strong>” in this context?",
    choices: [
      {
        label: "A",
        text: "When the rare coin's predicted value is approximately <strong>82</strong> dollars, it is <strong>8 % sign</strong> greater than the predicted value, in dollars, on January 1 of the previous year.",
      },
      {
        label: "B",
        text: "When the rare coin’s predicted value is approximately <strong>82</strong> dollars, it is <strong>8</strong> times the predicted value, in dollars, on January 1 of the previous year.",
      },
      {
        label: "C",
        text: "From the day Immanuel purchased the rare coin to <strong>8</strong> years after Immanuel purchased the coin, its predicted value increased by a total of approximately <strong>82</strong> dollars.",
      },
      {
        label: "D",
        text: "<strong>8</strong> years after Immanuel purchased the rare coin, its predicted value is approximately <strong>82</strong> dollars.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the function <strong>f (x) = 65 (1.03)^(x)</strong> gives the predicted value, in dollars, of a certain rare coin <strong>x</strong> years after Immanuel purchased it. It follows that <strong>f (x)</strong> represents the predicted value, in dollars, of the coin <strong>x</strong> years after Immanuel purchased it. Since the value of <strong>f (8)</strong> is the value of <strong>f (x)</strong> when <strong>x = 8</strong>, it follows that “<strong>f (8)</strong> is approximately equal to <strong>82</strong>” means that <strong>f (x)</strong> is approximately equal to <strong>82</strong> when <strong>x = 8</strong>. Therefore, the best interpretation of the statement “<strong>f (8)</strong> is approximately equal to <strong>82</strong>” in this context is <strong>8</strong> years after Immanuel purchased the rare coin, its predicted value is approximately <strong>82</strong> dollars.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "beca03de",
    type: "mcq",
    questionHtml:
      "A rectangle has a length that is <strong>15</strong> times its width. The function <strong>y = (15 w) (w)</strong> represents this situation, where <strong>y</strong> is the area, in square feet, of the rectangle and <strong>y > 0</strong>. Which of the following is the best interpretation of <strong>15 w</strong> in this context?",
    choices: [
      { label: "A", text: "The length of the rectangle, in feet" },
      { label: "B", text: "The area of the rectangle, in square feet" },
      {
        label: "C",
        text: "The difference between the length and the width of the rectangle, in feet",
      },
      { label: "D", text: "The width of the rectangle, in feet" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that a rectangle has a length that is <strong>15</strong> times its width. It's also given that the function <strong>y = (15 w) (w)</strong> represents this situation, where <strong>y</strong> is the area, in square feet, of the rectangle and <strong>y > 0</strong>. The area of a rectangle can be calculated by multiplying the rectangle's length by its width. Since the rectangle has a length that is <strong>15</strong> times its width, it follows that <strong>w</strong> represents the width of the rectangle, in feet, and <strong>15 w</strong> represents the length of the rectangle, in feet. Therefore, the best interpretation of <strong>15 w</strong> in this context is that it's the length of the rectangle, in feet.<br>Choice B is incorrect. This is the best interpretation of <strong>y</strong>, not <strong>15 w</strong>, in the given function.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect. This is the best interpretation of <strong>w</strong>, not <strong>15 w</strong>, in the given function.",
    hasFigure: false,
  },
  {
    id: "c048055c",
    type: "mcq",
    questionHtml:
      "A model predicts that the population of Springfield was <strong>15, 000</strong> in <strong>2005</strong>. The model also predicts that each year for the next <strong>5</strong> years, the population <strong>p</strong> increased by <strong>4 % sign</strong> of the previous year's population. Which equation best represents this model, where <strong>x</strong> is the number of years after <strong>2005</strong>, for <strong>x < or = 5</strong>?",
    choices: [
      { label: "A", text: "<strong>p = 0.96 (15, 000)^(x)</strong>" },
      { label: "B", text: "<strong>p = 1.04 (15, 000)^(x)</strong>" },
      { label: "C", text: "<strong>p = 15, 000 (0.96)^(x)</strong>" },
      { label: "D", text: "<strong>p = 15, 000 (1.04)^(x)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that a model predicts the population of Springfield in <strong>2005</strong> was <strong>15, 000</strong>. The model also predicts that each year for the next <strong>5</strong> years, the population increased by <strong>4 % sign</strong> of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by <strong>1.04</strong>. Since the predicted population in <strong>2005</strong> was <strong>15, 000</strong>, the predicted population <strong>1</strong> year later is <strong>15, 000 (1.04)</strong>. The predicted population <strong>2</strong> years later is this value times <strong>1.04</strong>, which is <strong>15, 000 (1.04) (1.04)</strong>, or <strong>15, 000 (1.04)²</strong>. The predicted population <strong>3</strong> years later is this value times <strong>1.04</strong>, or <strong>15, 000 (1.04)³</strong>. More generally, the predicted population, <strong>p</strong>, <strong>x</strong> years after <strong>2005</strong> is represented by the equation <strong>p = 15, 000 (1.04)^(x)</strong>. <br>Choice A is incorrect. Substituting <strong>0</strong> for <strong>x</strong> in this equation indicates the predicted population in <strong>2005</strong> was <strong>0.96</strong> rather than <strong>15, 000</strong>.<br>Choice B is incorrect. Substituting <strong>0</strong> for <strong>x</strong> in this equation indicates the predicted population in <strong>2005</strong> was <strong>1.04</strong> rather than <strong>15, 000</strong>.<br>Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by <strong>4 % sign</strong> each year.",
    hasFigure: false,
  },
  {
    id: "c4cd5bcc",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the y-coordinate of the y-intercept of the graph of the function f is c. Which of the following must be equal to c ?",
    choices: [
      { label: "A", text: "<strong>f of 0</strong>" },
      { label: "B", text: "<strong>f of 1</strong>" },
      { label: "C", text: "<strong>f of 2</strong>" },
      { label: "D", text: "<strong>f of 3</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A y-intercept is the point in the xy-plane where the graph of the function crosses the y-axis, which is where <strong>x = 0</strong>. It’s given that the y-coordinate of the y-intercept of the graph of function f is c. It follows that the coordinate pair representing the y-intercept must be <strong>0, c</strong>. Therefore, c must equal <strong>f of 0</strong>.Choices B, C, and D are incorrect because <strong>f of 1</strong>, <strong>f of 2</strong>, and <strong>f of 3</strong>would represent the y-value of the coordinate where <strong>x = 1</strong>, <strong>x = 2</strong>, and <strong>x = 3</strong>, respectively.",
    hasFigure: false,
  },
  {
    id: "c7a187a7",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = x² − 18 x − 360</strong><br>If the given function <strong>f</strong> is graphed in the xy-plane, where <strong>y = f (x)</strong>, what is an x-intercept of the graph?",
    choices: [
      { label: "A", text: "<strong>(−12, 0)</strong>" },
      { label: "B", text: "<strong>(−30, 0)</strong>" },
      { label: "C", text: "<strong>(−360, 0)</strong>" },
      { label: "D", text: "<strong>(12, 0)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>y = f (x)</strong>. The x-intercepts of a graph in the xy-plane are the points where <strong>y = 0</strong>. Thus, for an x-intercept of the graph of function <strong>f</strong>, <strong>0 = f (x)</strong>. Substituting <strong>0</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = x² − 18 x − 360</strong> yields <strong>0 = x² − 18 x − 360</strong>. Factoring the right-hand side of this equation yields <strong>0 = (x + 12) (x − 30)</strong>. By the zero product property, <strong>x + 12 = 0</strong> and <strong>x − 30 = 0</strong>. Subtracting <strong>12</strong> from both sides of the equation <strong>x + 12 = 0</strong> yields <strong>x = −12</strong>. Adding <strong>30</strong> to both sides of the equation <strong>x − 30 = 0</strong> yields <strong>x = 30</strong>. Therefore, the x-intercepts of the graph of <strong>y = f (x)</strong> are <strong>(−12, 0)</strong> and <strong>(30, 0)</strong>. Of these two x-intercepts, only <strong>(−12, 0)</strong> is given as a choice.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "cc6ccd71",
    type: "mcq",
    questionHtml:
      "In quadrant 2 the curve falls sharply to cross the x axis at point (negative 1 comma 0).<br>In quadrant 3 the curve falls sharply to cross the y axis at point (0 comma negative 5.5).<br>In quadrant 4:<br><br>The curve falls gradually to a relative minimum at point (1 comma negative 7).<br>The curve rises sharply to cross the x axis at point (4 comma 0).<br><br>In quadrant 1:<br><br>The curve rises sharply to a relative maximum at point (5.5 comma 3).<br>The curve falls sharply to cross the x axis at point (7 comma 0).<br><br>In quadrant 4 the curve falls sharply. <br><br>The graph of <strong>y = f (x)</strong> is shown, where the function <strong>f</strong> is defined by <strong>f (x) = a x³ + b x² + c x + d</strong> and <strong>a</strong>, <strong>b</strong>, <strong>c</strong>, and <strong>d</strong> are constants. For how many values of <strong>x</strong> does <strong>f (x) = 0</strong>?",
    choices: [
      { label: "A", text: "One" },
      { label: "B", text: "Two" },
      { label: "C", text: "Three" },
      { label: "D", text: "Four" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. If a value of <strong>x</strong> satisfies <strong>f (x) = 0</strong>, the graph of <strong>y = f (x)</strong> will contain a point <strong>(x, 0)</strong> and thus touch the x-axis. Since there are <strong>3</strong> points at which this graph touches the x-axis, there are <strong>3</strong> values of <strong>x</strong> for which <strong>f (x) = 0</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/cc6ccd71_svg1.svg",
  },
  {
    id: "d4950429",
    type: "mcq",
    questionHtml:
      "A rectangle has a length of <strong>x</strong> units and a width of <strong>(x − 15)</strong> units. If the rectangle has an area of <strong>76</strong> square units, what is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>4</strong>" },
      { label: "B", text: "<strong>19</strong>" },
      { label: "C", text: "<strong>23</strong>" },
      { label: "D", text: "<strong>76</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, <strong>x</strong> units, by the given width, <strong>(x − 15)</strong> units, yields <strong>x (x − 15)</strong> square units. If the rectangle has an area of <strong>76</strong> square units, it follows that <strong>x (x − 15) = 76</strong>, or <strong>x² − 15 x = 76</strong>. Subtracting <strong>76</strong> from both sides of this equation yields <strong>x² − 15 x − 76 = 0</strong>. Factoring the left-hand side of this equation yields <strong>(x − 19) (x + 4) = 0</strong>. Applying the zero product property to this equation yields two solutions: <strong>x = 19</strong> and <strong>x = −4</strong>. Since <strong>x</strong> is the rectangle’s length, in units, which must be positive, the value of <strong>x</strong> is <strong>19</strong>.<br>Choice A is incorrect. This is the width, in units, of the rectangle, not the value of <strong>x</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of <strong>x</strong>.",
    hasFigure: false,
  },
  {
    id: "d675744f",
    type: "mcq",
    questionHtml:
      "Which of the following is the graph in the xy-plane of the given equation?",
    choices: [
      {
        label: "A",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 4 through 4, in increments of 1, are indicated on the x axis. The numbers − 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, high above the x-axis. As it moves rightward it goes downward, at first steeply, then more and more slowly as it gets closer and closer to the x axis. Eventually it goes downward toward the x axis so slowly it is almost horizontal. The curve passes through − 2, 32, and − 1, 8, and crosses the y-axis a little above the origin. It ends to the right of the y-axis, just above the x-axis.</strong>",
      },
      {
        label: "B",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 4 through 4, in increments of 1, are indicated on the x axis. The numbers − 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, high above the x axis. As it moves rightward it goes downward, at first steeply, then more and more slowly as it gets closer and closer to the x axis. Eventually it goes downward toward the x axis so slowly it is almost horizontal. It ends to the right of the y axis, just above the x axis. The curve passes through − 3, 32 −2, 16, and − 1, 8. It crosses the y-axis a little above the origin and ends to the right of the y-axis, just above the x-axis.</strong>",
      },
      {
        label: "C",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 4 through 4, in increments of 1, are indicated on the x axis. The numbers − 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, just above the x axis. As it moves rightward, it goes upward, at first so slowly it is almost horizontal, then more and more quickly. The curve crosses the y-axis a little above the origin, and passes through 1, 8 and 2, 32. It ends to the right of the y-axis, high above the x-axis.</strong>",
      },
      {
        label: "D",
        text: "<strong>The answer choice presents the graph of a curve in the xy plane, with the origin labeled O. The numbers − 4 through 4, in increments of 1, are indicated on the x axis. The numbers − 25 through 100, in increments of 25, are indicated on the y axis. The curve begins to the left of the y axis, just above the x axis. As it moves rightward, it goes upward, at first so slowly it is almost horizontal, then more and more quickly. It ends to the right of the y axis, high above the x axis. The curve crosses the y axis a little above the origin, and it passes through 1, 8, 2, 16, and 3, 32. It ends to the right of the y-axis, high above the x-axis.</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of the graph of an equation is the point <strong>with coordinates 0, b</strong>, where b is the value of y when <strong>x = 0</strong>. For the given equation, <strong>y = 4</strong> when <strong>x = 0</strong>. It follows that the y-intercept of the graph of the given equation is <strong>0, 4</strong>. Additionally, for the given equation, the value of y doubles for each increase of 1 in the value of x. Therefore, the graph contains the points <strong>with coordinates 1, 8</strong>, <strong>2, 16</strong>, <strong>3, 32</strong>, and <strong>4, 64</strong>. Only the graph shown in choice D passes through these points.Choices A and B are incorrect because these are graphs of decreasing, not increasing, exponential functions. Choice C is incorrect because the value of y increases by a growth factor greater than 2 for each increase of 1 in the value of x.",
    hasFigure: true,
    figureUrl: "/practice-images/d675744f_img2.png",
  },
  {
    id: "d71f6dbf",
    type: "mcq",
    questionHtml:
      "The height, in feet, of an object x seconds after it is thrown straight up in the air can be modeled by the function <strong>h of x = −16 x² + 20 x + 5</strong>. Based on the model, which of the following statements best interprets the equation <strong>h of 1 . 4 = 1 . 6 4</strong> ?",
    choices: [
      {
        label: "A",
        text: "The height of the object 1.4 seconds after being thrown straight up in the air is 1.64 feet.",
      },
      {
        label: "B",
        text: "The height of the object 1.64 seconds after being thrown straight up in the air is 1.4 feet.",
      },
      {
        label: "C",
        text: "The height of the object 1.64 seconds after being thrown straight up in the air is approximately 1.4 times as great as its initial height.",
      },
      {
        label: "D",
        text: "The speed of the object 1.4 seconds after being thrown straight up in the air is approximately 1.64 feet per second.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The value 1.4 is the value of x, which represents the number of seconds after the object was thrown straight up in the air. When the function h is evaluated for x = 1.4, the function has a value of 1.64, which is the height, in feet, of the object.Choices B and C are incorrect and may result from misidentifying seconds as feet and feet as seconds. Additionally, choice C may result from incorrectly including the initial height of the object as the input x. Choice D is incorrect and may result from misidentifying height as speed.",
    hasFigure: false,
  },
  {
    id: "d8ace155",
    type: "mcq",
    questionHtml:
      "A company opens an account with an initial balance of <strong>dollar sign 36, 100.00</strong>. The account earns interest, and no additional deposits or withdrawals are made. The account balance is given by an exponential function <strong>A</strong>, where <strong>A (t)</strong> is the account balance, in dollars, <strong>t</strong> years after the account is opened. The account balance after <strong>13</strong> years is <strong>dollar sign 68, 071.93</strong>. Which equation could define <strong>A</strong>?",
    choices: [
      { label: "A", text: "<strong>A (t) = 36, 100.00 (1.05)^(t)</strong>" },
      { label: "B", text: "<strong>A (t) = 31, 971.93 (1.05)^(t)</strong>" },
      { label: "C", text: "<strong>A (t) = 31, 971.93 (0.05)^(t)</strong>" },
      { label: "D", text: "<strong>A (t) = 36, 100.00 (0.05)^(t)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Since it's given that the account balance, <strong>A (t)</strong>, in dollars, after <strong>t</strong> years can be modeled by an exponential function, it follows that function <strong>A</strong> can be written in the form <strong>A (t) = N r^(t)</strong>, where <strong>N</strong> is the initial value of the function and <strong>r</strong> is a constant related to the growth of the function. It's given that the initial balance of the account is <strong>dollar sign 36, 100.00</strong>, so it follows that the initial value of the function, or <strong>N</strong>, must be <strong>36, 100.00</strong>. Substituting <strong>36, 100.00</strong> for <strong>N</strong> in the equation <strong>A (t) = N r^(t)</strong> yields <strong>A (t) = 36, 100.00 r^(t)</strong>. It's given that the account balance after <strong>13</strong> years, or when <strong>t = 13</strong>, is <strong>dollar sign 68, 071.93</strong>. It follows that <strong>A (13) = 68, 071.93</strong>, or <strong>36, 100.00 r¹³ = 68, 071.93</strong>. Dividing each side of the equation <strong>36, 100.00 r¹³ = 68, 071.93</strong> by <strong>36, 100.00</strong> yields <strong>r¹³ = (68, 071.93) / (36, 100.00)</strong>. Taking the <strong>13</strong>th root of both sides of this equation yields <strong>r = RootIndex 13 √((68, 071.93) / (36, 100.00))</strong>, or <strong>r</strong> is approximately equal to <strong>1.05</strong>. Substituting <strong>1.05</strong> for <strong>r</strong> in the equation <strong>A (t) = 36, 100.00 r^(t)</strong> yields <strong>A (t) = 36, 100.00 (1.05)^(t)</strong>, so the equation <strong>A (t) = 36, 100.00 (1.05)^(t)</strong> could define <strong>A</strong>. <br>Choice B is incorrect. Substituting <strong>0</strong> for <strong>t</strong> in this function indicates an initial balance of <strong>dollar sign 31, 971.93</strong>, rather than <strong>dollar sign 36, 100.00</strong>.<br>Choice C is incorrect. Substituting <strong>0</strong> for <strong>t</strong> in this function indicates an initial balance of <strong>dollar sign 31, 971.93</strong>, rather than <strong>dollar sign 36, 100.00</strong>. Additionally, this function indicates the account balance is decreasing, rather than increasing, over time. <br>Choice D is incorrect. This function indicates the account balance is decreasing, rather than increasing, over time.",
    hasFigure: false,
  },
  {
    id: "db888cd6",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve is in quadrant 1.<br>From point (0 comma 9), the curve trends down sharply.<br>From point (5 comma 1), the curve trends down gradually.<br><br>The graph gives the estimated number of catalogs <strong>y</strong>, in thousands, a company sent to its customers at the end of each year, where <strong>x</strong> represents the number of years since the end of <strong>1992</strong>, where <strong>0 < or = x < or = 10</strong>. Which statement is the best interpretation of the y-intercept in this context?",
    choices: [
      {
        label: "A",
        text: "The estimated total number of catalogs the company sent to its customers during the first <strong>10</strong> years was <strong>9, 000</strong>.",
      },
      {
        label: "B",
        text: "The estimated total number of catalogs the company sent to its customers from the end of <strong>1992</strong> to the end of <strong>2002</strong> was <strong>90</strong>.",
      },
      {
        label: "C",
        text: "The estimated number of catalogs the company sent to its customers at the end of <strong>1992</strong> was <strong>9</strong>.",
      },
      {
        label: "D",
        text: "The estimated number of catalogs the company sent to its customers at the end of <strong>1992</strong> was <strong>9, 000</strong>.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of the graph is the point at which the graph crosses the y-axis, or the point for which the value of <strong>x</strong> is <strong>0</strong>. Therefore, the y-intercept of the given graph is the point <strong>(0, 9)</strong>. It's given that <strong>x</strong> represents the number of years since the end of <strong>1992</strong>. Therefore, <strong>x = 0</strong> represents <strong>0</strong> years since the end of <strong>1992</strong>, which is the same as the end of <strong>1992</strong>. It's also given that <strong>y</strong> represents the estimated number of catalogs, in thousands, that the company sent to its customers at the end of the year. Therefore, <strong>y = 9</strong> represents <strong>9, 000</strong> catalogs. It follows that the y-intercept <strong>(0, 9)</strong> means that the estimated number of catalogs the company sent to its customers at the end of <strong>1992</strong> was <strong>9, 000</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/db888cd6_svg1.svg",
  },
  {
    id: "dba7432e",
    type: "mcq",
    questionHtml:
      "The table above gives the values of the function f for some values of x. Which of the following equations could define f ?",
    choices: [
      {
        label: "A",
        text: "<strong>f of x = 5 · (2 raised to the x + 1 power, )</strong>",
      },
      {
        label: "B",
        text: "<strong>f of x = 5 · (2 to the x power, )</strong>",
      },
      {
        label: "C",
        text: "<strong>f of x = 5 · (2 raised to the negative, (x + 1, ), power, )</strong>",
      },
      {
        label: "D",
        text: "<strong>f of x = 5 · (2 to the − x power, )</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Each choice has a function with coefficient 5 and base 2, so the exponents must be analyzed. When the input value of x increases, the output value of f(x) decreases, so the exponent must be negative. An exponent of –x yields the values in the table: <strong>first value, 5, which = 5 · (2 to the − 0 power, )</strong>, <strong>second value, five halves, which = 5 · (2 to the − 1 power, )</strong>, <strong>third value, five fourths, which = 2 to the − 2 power</strong>, and <strong>fourth value, five eighths, which = 5 · (2 to the − 3 power, )</strong>.Choices A and B are incorrect and may result from choosing equations that yield an increasing, rather than decreasing, output value of f(x) when the input value of x increases. Choice C is incorrect and may result from choosing an equation that doesn’t yield the values in the table.",
    hasFigure: false,
  },
  {
    id: "dd8ac009",
    type: "mcq",
    questionHtml:
      "Time (years)<br>Total amount (dollars)<br><br><strong>0</strong><br><strong>670.00</strong><br><br><strong>1</strong><br><strong>674.02</strong><br><br><strong>2</strong><br><strong>678.06</strong><br><br>Sara opened a savings account at a bank. The table shows the exponential relationship between the time <strong>t</strong>, in years, since Sara opened the account and the total amount <strong>d</strong>, in dollars, in the account. If Sara made no additional deposits or withdrawals, which of the following equations best represents the relationship between <strong>t</strong> and <strong>d</strong>?",
    choices: [
      { label: "A", text: "<strong>d = 0.006 (1 + 670)^(t)</strong>" },
      { label: "B", text: "<strong>d = 670 (1 + 0.006)^(t)</strong>" },
      { label: "C", text: "<strong>d = (1 + 0.006)^(t)</strong>" },
      { label: "D", text: "<strong>d = (1 + 670)^(t)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that the relationship between <strong>t</strong> and <strong>d</strong> is exponential. The table shows that the value of <strong>d</strong> increases as the value of <strong>t</strong> increases. Therefore, the relationship between <strong>t</strong> and <strong>d</strong> can be represented by an increasing exponential equation of the form <strong>d = a (1 + b)^(t)</strong>, where <strong>a</strong> and <strong>b</strong> are positive constants. The table shows that when <strong>t = 0</strong>, <strong>d = 670</strong>. Substituting <strong>0</strong> for <strong>t</strong> and <strong>670</strong> for <strong>d</strong> in the equation <strong>d = a (1 + b)^(t)</strong> yields <strong>670 = a (1 + b)⁰</strong>, which is equivalent to <strong>670 = a (1)</strong>, or <strong>670 = a</strong>. Substituting <strong>670</strong> for <strong>a</strong> in the equation <strong>d = a (1 + b)^(t)</strong> yields <strong>d = 670 (1 + b)^(t)</strong>. The table also shows that when <strong>t = 1</strong>, <strong>d = 674.02</strong>. Substituting <strong>1</strong> for <strong>t</strong> and <strong>674.02</strong> for <strong>d</strong> in the equation <strong>d = 670 (1 + b)^(t)</strong> yields <strong>674.02 = 670 (1 + b)¹</strong>, or <strong>674.02 = 670 (1 + b)</strong>. Dividing both sides of this equation by <strong>670</strong> yields <strong>1.006 = 1 + b</strong>. Subtracting <strong>1</strong> from both sides of this equation yields <strong>b = 0.006</strong>. Substituting <strong>0.006</strong> for <strong>b</strong> in the equation <strong>d = 670 (1 + b)^(t)</strong> yields <strong>d = 670 (1 + 0.006)^(t)</strong>. Therefore, of the choices, choice B best represents the relationship between <strong>t</strong> and <strong>d</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "e1391dd6",
    type: "mcq",
    questionHtml:
      "According to Moore’s law, the number of transistors included on microprocessors doubles every 2 years. In 1985, a microprocessor was introduced that had 275,000 transistors. Based on this information, in which of the following years does Moore’s law estimate the number of transistors to reach 1.1 million?",
    choices: [
      { label: "A", text: "1987" },
      { label: "B", text: "1989" },
      { label: "C", text: "1991" },
      { label: "D", text: "1994" },
    ],
    correctAnswer: "",
    explanation:
      "Choice B is correct. Let x be the number of years after 1985. It follows that <strong>x over 2</strong> represents the number of 2-year periods that will occur within an x-year period. According to Moore’s law, every 2 years, the number of transistors included on microprocessors is estimated to double. Therefore, x years after 1985, the number of transistors will double <strong>x over 2</strong> times. Since the number of transistors included on a microprocessor was 275,000, or .275 million, in 1985, the estimated number of transistors, in millions, included x years after 1985 can be modeled as <strong>0 . 2 7 5 · 2 raised to the x over 2 power</strong>. The year in which the number of transistors is estimated to be 1.1 million is represented by the value of x when <strong>1 . 1 = 0 . 2 7 5 · 2 raised to the x over 2 power</strong>. Dividing both sides of this equation by .275 yields <strong>4 = 2 raised to the x over 2 power</strong>, which can be rewritten as <strong>2² = 2 raised to the x over 2 power</strong>. Since the exponential equation has equal bases on each side, it follows that the exponents must also be equal: <strong>2 = x over 2</strong>. Multiplying both sides of the equation <strong>2 = x over 2</strong> by 2 yields <strong>x = 4</strong>. Therefore, according to Moore’s law, 4 years after 1985, or in 1989, the number of transistors included on microprocessors is estimated to reach 1.1 million.Alternate approach: According to Moore’s law, 2 years after 1985 (in 1987), the number of transistors included on a microprocessor is estimated to be <strong>2 · 275, 000</strong>, or 550,000, and 2 years after 1987 (in 1989), the number of transistors included on microprocessors is estimated to be <strong>2 · 550, 000</strong>, or 1,100,000. Therefore, the year that Moore’s law estimates the number of transistors on microprocessors to reach 1.1 million is 1989.<br>Choices A, C, and D are incorrect. According to Moore’s law, the number of transistors included on microprocessors is estimated to reach 550,000 in 1987, 2.2 million in 1991, and about 6.2 million in 1994.",
    hasFigure: false,
  },
  {
    id: "e53add44",
    type: "mcq",
    questionHtml:
      "The function S above models the annual salary, in dollars, of an employee n years after starting a job, where a is a constant. If the employee’s salary increases by 4% each year, what is the value of a ?",
    choices: [
      { label: "A", text: "0.04" },
      { label: "B", text: "0.4" },
      { label: "C", text: "1.04" },
      { label: "D", text: "1.4" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. A model for a quantity S that increases by a certain percentage per time period n is an exponential function in the form <strong>S of n = I · (1 + r over 100, ), to the n power</strong>, where I is the initial value at time <strong>n = 0</strong> for r% annual increase. It’s given that the annual increase in an employee’s salary is 4%, so <strong>r = 4</strong>. The initial value can be found by substituting 0 for n in the given function, which yields <strong>S of 0 = 38, 000</strong>. Therefore, <strong>I = 38, 000</strong>. Substituting these values for r and I into the form of the exponential function <strong>S of n = I · (1 + r over 100, ), to the n power</strong> yields <strong>S of n = 38, 000 · (1 + 4 over 100, ), to the n power</strong>, or <strong>S of n = 38, 000 · 1 . 0 4 to the n power</strong>. Therefore, the value of a in the given function is 1.04.Choices A, B, and D are incorrect and may result from incorrectly representing the annual increase in the exponential function.",
    hasFigure: false,
  },
  {
    id: "f1c81b3b",
    type: "spr",
    questionHtml:
      "The exponential function <strong>g</strong> is defined by <strong>g (x) = 19 dot a^(x)</strong>, where <strong>a</strong> is a positive constant. If <strong>g (3) = 2, 375</strong>, what is the value of <strong>g (4)</strong>?",
    choices: [],
    correctAnswer: "11875",
    explanation:
      "The correct answer is <strong>11, 875</strong>. It's given that the exponential function <strong>g</strong> is defined by <strong>g (x) = 19 dot a^(x)</strong>, where <strong>a</strong> is a positive constant, and <strong>g (3) = 2, 375</strong>. It follows that when <strong>x = 3</strong>, <strong>g (x) = 2, 375</strong>. Substituting <strong>3</strong> for <strong>x</strong> and <strong>2, 375</strong> for <strong>g (x)</strong> in the given equation yields <strong>2, 375 = 19 dot a³</strong>. Dividing each side of this equation by <strong>19</strong> yields <strong>125 = a³</strong>. Taking the cube root of both sides of this equation gives <strong>a = 5</strong>. Substituting <strong>4</strong> for <strong>x</strong> and <strong>5</strong> for <strong>a</strong> in the equation <strong>g (x) = 19 dot a^(x)</strong> yields <strong>g (4) = 19 dot 5⁴</strong>, or <strong>g (4) = 11, 875</strong>. Therefore, the value of <strong>g (4)</strong> is <strong>11, 875</strong>.",
    hasFigure: false,
  },
  {
    id: "f44a29a8",
    type: "mcq",
    questionHtml:
      "An object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. What is the speed, in meters per second, of an object with a mass of 4 kilograms and kinetic energy of 18 joules?",
    choices: [
      { label: "A", text: "3" },
      { label: "B", text: "6" },
      { label: "C", text: "9" },
      { label: "D", text: "36" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that an object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in meters per second. This relationship can be represented by the equation <strong>K = one half m v²</strong>, where K is the kinetic energy, m is the mass, and v is the speed. Substituting a mass of 4 kilograms for m and a kinetic energy of 18 joules for K results in the equation <strong>18 = one half · 4 · v²</strong>, or <strong>18 = 2 v²</strong>. Dividing both sides of this equation by 2 yields <strong>9 = v²</strong>. Taking the square root of both sides yields <strong>v = −3</strong> and <strong>v = 3</strong>. Since speed can’t be expressed as a negative number, the speed of the object is 3 meters per second.Choice B is incorrect and may result from computation errors. Choice C is incorrect. This is the value of <strong>v²</strong> rather than v. Choice D is incorrect. This is the value of <strong>4 v²</strong> rather than v.",
    hasFigure: false,
  },
  {
    id: "f5e8ccf1",
    type: "mcq",
    questionHtml:
      "The function f is defined above. Which of the following is NOT an x-intercept of the graph of the function in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>−4, 0</strong>" },
      { label: "B", text: "<strong>−two thirds, 0</strong>" },
      { label: "C", text: "<strong>1, 0</strong>" },
      { label: "D", text: "<strong>three halves, 0</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The graph of the function f in the xy-plane has x-intercepts at the points <strong>with coordinates x, y</strong>, where <strong>y = f of x, which = 0</strong>. Substituting 0 for <strong>f of x</strong> in the given equation yields <strong>0 = (x + 4, ) · (x − 1, ) · (2 x − 3, )</strong>. By the zero product property, if <strong>0 = (x + 4, ) · (x − 1, ) · (2 x − 3, )</strong>, then <strong>x + 4 = 0</strong>, <strong>x − 1 = 0</strong>, or <strong>2 x − 3 = 0</strong>. Solving each of these linear equations for x, it follows that <strong>x = −4</strong>, <strong>x = 1</strong>, and <strong>x = three halves</strong>, respectively. This means that the graph of the function f in the xy-plane has three x-intercepts: <strong>−4, 0</strong>, <strong>1, 0</strong>, and <strong>three halves, 0</strong>. Therefore, <strong>−two thirds, 0</strong> isn’t an x-intercept of the graph of the function f.<br><br>Alternate approach: Substitution may be used. Since by definition an x-intercept of any graph is a point in the form <strong>k, 0</strong> where k is a constant, and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function f. Substituting <strong>−two thirds</strong> for x and 0 for <strong>f of x</strong> in the given equation yields <strong>0 = (−two thirds + 4, ) · (−two thirds − 1, ) · (2 · −two thirds − 3, )</strong>, or <strong>0 = the fraction 650 over 27</strong>. Therefore, the point <strong>with coordinates − two thirds, 0</strong> doesn’t lie on the graph of the function f and can’t be an x-intercept of the graph.Choices A, C, and D are incorrect because each of these points is an x-intercept of the graph of the function f in the xy-plane. By definition, an x-intercept is a point on the graph of the form <strong>k, 0</strong>, where k is a constant. Substituting <strong>−4</strong> for x and 0 for <strong>f of x</strong> in the given equation yields <strong>0 = (−4 + 4, ) · (−4 − 1, ) · (2 · −4 − 3, )</strong>, or <strong>0 = 0</strong>. Since this is a true statement, the point <strong>with coordinates − 4, 0</strong> lies on the graph of the function f and is an x-intercept of the graph. Performing similar substitution using the points <strong>with coordinates 1, 0</strong> and <strong>three halves, 0</strong> also yields the true statement <strong>0 = 0</strong>, illustrating that these points also lie on the graph of the function f and are x-intercepts of the graph.",
    hasFigure: false,
  },
  {
    id: "f89af023",
    type: "mcq",
    questionHtml:
      "A rectangular volleyball court has an area of 162 square meters. If the length of the court is twice the width, what is the width of the court, in meters?",
    choices: [
      { label: "A", text: "9" },
      { label: "B", text: "18" },
      { label: "C", text: "27" },
      { label: "D", text: "54" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the volleyball court is rectangular and has an area of 162 square meters. The formula for the area of a rectangle is <strong>A = l w</strong>, where A is the area, <strong>l</strong> is the length, and w is the width of the rectangle. It’s also given that the length of the volleyball court is twice the width, thus <strong>l = 2 w</strong>. Substituting the given value into the formula for the area of a rectangle and using the relationship between length and width for this rectangle yields <strong>162 = 2 w · w</strong>. This equation can be rewritten as <strong>162 = 2 w²</strong>. Dividing both sides of this equation by 2 yields <strong>81 = w²</strong>. Taking the square root of both sides of this equation yields <strong>+ or − 9 = w</strong>. Since the width of a rectangle is a positive number, the width of the volleyball court is 9 meters.<br><br> Choice B is incorrect because this is the length of the rectangle. Choice C is incorrect because this is the result of using 162 as the perimeter rather than the area. Choice D is incorrect because this is the result of calculating w in the equation <strong>162 = 2 w + w</strong> instead of <strong>162 = 2 w · w</strong>.",
    hasFigure: false,
  },
];

export const NONLINEAR_FUNC_HARD: PracticeQuestion[] = [
  {
    id: "0121a235",
    type: "mcq",
    questionHtml:
      "The table above gives selected values of a polynomial function p. Based on the values in the table, which of the following must be a factor of p ?",
    choices: [
      { label: "A", text: "<strong>(x − 3, )</strong>" },
      { label: "B", text: "<strong>(x + 3, )</strong>" },
      { label: "C", text: "<strong>(x − 1, ) · (x + 2, )</strong>" },
      { label: "D", text: "<strong>(x + 1, ) · (x − 2, )</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. According to the table, when x is <strong>−1</strong> or 2, <strong>p of x = 0</strong>. Therefore, two x-intercepts of the graph of p are <strong>−1, 0</strong> and <strong>2, 0</strong>. Since <strong>−1, 0</strong> and <strong>2, 0</strong> are x-intercepts, it follows that <strong>(x + 1, )</strong> and <strong>(x − 2, )</strong> are factors of the polynomial equation. This is because when <strong>x = −1</strong>, the value of <strong>x + 1</strong> is 0. Similarly, when <strong>x = 2</strong>, the value of <strong>x − 2</strong> is 0. Therefore, the product <strong>(x + 1, ) · (x − 2, )</strong> is a factor of the polynomial function p.Choice A is incorrect. The factor <strong>x − 3</strong> corresponds to an x-intercept of <strong>3, 0</strong>, which isn’t present in the table. Choice B is incorrect. The factor <strong>x + 3</strong> corresponds to an x-intercept of <strong>−3, 0</strong>, which isn’t present in the table. Choice C is incorrect. The factors <strong>x − 1</strong> and <strong>x + 2</strong> correspond to x-intercepts <strong>with coordinates 1, 0</strong> and <strong>−2, 0</strong>, respectively, which aren’t present in the table.",
    hasFigure: false,
  },
  {
    id: "01668cd6",
    type: "mcq",
    questionHtml:
      "The functions <strong>f</strong> and <strong>g</strong> are defined by the given equations, where <strong>x > or = 0</strong>. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where <strong>x > or = 0</strong>?<br><br><strong>f (x) = 33 (0.4)^(x + 3)</strong><br><strong>g (x) = 33 (0.16) (0.4)^(x − 2)</strong>",
    choices: [
      { label: "A", text: "I only" },
      { label: "B", text: "II only" },
      { label: "C", text: "I and II" },
      { label: "D", text: "Neither I nor II" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Functions <strong>f</strong> and <strong>g</strong> are both exponential functions with a base of <strong>0.40</strong>. Since <strong>0.40</strong> is less than <strong>1</strong>, functions <strong>f</strong> and <strong>g</strong> are both decreasing exponential functions. This means that <strong>f (x)</strong> and <strong>g (x)</strong> decrease as <strong>x</strong> increases. Since <strong>f (x)</strong> and <strong>g (x)</strong> decrease as <strong>x</strong> increases, the maximum value of each function occurs at the least value of <strong>x</strong> for which the function is defined. It's given that functions <strong>f</strong> and <strong>g</strong> are defined for <strong>x > or = 0</strong>. Therefore, the maximum value of each function occurs at <strong>x = 0</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the equation defining <strong>f</strong> yields <strong>f (0) = 33 (0.4)^(0 + 3)</strong>, which is equivalent to <strong>f (0) = 33 (0.4)³</strong>, or <strong>f (0) = 2.112</strong>. Therefore, the maximum value of <strong>f</strong> is <strong>2.112</strong>. Since the equation <strong>f (x) = 33 (0.4)^(x + 3)</strong> doesn't display the value <strong>2.112</strong>, the equation defining <strong>f</strong> doesn't display the maximum value of <strong>f</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the equation defining <strong>g</strong> yields <strong>g (0) = 33 (0.16) (0.4)^(0 − 2)</strong>, which can be rewritten as <strong>g (0) = 33 (0.16) ((1) / (0.4²))</strong>, or <strong>g (0) = 33 (0.16) ((1) / (0.16))</strong>, which is equivalent to <strong>g (0) = 33</strong>. Therefore, the maximum value of <strong>g</strong> is <strong>33</strong>. Since the equation <strong>g (x) = 33 (0.16) (0.4)^(x − 2)</strong> displays the value <strong>33</strong>, the equation defining <strong>g</strong> displays the maximum value of <strong>g</strong>. Thus, only equation II displays, as a constant or coefficient, the maximum value of the function it defines.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "04bbce67",
    type: "spr",
    questionHtml:
      "<strong>f (x) = (x + 7)² + 4</strong><br>The function <strong>f</strong> is defined by the given equation. For what value of <strong>x</strong> does <strong>f (x)</strong> reach its minimum?",
    choices: [],
    correctAnswer: "-7",
    explanation:
      "The correct answer is <strong>−7</strong>. For a quadratic function defined by an equation of the form  <strong>f (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants and <strong>a > 0</strong>, the function reaches its minimum when <strong>x = h</strong>. In the given function, <strong>a = 1</strong>, <strong>h = −7</strong>, and <strong>k = 4</strong>. Therefore, the value of <strong>x</strong> for which <strong>f (x)</strong> reaches its minimum is <strong>−7</strong>.",
    hasFigure: false,
  },
  {
    id: "08d03fe4",
    type: "mcq",
    questionHtml:
      "For the exponential function <strong>f</strong>, the value of <strong>f (1)</strong> is <strong>k</strong>, where <strong>k</strong> is a constant. Which of the following equivalent forms of the function <strong>f</strong> shows the value of <strong>k</strong> as the coefficient or the base?",
    choices: [
      { label: "A", text: "<strong>f (x) = 50 (2)^(x + 1)</strong>" },
      { label: "B", text: "<strong>f (x) = 80 (2)^(x)</strong>" },
      { label: "C", text: "<strong>f (x) = 128 (2)^(x − 1)</strong>" },
      { label: "D", text: "<strong>f (x) = 205 (2)^(x − 2)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. For the form of the function in choice C, <strong>f (x) = 128 (1.6)^(x − 1)</strong>, the value of <strong>f (1)</strong> can be found as <strong>128 (1.6)^(1 − 1)</strong>, which is equivalent to <strong>128 (1.6)⁰</strong>, or <strong>128</strong>. Therefore, <strong>k = 128</strong>, which is shown in <strong>f (x) = 128 (1.6)^(x − 1)</strong> as the coefficient.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "0e61101e",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 9 (4)^(x)</strong><br>The function <strong>f</strong> is defined by the given equation. If <strong>g (x) = f (x + 2)</strong>, which of the following equations defines the function <strong>g</strong>?",
    choices: [
      { label: "A", text: "<strong>g (x) = 18 (4)^(x)</strong>" },
      { label: "B", text: "<strong>g (x) = 144 (4)^(x)</strong>" },
      { label: "C", text: "<strong>g (x) = 18 (8)^(x)</strong>" },
      { label: "D", text: "<strong>g (x) = 81 (16)^(x)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>f (x) = 9 (4)^(x)</strong> and <strong>g (x) = f (x + 2)</strong>. Substituting <strong>x + 2</strong> for <strong>x</strong> in <strong>f (x) = 9 (4)^(x)</strong> gives <strong>f (x + 2) = 9 (4)^(x + 2)</strong>. Rewriting this equation using properties of exponents gives <strong>f (x + 2) = 9 (4)^(x) (4)²</strong>, which is equivalent to <strong>f (x + 2) = 9 (4)^(x) (16)</strong>. Multiplying <strong>9</strong> and <strong>16</strong> in this equation gives <strong>f (x + 2) = 144 (4)^(x)</strong>. Since <strong>g (x) = f (x + 2)</strong>, <strong>g (x) = 144 (4)^(x)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "1178f2df",
    type: "spr",
    questionHtml:
      "<strong>x</strong><br><strong>y</strong><br><br><strong>21</strong><br><strong>−8</strong><br><br><strong>23</strong><br><strong>8</strong><br><br><strong>25</strong><br><strong>−8</strong><br><br>The table shows three values of <strong>x</strong> and their corresponding values of <strong>y</strong>, where <strong>y = f (x) + 4</strong> and <strong>f</strong> is a quadratic function. What is the y-coordinate of the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane?",
    choices: [],
    correctAnswer: "-2112",
    explanation:
      "The correct answer is <strong>−2, 112</strong>. It's given that <strong>f</strong> is a quadratic function. It follows that <strong>f</strong> can be defined by an equation of the form <strong>f (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants. It's also given that the table shows three values of <strong>x</strong> and their corresponding values of <strong>y</strong>, where <strong>y = f (x) + 4</strong>. Substituting <strong>a (x − h)² + k</strong> for <strong>f (x)</strong> in this equation yields <strong>y = a (x − h)² + k + 4</strong>. This equation represents a quadratic relationship between <strong>x</strong> and <strong>y</strong>, where <strong>k + 4</strong> is either the maximum or the minimum value of <strong>y</strong>, which occurs when <strong>x = h</strong>. For quadratic relationships between <strong>x</strong> and <strong>y</strong>, the maximum or minimum value of <strong>y</strong> occurs at the value of <strong>x</strong> halfway between any two values of <strong>x</strong> that have the same corresponding value of <strong>y</strong>. The table shows that x-values of <strong>21</strong> and <strong>25</strong> correspond to the same y-value, <strong>−8</strong>. Since <strong>23</strong> is halfway between <strong>21</strong> and <strong>25</strong>, the maximum or minimum value of <strong>y</strong> occurs at an x-value of <strong>23</strong>. The table shows that when <strong>x = 23</strong>, <strong>y = 8</strong>. It follows that <strong>h = 23</strong> and <strong>k + 4 = 8</strong>. Subtracting <strong>4</strong> from both sides of the equation <strong>k + 4 = 8</strong> yields <strong>k = 4</strong>. Substituting <strong>23</strong> for <strong>h</strong> and <strong>4</strong> for <strong>k</strong> in the equation <strong>y = a (x − h)² + k + 4</strong> yields <strong>y = a (x − 23)² + 4 + 4</strong>, or <strong>y = a (x − 23)² + 8</strong>. The value of <strong>a</strong> can be found by substituting any x-value and its corresponding y-value for <strong>x</strong> and <strong>y</strong>, respectively, in this equation. Substituting <strong>25</strong> for <strong>x</strong> and <strong>−8</strong> for <strong>y</strong> in this equation yields <strong>−8 = a (25 − 23)² + 8</strong>, or <strong>−8 = a (2)² + 8</strong>. Subtracting <strong>8</strong> from both sides of this equation yields <strong>−16 = a (2)²</strong>, or <strong>−16 = 4 a</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>−4 = a</strong>. Substituting <strong>−4</strong> for <strong>a</strong>, <strong>23</strong> for <strong>h</strong>, and <strong>4</strong> for <strong>k</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> yields <strong>f (x) = − 4 (x − 23)² + 4</strong>. The y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane is the point on the graph where <strong>x = 0</strong>. Substituting <strong>0</strong> for <strong>x</strong> in the equation <strong>f (x) = − 4 (x − 23)² + 4</strong> yields <strong>f (0) = − 4 (0 − 23)² + 4</strong>, or <strong>f (0) = − 4 (−23)² + 4</strong>. This is equivalent to <strong>f (0) = −2, 112</strong>, so the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane is <strong>(0 −2, 112)</strong>. Thus, the y-coordinate of the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane is <strong>−2, 112</strong>.",
    hasFigure: false,
  },
  {
    id: "1853bb35",
    type: "mcq",
    questionHtml:
      "For the function <strong>q</strong>, the value of <strong>q (x)</strong> decreases by <strong>45 % sign</strong> for every increase in the value of <strong>x</strong> by <strong>1</strong>. If <strong>q (0) = 14</strong>, which equation defines <strong>q</strong>?",
    choices: [
      { label: "A", text: "<strong>q (x) = 0.55 (14)^(x)</strong>" },
      { label: "B", text: "<strong>q (x) = 1.45 (14)^(x)</strong>" },
      { label: "C", text: "<strong>q (x) = 14 (0.55)^(x)</strong>" },
      { label: "D", text: "<strong>q (x) = 14 (1.45)^(x)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since the value of <strong>q (x)</strong> decreases by a fixed percentage, <strong>45 % sign</strong>, for every increase in the value of <strong>x</strong> by <strong>1</strong>, the function <strong>q</strong> is a decreasing exponential function. A decreasing exponential function can be written in the form <strong>q (x) = a (1 − (p) / (100))^(x)</strong>, where <strong>a</strong> is the value of <strong>q (0)</strong> and the value of <strong>q (x)</strong> decreases by <strong>p % sign</strong> for every increase in the value of <strong>x</strong> by <strong>1</strong>. If <strong>q (0) = 14</strong>, then <strong>a = 14</strong>. Since the value of <strong>q (x)</strong> decreases by <strong>45 % sign</strong> for every increase in the value of <strong>x</strong> by <strong>1</strong>, <strong>p = 45</strong>. Substituting <strong>14</strong> for <strong>a</strong> and <strong>45</strong> for <strong>p</strong> in the equation <strong>q (x) = a (1 − (p) / (100))^(x)</strong> yields <strong>q (x) = 14 (1 − (45) / (100))^(x)</strong>, which is equivalent to <strong>q (x) = 14 (1 − 0.45)^(x)</strong>, or <strong>q (x) = 14 (0.55)^(x)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. For this function, the value of <strong>q (x)</strong> increases, rather than decreases, by <strong>45 % sign</strong> for every increase in the value of <strong>x</strong> by <strong>1</strong>.",
    hasFigure: false,
  },
  {
    id: "18e35375",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = (x − 14) (x + 19)</strong><br>The function <strong>f</strong> is defined by the given equation. For what value of <strong>x</strong> does <strong>f (x)</strong> reach its minimum?",
    choices: [
      { label: "A", text: "<strong>−266</strong>" },
      { label: "B", text: "<strong>−19</strong>" },
      { label: "C", text: "<strong>−(33) / (2)</strong>" },
      { label: "D", text: "<strong>−five halves</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that <strong>f (x) = (x − 14) (x + 19)</strong>, which can be rewritten as <strong>f (x) = x² + 5 x − 266</strong>. Since the coefficient of the <strong>x²</strong>-term is positive, the graph of <strong>y = f (x)</strong> in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of <strong>x</strong> such that <strong>f (x)</strong> reaches its minimum. For an equation in the form <strong>f (x) = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, the x-coordinate of the vertex is <strong>− (b) / (2 a)</strong>. For the equation <strong>f (x) = x² + 5 x − 266</strong>, <strong>a = 1</strong>, <strong>b = 5</strong>, and <strong>c = −266</strong>. It follows that the x-coordinate of the vertex is <strong>− (5) / (2 (1))</strong>, or <strong>−five halves</strong>. Therefore, <strong>f (x)</strong> reaches its minimum when the value of <strong>x</strong> is <strong>−five halves</strong>.<br>Alternate approach: The value of <strong>x</strong> for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that <strong>f (x) = (x − 14) (x + 19)</strong>, it follows that the two x-intercepts of the graph of <strong>y = f (x)</strong> in the xy-plane occur when <strong>x = 14</strong> and <strong>x = −19</strong>, or at the points <strong>(14, 0)</strong> and <strong>(−19, 0)</strong>. The midpoint between two points, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, is <strong>((x 1 + x 2) / (2), (y 1 + y 2) / (2))</strong>. Therefore, the midpoint between <strong>(14, 0)</strong> and <strong>(−19, 0)</strong> is <strong>((14 + (−19)) / (2), (0 + 0) / (2))</strong>, or <strong>(−five halves, 0)</strong>. It follows that <strong>f (x)</strong> reaches its minimum when the value of <strong>x</strong> is <strong>−five halves</strong>.<br>Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane.<br>Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of <strong>y = f (x)</strong> in the xy-plane.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "1a722d7d",
    type: "mcq",
    questionHtml:
      "Let the function p be defined as <strong>p of x = the fraction with numerator, (x − c, ), ² + 160, and denominator, 2 c, end fraction</strong>, where c is a constant. If <strong>p of c = 10</strong>, what is the value of <strong>p of 12</strong> ?",
    choices: [
      { label: "A", text: "10.00" },
      { label: "B", text: "10.25" },
      { label: "C", text: "10.75" },
      { label: "D", text: "11.00" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The value of p(12) depends on the value of the constant c, so the value of c must first be determined. It is given that p(c) = 10. Based on the definition of p, it follows that:<br> <strong>p of c = the fraction with numerator (c − c, ), ² + 160, and denominator 2 c, end fraction, which = 10</strong><br><br> <strong>the fraction 160 over 2 c, end fraction = 10</strong><br><br> <strong>2 c = 16</strong><br><br> <strong>c = 8</strong><br>This means that <strong>p of x = the fraction with numerator (x − 8, ), ² + 160, and denominator 16</strong> for all values of x. Therefore:<br><br> <strong>p of 12 = the fraction with numerator (12 − 8, ), ² + 160, and denominator 16</strong><br><br> <strong>which = the fraction with numerator 16 + 160, and denominator 16</strong><br><br> <strong>which = 11</strong><br>Choice A is incorrect. It is the value of p(8), not p(12). Choices B and C are incorrect. If one of these values were correct, then x = 12 and the selected value of p(12) could be substituted into the equation to solve for c. However, the values of c that result from choices B and C each result in p(c) < 10.",
    hasFigure: false,
  },
  {
    id: "1f353a9e",
    type: "mcq",
    questionHtml:
      "<strong>f (t) = 8, 000 (0.65)^(t)</strong><br>The given function <strong>f</strong> models the number of coupons a company sent to their customers at the end of each year, where <strong>t</strong> represents the number of years since the end of <strong>1998</strong>, and <strong>0 < or = t < or = 5</strong>. If <strong>y = f (t)</strong> is graphed in the ty-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?",
    choices: [
      {
        label: "A",
        text: "The minimum estimated number of coupons the company sent to their customers during the <strong>5</strong> years was <strong>1, 428</strong>.",
      },
      {
        label: "B",
        text: "The minimum estimated number of coupons the company sent to their customers during the <strong>5</strong> years was <strong>8, 000</strong>.",
      },
      {
        label: "C",
        text: "The estimated number of coupons the company sent to their customers at the end of <strong>1998</strong> was <strong>1, 428</strong>.",
      },
      {
        label: "D",
        text: "The estimated number of coupons the company sent to their customers at the end of <strong>1998</strong> was <strong>8, 000</strong>.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of a graph in the ty-plane is the point where <strong>t = 0</strong>. For the given function <strong>f</strong>, the y-intercept of the graph of <strong>y = f (t)</strong> in the ty-plane can be found by substituting <strong>0</strong> for <strong>t</strong> in the equation <strong>y = 8, 000 (0 . 65)^(t)</strong>, which gives <strong>y = 8, 000 (0 . 65)⁰</strong>. This is equivalent to <strong>y = 8, 000 (1)</strong>, or <strong>y = 8, 000</strong>. Therefore, the y-intercept of the graph of <strong>y = f (t)</strong> is <strong>(0, 8, 000)</strong>. It’s given that the function <strong>f</strong> models the number of coupons a company sent to their customers at the end of each year. Therefore, <strong>f (t)</strong> represents the estimated number of coupons the company sent to their customers at the end of each year. It's also given that <strong>t</strong> represents the number of years since the end of <strong>1998</strong>. Therefore, <strong>t = 0</strong> represents <strong>0</strong> years since the end of <strong>1998</strong>, or the end of <strong>1998</strong>. Thus, the best interpretation of the y-intercept of the graph of <strong>y = f (t)</strong> is that the estimated number of coupons the company sent to their customers at the end of <strong>1998</strong> was <strong>8, 000</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors. <br>Choice B is incorrect and may result from conceptual or calculation errors. <br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "1fe10d97",
    type: "mcq",
    questionHtml:
      "<strong>p (t) = 90, 000 (1.06)^(t)</strong><br>The given function <strong>p</strong> models the population of Lowell <strong>t</strong> years after a census. Which of the following functions best models the population of Lowell <strong>m</strong> months after the census?",
    choices: [
      {
        label: "A",
        text: "<strong>r (m) = (90, 000) / (12) (1.06)^(m)</strong>",
      },
      {
        label: "B",
        text: "<strong>r (m) = 90, 000 ((1.06) / (12))^(m)</strong>",
      },
      {
        label: "C",
        text: "<strong>r (m) = 90, 000 ((1.06) / (12))^((m) / (12))</strong>",
      },
      {
        label: "D",
        text: "<strong>r (m) = 90, 000 (1.06)^((m) / (12))</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the function <strong>p</strong> models the population of Lowell <strong>t</strong> years after a census. Since there are <strong>12</strong> months in a year, <strong>m</strong> months after the census is equivalent to <strong>(m) / (12)</strong> years after the census. Substituting <strong>(m) / (12)</strong> for <strong>t</strong> in the equation <strong>p (t) = 90, 000 (1.06)^(t)</strong> yields <strong>p ((m) / (12)) = 90, 000 (1.06)^((m) / (12))</strong>. Therefore, the function <strong>r</strong> that best models the population of Lowell <strong>m</strong> months after the census is <strong>r (m) = 90, 000 (1.06)^((m) / (12))</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "263f9937",
    type: "mcq",
    questionHtml:
      "A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per milliliter reach <strong>5 . 1 2 · 10, to the power 8</strong>?",
    choices: [
      { label: "A", text: "Day 5" },
      { label: "B", text: "Day 9" },
      { label: "C", text: "Day 11" },
      { label: "D", text: "Day 12" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day 1 to day 2, the number of bacteria increased from 2.5 × 105 to 5.0 × 105. At the end of day 3 there are 106 bacteria per milliliter. At the end of day 4, there will be 106 × 2 bacteria per milliliter. At the end of day 5, there will be (106 × 2) × 2, or 106 × (22) bacteria per milliliter, and so on. At the end of day d, the number of bacteria will be 106 × (2d – 3). If the number of bacteria per milliliter will reach 5.12 × 108 at the end of day d, then the equation <strong>10 to the power 6, end power · 2 to the power d − 3, end power = 5 . 1 2 · 10 to the power 8</strong> must hold. Since 5.12 × 108 can be rewritten as 512 × 106,  the equation is equivalent to <strong>2 to the power d − 3, end power = 512</strong>. Rewriting 512 as 29 gives d – 3 = 9, so d = 12. The number of bacteria per milliliter would reach 5.12 × 108 at the end of day 12. Choices A, B, and C are incorrect. Given the growth rate of the bacteria, the number of bacteria will not reach 5.12 × 108 per milliliter by the end of any of these days.",
    hasFigure: false,
  },
  {
    id: "271ffad7",
    type: "mcq",
    questionHtml:
      "A quadratic function models a projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. The model estimates that the projectile was launched from an initial height of <strong>7</strong> meters above the ground and reached a maximum height of <strong>51.1</strong> meters above the ground <strong>3</strong> seconds after the launch. How many seconds after the launch does the model estimate that the projectile will return to a height of <strong>7</strong> meters?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>6</strong>" },
      { label: "C", text: "<strong>7</strong>" },
      { label: "D", text: "<strong>9</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It follows that an equation representing the model can be written in the form <strong>f (x) = a (x − h)² + k</strong>, where <strong>f (x)</strong> is the projectile's estimated height above the ground, in meters, <strong>x</strong> seconds after the launch, <strong>a</strong> is a constant, and <strong>k</strong> is the maximum height above the ground, in meters, the model estimates the projectile reached <strong>h</strong> seconds after the launch. It's given that the model estimates the projectile reached a maximum height of <strong>51.1</strong> meters above the ground <strong>3</strong> seconds after the launch. Therefore, <strong>k = 51.1</strong> and <strong>h = 3</strong>. Substituting <strong>51.1</strong> for <strong>k</strong> and <strong>3</strong> for <strong>h</strong> in the equation <strong>f (x) = a (x − h)² + k</strong> yields <strong>f (x) = a (x − 3)² + 51.1</strong>. It's also given that the model estimates that the projectile was launched from an initial height of <strong>7</strong> meters above the ground. Therefore, when <strong>x = 0</strong>, <strong>f (x) = 7</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>7</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a (x − 3)² + 51.1</strong> yields <strong>7 = a (0 − 3)² + 51.1</strong>, or <strong>7 = 9 a + 51.1</strong>. Subtracting <strong>51.1</strong> from both sides of this equation yields <strong>−44.1 = 9 a</strong>. Dividing both sides of this equation by <strong>9</strong> yields <strong>−4.9 = a</strong>. Substituting <strong>−4.9</strong> for <strong>a</strong> in the equation <strong>f (x) = a (x − 3)² + 51.1</strong> yields <strong>f (x) = − 4.9 (x − 3)² + 51.1</strong>. Therefore, the equation <strong>f (x) = − 4.9 (x − 3)² + 51.1</strong> models the projectile's height, in meters, above the ground <strong>x</strong> seconds after it was launched. The number of seconds after the launch that the model estimates that the projectile will return to a height of <strong>7</strong> meters is the value of <strong>x</strong> when <strong>f (x) = 7</strong>. Substituting <strong>7</strong> for <strong>f (x)</strong> in <strong>f (x) = − 4.9 (x − 3)² + 51.1</strong> yields <strong>7 = − 4.9 (x − 3)² + 51.1</strong>. Subtracting <strong>51.1</strong> from both sides of this equation yields <strong>−44.1 = − 4.9 (x − 3)²</strong>. Dividing both sides of this equation by <strong>−4.9</strong> yields <strong>9 = (x − 3)²</strong>. Taking the square root of both sides of this equation yields two equations: <strong>3 = x − 3</strong> and <strong>−3 = x − 3</strong>. Adding <strong>3</strong> to both sides of the equation <strong>3 = x − 3</strong> yields <strong>6 = x</strong>. Adding <strong>3</strong> to both sides of the equation <strong>−3 = x − 3</strong> yields <strong>0 = x</strong>. Since <strong>0</strong> seconds after the launch represents the time at which the projectile was launched, <strong>6</strong> must be the number of seconds the model estimates that the projectile will return to a height of <strong>7</strong> meters.<br>Alternate approach: It's given that a quadratic function models the projectile's height, in meters, above the ground in terms of the time, in seconds, after it was launched. It's also given that the model estimates that the projectile was launched from an initial height of <strong>7</strong> meters above the ground and reached a maximum height of <strong>51.1</strong> meters above the ground <strong>3</strong> seconds after the launch. Since the model is quadratic, and quadratic functions are symmetric, the model estimates that for any given height less than the maximum height, the time the projectile takes to travel from the given height to the maximum height is the same as the time the projectile takes to travel from the maximum height back to the given height. Thus, since the model estimates the projectile took <strong>3</strong> seconds to travel from <strong>7</strong> meters above the ground to its maximum height of <strong>51.1</strong> meters above the ground, the model also estimates the projectile will take <strong>3</strong> more seconds to travel from its maximum height of <strong>51.1</strong> meters above the ground back to <strong>7</strong> meters above the ground. Thus, the model estimates that the projectile will return to a height of <strong>7</strong> meters <strong>3</strong> seconds after it reaches its maximum height, which is <strong>6</strong> seconds after the launch.<br>Choice A is incorrect. The model estimates that <strong>3</strong> seconds after the launch, the projectile reached a height of <strong>51.1</strong> meters, not <strong>7</strong> meters.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "2c6f214f",
    type: "mcq",
    questionHtml:
      "The first term of a sequence is <strong>9</strong>. Each term after the first is <strong>4</strong> times the preceding term. If <strong>w</strong> represents the <strong>n</strong>th term of the sequence, which equation gives <strong>w</strong> in terms of <strong>n</strong>?",
    choices: [
      { label: "A", text: "<strong>w = 4 (9^(n))</strong>" },
      { label: "B", text: "<strong>w = 4 (9^(n − 1))</strong>" },
      { label: "C", text: "<strong>w = 9 (4^(n))</strong>" },
      { label: "D", text: "<strong>w = 9 (4^(n − 1))</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since <strong>w</strong> represents the <strong>n</strong>th term of the sequence and <strong>9</strong> is the first term of the sequence, the value of <strong>w</strong> is <strong>9</strong> when the value of <strong>n</strong> is <strong>1</strong>. Since each term after the first is <strong>4</strong> times the preceding term, the value of <strong>w</strong> is <strong>9 (4)</strong> when the value of <strong>n</strong> is <strong>2</strong>. Therefore, the value of <strong>w</strong> is <strong>9 (4) (4)</strong>, or <strong>9 (4)²</strong>, when the value of <strong>n</strong> is <strong>3</strong>. More generally, the value of <strong>w</strong> is <strong>9 (4^(n − 1))</strong> for a given value of <strong>n</strong>. Therefore, the equation <strong>w = 9 (4^(n − 1))</strong> gives <strong>w</strong> in terms of <strong>n</strong>.<br>Choice A is incorrect. This equation describes a sequence for which the first term is <strong>36</strong>, rather than <strong>9</strong>, and each term after the first is <strong>9</strong>, rather than <strong>4</strong>, times the preceding term. <br>Choice B is incorrect. This equation describes a sequence for which the first term is <strong>4</strong>, rather than <strong>9</strong>, and each term after the first is <strong>9</strong>, rather than <strong>4</strong>, times the preceding term.<br>Choice C is incorrect. This equation describes a sequence for which the first term is <strong>36</strong>, rather than <strong>9</strong>.",
    hasFigure: false,
  },
  {
    id: "2d1614a1",
    type: "mcq",
    questionHtml:
      "<strong>P (t) = 290 (1.04)^((four sixths) t)</strong><br>The function <strong>P</strong> models the population, in thousands, of a certain city <strong>t</strong> years after <strong>2005</strong>. According to the model, the population is predicted to increase by <strong>n % sign</strong> every <strong>18</strong> months. What is the value of <strong>n</strong>?",
    choices: [
      { label: "A", text: "<strong>0.38</strong>" },
      { label: "B", text: "<strong>1.04</strong>" },
      { label: "C", text: "<strong>4</strong>" },
      { label: "D", text: "<strong>6</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that the function <strong>P</strong> models the population of the city <strong>t</strong> years after <strong>2005</strong>. Since there are <strong>12</strong> months in a year, <strong>18</strong> months is equivalent to <strong>(18) / (12)</strong> years. Therefore, the expression <strong>(18) / (12) x</strong> can represent the number of years in <strong>x</strong> <strong>18</strong>-month periods. Substituting <strong>(18) / (12) x</strong> for <strong>t</strong> in the given equation yields <strong>P ((18) / (12) x) = 290 (1.04)^((four sixths) ((18) / (12) x))</strong>, which is equivalent to <strong>P ((18) / (12) x) = 290 (1.04)^(x)</strong>. Therefore, for each <strong>18</strong>-month period, the predicted population of the city is <strong>1.04</strong> times, or <strong>104 % sign</strong> of, the previous population. This means that the population is predicted to increase by <strong>4 % sign</strong> every <strong>18</strong> months.<br>Choice A is incorrect and may result from conceptual or calculation errors. <br>Choice B is incorrect. Each year, the predicted population of the city is <strong>1.04</strong> times the previous year's predicted population, which is not the same as an increase of <strong>1.04 % sign</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "2f51abc2",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = |59 − 2 x|</strong><br>The function <strong>f</strong> is defined by the given equation. For which of the following values of <strong>k</strong> does <strong>f (k) = 3 k</strong>?",
    choices: [
      { label: "A", text: "<strong>(59) / (5)</strong>" },
      { label: "B", text: "<strong>(59) / (2)</strong>" },
      { label: "C", text: "<strong>(177) / (5)</strong>" },
      { label: "D", text: "<strong>59</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The value of <strong>k</strong> for which <strong>f (k) = 3 k</strong> can be found by substituting <strong>k</strong> for <strong>x</strong> and <strong>3 k</strong> for <strong>f (x)</strong> in the given equation, <strong>f (x) = |59 − 2 x|</strong>, which yields <strong>3 k = |59 − 2 k|</strong>. For this equation to be true, either <strong>− 3 k = 59 − 2 k</strong> or <strong>3 k = 59 − 2 k</strong>. Adding <strong>2 k</strong> to both sides of the equation <strong>− 3 k = 59 − 2 k</strong> yields <strong>−k = 59</strong>. Dividing both sides of this equation by <strong>−1</strong> yields <strong>k = −59</strong>. To check whether <strong>−59</strong> is the value of <strong>k</strong>, substituting <strong>−59</strong> for <strong>k</strong> in the equation <strong>3 k = |59 − 2 k|</strong> yields <strong>3 (−59) = |59 − 2 (−59)|</strong>, which is equivalent to <strong>−177 = |177|</strong>, or <strong>−177 = 177</strong>, which isn't a true statement. Therefore, <strong>−59</strong> isn't the value of <strong>k</strong>. Adding <strong>2 k</strong> to both sides of the equation <strong>3 k = 59 − 2 k</strong> yields <strong>5 k = 59</strong>. Dividing both sides of this equation by <strong>5</strong> yields <strong>k = (59) / (5)</strong>. To check whether <strong>(59) / (5)</strong> is the value of <strong>k</strong>, substituting <strong>(59) / (5)</strong> for <strong>k</strong> in the equation <strong>3 k = |59 − 2 k|</strong> yields <strong>3 ((59) / (5)) = |59 − 2 ((59) / (5))|</strong>, which is equivalent to <strong>(177) / (5) = |(177) / (5)|</strong>, or <strong>(177) / (5) = (177) / (5)</strong>, which is a true statement. Therefore, the value of <strong>k</strong> for which <strong>f (k) = 3 k</strong> is <strong>(59) / (5)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "43926bd9",
    type: "spr",
    questionHtml:
      "<strong>x</strong><br> <strong>f of x</strong><br> <strong>1</strong><br> <strong>a</strong><br> <strong>2</strong><br> <strong>a, to the fifth power.</strong><br> <strong>3</strong><br> <strong>a, to the ninth power</strong><br> <br><br> <br><br> For the exponential function f, the table above shows several values of x and their corresponding values of <strong>f of x</strong>, where a is a constant greater than 1. If k is a constant and <strong>f of k = a, to the twenty ninth power</strong>, what is the value of k ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 8. The values of <strong>f of x</strong> for the exponential function f shown in the table increase by a factor of <strong>a, to the fourth power</strong> for each increase of 1 in x. This relationship can be represented by the equation <strong>f of x = a, raised to the 4 x + b power</strong>, where b is a constant. It’s given that when <strong>x = 2</strong>, <strong>f of x = a, to the fifth power</strong>.  Substituting 2 for x and <strong>a, to the fifth power</strong> for <strong>f of x</strong> into <strong>f of x = a, raised to the 4 x + b power</strong> yields <strong>a, to the fifth power = a, raised to the 4 · 2 + b power</strong>. Since <strong>4 · 2 + b = 5</strong>, it follows that <strong>b = −3</strong>. Thus, an equation that defines the function f is <strong>f of x = a, raised to the 4 x − 3 power</strong>. It follows that the value of k such that <strong>f of k = a, to the twenty ninth power</strong> can be found by solving the equation <strong>4 k − 3 = 29</strong>, which yields <strong>k = 8</strong>.",
    hasFigure: false,
  },
  {
    id: "4a0d0399",
    type: "spr",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = a^(x) + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. In the <strong>xy</strong>-plane, the graph of <strong>y = f (x)</strong> has an <strong>x</strong>-intercept at <strong>(2, 0)</strong> and a <strong>y</strong>-intercept at <strong>(0 −323)</strong>. What is the value of <strong>b</strong>?",
    choices: [],
    correctAnswer: "-324",
    explanation:
      "The correct answer is <strong>−324</strong>. It's given that the function <strong>f</strong> is defined by <strong>f (x) = a^(x) + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. It's also given that the graph of <strong>y = f (x)</strong> has a y-intercept at <strong>(0 −323)</strong>. It follows that <strong>f (0) = −323</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>−323</strong> for <strong>f (x)</strong> in <strong>f (x) = a^(x) + b</strong> yields <strong>−323 = a⁰ + b</strong>, or <strong>−323 = 1 + b</strong>. Subtracting <strong>1</strong> from each side of this equation yields <strong>−324 = b</strong>. Therefore, the value of <strong>b</strong> is <strong>−324</strong>.",
    hasFigure: false,
  },
  {
    id: "4b642eef",
    type: "spr",
    questionHtml:
      "The total distance d, in meters, traveled by an object moving in a straight line can be modeled by a quadratic function that is defined in terms of t, where t is the time in seconds. At a time of 10.0 seconds, the total distance traveled by the object is 50.0 meters, and at a time of 20.0 seconds, the total distance traveled by the object is 200.0 meters. If the object was at a distance of 0 meters when <strong>t = 0</strong>, then what is the total distance traveled, in meters, by the object after 30.0 seconds?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 450. The quadratic equation that models this situation can be written in the form <strong>d = a, t² + b t + c</strong>, where a, b, and c are constants. It’s given that the distance, d, the object traveled was 0 meters when <strong>t = 0</strong> seconds. These values can be substituted into the equation to solve for a, b, and c: <strong>0 = a · 0² + b · 0 + c</strong>. Therefore, <strong>c = 0</strong>, and it follows that <strong>d = a, t² + b t</strong>. Since it’s also given that d is 50 when t is 10 and d is 200 when t is 20, these values for d and t can be substituted to create a system of two linear equations: <strong>50 = a · 10² + b · 10</strong> and <strong>200 = a · 20² + b · 20</strong>, or <strong>10 a + b = 5</strong> and <strong>20 a + b = 10</strong>. Subtracting the first equation from the second equation yields <strong>10 a = 5</strong>, or <strong>a = one half</strong>. Substituting <strong>one half</strong> for a in the first equation and solving for b yields <strong>b = 0</strong>. Therefore, the equation that represents this situation is <strong>d = one half t²</strong>. Evaluating this function when <strong>t = 30</strong> seconds yields <strong>d = one half (30, ), ² = 450</strong>, or <strong>d = 450</strong> meters.",
    hasFigure: false,
  },
  {
    id: "4d037075",
    type: "mcq",
    questionHtml:
      "Moving from left to right:<br><br>The curve is in quadrant 3.<br>The curve trends down sharply.<br><br>As x increases, the curve approaches the line x equals negative 4.<br>As x decreases, the curve approaches the line y equals 0.<br>The curve passes through the following points:<br><br>(negative 7 comma negative 2)<br>(negative 6 comma negative 3)<br>(negative 5 comma negative 6)<br><br>The rational function <strong>f</strong> is defined by an equation in the form <strong>f (x) = (a) / (x + b)</strong>, where <strong>a</strong> and <strong>b</strong> are constants. The partial graph of <strong>y = f (x)</strong> is shown. If  <strong>g (x) = f (x + 4)</strong>, which equation could define function <strong>g</strong>?",
    choices: [
      { label: "A", text: "<strong>g (x) = (6) / (x)</strong>" },
      { label: "B", text: "<strong>g (x) = (6) / (x + 4)</strong>" },
      { label: "C", text: "<strong>g (x) = (6) / (x + 8)</strong>" },
      { label: "D", text: "<strong>g (x) = (6 (x + 4)) / (x + 4)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>f (x) = (a) / (x + b)</strong> and that the graph shown is a partial graph of <strong>y = f (x)</strong>. Substituting <strong>y</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = (a) / (x + b)</strong> yields <strong>y = (a) / (x + b)</strong>. The graph passes through the point <strong>(−7 −2)</strong>. Substituting <strong>−7</strong> for <strong>x</strong> and <strong>−2</strong> for <strong>y</strong> in the equation <strong>y = (a) / (x + b)</strong> yields <strong>−2 = (a) / (−7 + b)</strong>. Multiplying each side of this equation by <strong>−7 + b</strong> yields <strong>− 2 (−7 + b) = a</strong>, or <strong>14 − 2 b = a</strong>. The graph also passes through the point <strong>(−5 −6)</strong>. Substituting <strong>−5</strong> for <strong>x</strong> and <strong>−6</strong> for <strong>y</strong> in the equation <strong>y = (a) / (x + b)</strong> yields <strong>−6 = (a) / (−5 + b)</strong>. Multiplying each side of this equation by <strong>−5 + b</strong> yields <strong>− 6 (−5 + b) = a</strong>, or <strong>30 − 6 b = a</strong>. Substituting <strong>14 − 2 b</strong> for <strong>a</strong> in this equation yields <strong>30 − 6 b = 14 − 2 b</strong>. Adding <strong>6 b</strong> to each side of this equation yields <strong>30 = 14 + 4 b</strong>. Subtracting <strong>14</strong> from each side of this equation yields <strong>16 = 4 b</strong>. Dividing each side of this equation by <strong>4</strong> yields <strong>4 = b</strong>. Substituting <strong>4</strong> for <strong>b</strong> in the equation <strong>14 − 2 b = a</strong> yields <strong>14 − 2 (4) = a</strong>, or <strong>6 = a</strong>. Substituting <strong>6</strong> for <strong>a</strong> and <strong>4</strong> for <strong>b</strong> in the equation <strong>f (x) = (a) / (x + b)</strong> yields <strong>f (x) = (6) / (x + 4)</strong>. It's given that <strong>g (x) = f (x + 4)</strong>. Substituting <strong>x + 4</strong> for <strong>x</strong> in the equation <strong>f (x) = (6) / (x + 4)</strong> yields <strong>f (x + 4) = (6) / (x + 4 + 4)</strong>, which is equivalent to <strong>f (x + 4) = (6) / (x + 8)</strong>. It follows that <strong>g (x) = (6) / (x + 8)</strong>.<br>Choice A is incorrect. This could define function <strong>g</strong> if <strong>g (x) = f (x − 4)</strong>.<br>Choice B is incorrect. This could define function <strong>g</strong> if <strong>g (x) = f (x)</strong>.<br>Choice D is incorrect. This could define function <strong>g</strong> if <strong>g (x) = f (x) dot (x + 4)</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/4d037075_svg1.svg",
  },
  {
    id: "4d7064a6",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = (x − 10) (x + 13)</strong><br>The function <strong>f</strong> is defined by the given equation. For what value of <strong>x</strong> does <strong>f (x)</strong> reach its minimum?",
    choices: [
      { label: "A", text: "<strong>−130</strong>" },
      { label: "B", text: "<strong>−13</strong>" },
      { label: "C", text: "<strong>−(23) / (2)</strong>" },
      { label: "D", text: "<strong>−three halves</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that <strong>f (x) = (x − 10) (x + 13)</strong>, which can be rewritten as  <strong>f (x) = x² + 3 x − 130</strong>. Since the coefficient of the <strong>x²</strong>-term is positive, the graph of <strong>y = f (x)</strong> in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of <strong>x</strong> such that <strong>f (x)</strong> reaches its minimum. For an equation in the form <strong>f (x) = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, the x-coordinate of the vertex is <strong>− (b) / (2 a)</strong>. For the equation <strong>f (x) = x² + 3 x − 130</strong>, <strong>a = 1</strong>, <strong>b = 3</strong>, and <strong>c = −130</strong>. It follows that the x-coordinate of the vertex is <strong>− (3) / (2 (1))</strong>, or <strong>−three halves</strong>. Therefore, <strong>f (x)</strong> reaches its minimum when the value of <strong>x</strong> is <strong>−three halves</strong>.<br>Alternate approach: The value of <strong>x</strong> for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that <strong>f (x) = (x − 10) (x + 13)</strong>, it follows that the two x-intercepts of the graph of <strong>y = f (x)</strong> in the xy-plane occur when <strong>x = 10</strong> and <strong>x = −13</strong>, or at the points <strong>(10, 0)</strong> and <strong>(−13, 0)</strong>. The midpoint between two points, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, is <strong>((x 1 + x 2) / (2), (y 1 + y 2) / (2))</strong>. Therefore, the midpoint between <strong>(10, 0)</strong> and <strong>(−13, 0)</strong> is <strong>((10 + (−13)) / (2), (0 + 0) / (2))</strong>, or <strong>(−three halves, 0)</strong>. It follows that <strong>f (x)</strong> reaches its minimum when the value of <strong>x</strong> is <strong>−three halves</strong>.<br>Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane.<br>Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of <strong>y = f (x)</strong> in the xy-plane.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "4dd4efcf",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = a x² + 4 x + c</strong><br>In the given quadratic function, <strong>a</strong> and <strong>c</strong> are constants. The graph of <strong>y = f (x)</strong> in the xy-plane is a parabola that opens upward and has a vertex at the point <strong>(h, k)</strong>, where <strong>h</strong> and <strong>k</strong> are constants. If <strong>k < 0</strong> and <strong>f (−9) = f (3)</strong>, which of the following must be true?<br><br><strong>c < 0</strong><br><strong>a > or = 1</strong>",
    choices: [
      { label: "A", text: "I only" },
      { label: "B", text: "II only" },
      { label: "C", text: "I and II" },
      { label: "D", text: "Neither I nor II" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the graph of <strong>y = f (x)</strong> in the xy-plane is a parabola with vertex <strong>(h, k)</strong>. If <strong>f (−9) = f (3)</strong>, then for the graph of <strong>y = f (x)</strong>, the point with an x-coordinate of <strong>−9</strong> and the point with an x-coordinate of <strong>3</strong> have the same y-coordinate. In the xy-plane, a parabola is a symmetric graph such that when two points have the same y-coordinate, these points are equidistant from the vertex, and the x-coordinate of the vertex is halfway between the x-coordinates of these two points. Therefore, for the graph of <strong>y = f (x)</strong>, the points with x-coordinates <strong>−9</strong> and <strong>3</strong> are equidistant from the vertex, <strong>(h, k)</strong>, and <strong>h</strong> is halfway between <strong>−9</strong> and <strong>3</strong>. The value that is halfway between <strong>−9</strong> and <strong>3</strong> is <strong>(−9 + 3) / (2)</strong>, or <strong>−3</strong>. Therefore, <strong>h = −3</strong>. The equation defining <strong>f</strong> can also be written in vertex form, <strong>f (x) = a (x − h)² + k</strong>. Substituting <strong>−3</strong> for <strong>h</strong> in this equation yields <strong>f (x) = a (x − (−3))² + k</strong>, or <strong>f (x) = a (x + 3)² + k</strong>. This equation is equivalent to <strong>f (x) = a (x² + 6 x + 9) + k</strong>, or <strong>f (x) = a x² + 6 a x + 9 a + k</strong>. Since <strong>f (x) = a x² + 4 x + c</strong>, it follows that <strong>6 a = 4</strong> and <strong>9 a + k = c</strong>. Dividing both sides of the equation <strong>6 a = 4</strong> by <strong>6</strong> yields <strong>a = four sixths</strong>, or <strong>a = two thirds</strong>. Since <strong>two thirds < 1</strong>, it's not true that <strong>a > or = 1</strong>. Therefore, statement II isn't true. Substituting <strong>two thirds</strong> for <strong>a</strong> in the equation <strong>9 a + k = c</strong> yields <strong>9 (two thirds) + k = c</strong>, or <strong>6 + k = c</strong>. Subtracting <strong>6</strong> from both sides of this equation yields <strong>k = c − 6</strong>. If <strong>k < 0</strong>, then <strong>c − 6 < 0</strong>, or <strong>c < 6</strong>. Since <strong>c</strong> could be any value less than <strong>6</strong>, it's not necessarily true that <strong>c < 0</strong>. Therefore, statement I isn't necessarily true. Thus, neither I nor II must be true.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "58dcc59f",
    type: "spr",
    questionHtml:
      "A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be 104 square feet, what will be the length, in feet, of the garden?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 13. Let w represent the width of the rectangular garden, in feet. Since the length of the garden will be 5 feet longer than the width of the garden, the length of the garden will be <strong>w + 5</strong> feet. Thus the area of the garden will be <strong>w · (w + 5, )</strong>. It is also given that the area of the garden will be 104 square feet. Therefore, <strong>w · (w + 5, ) = 104</strong>, which is equivalent to <strong>w² + 5 w − 104 = 0</strong>. Factoring this equation results in <strong>(w + 13, ) · (w − 8, ) = 0</strong>. Therefore, <strong>w = 8</strong> and <strong>w = −13</strong>. Because width cannot be negative, the width of the garden must be 8 feet. This means the length of the garden must be <strong>8 + 5 = 13</strong> feet.",
    hasFigure: false,
  },
  {
    id: "59d1f4b5",
    type: "mcq",
    questionHtml:
      "The equation above models the number of members, M, of a gym t years after the gym opens. Of the following, which equation models the number of members of the gym q quarter years after the gym opens?",
    choices: [
      {
        label: "A",
        text: "<strong>M = 1, 800 · 1 . 0 2, to the power of the fraction q over 4</strong>",
      },
      {
        label: "B",
        text: "<strong>M = 1, 800 · 1 . 0 2, to the power 4 q</strong>",
      },
      {
        label: "C",
        text: "<strong>M = 1, 800 · 1 . 0 0 5, to the power 4 q</strong>",
      },
      {
        label: "D",
        text: "<strong>M = 1, 800 · 1 . 0 8 2, to the power q</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. In 1 year, there are 4 quarter years, so the number of quarter years, q, is 4 times the number of years, t ; that is, <strong>q = 4, t</strong>. This is equivalent to <strong>t = q over 4</strong>, and substituting this into the expression for M in terms of t gives <strong>M = 1, 800 · 1 . 0 2 raised to the fraction q over 4 power</strong>.Choices B and D are incorrect and may be the result of incorrectly using <strong>t = 4 q</strong> instead of <strong>q = 4, t</strong>. (Choices B and D are nearly the same since <strong>1 . 0 2 raised to the 4 q power</strong> is equivalent to <strong>(1 . 0 2 to the fourth power, ), raised to the q power</strong>, which is approximately <strong>1 . 0 8 2 to the q power</strong>.) Choice C is incorrect and may be the result of incorrectly using <strong>t = 4 q</strong> and unnecessarily dividing 0.02 by 4.",
    hasFigure: false,
  },
  {
    id: "635f54ee",
    type: "mcq",
    questionHtml:
      "The surface area of a cube is <strong>6 · (the fraction a, over 4, ), ²</strong>, where a is a positive constant. Which of the following gives the perimeter of one face of the cube?",
    choices: [
      { label: "A", text: "<strong>the fraction a, over 4</strong>" },
      { label: "B", text: "<strong>a</strong>" },
      { label: "C", text: "<strong>4 a</strong>" },
      { label: "D", text: "<strong>6 a</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A cube has 6 faces of equal area, so if the total surface area of a cube is <strong>6 · (a, over 4, ), ²</strong>, then the area of one face is <strong>(a, over 4, ), ²</strong>. Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is <strong>(a, over 4, ), ²</strong>, then the length of one edge of the cube is <strong>a, over 4</strong>. Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is <strong>4 · the fraction a, over 4 = a</strong>.<br><br>Choice A is incorrect because if the perimeter of one face of the cube is <strong>a, over 4</strong>, then the total surface area of the cube is <strong>6 · (the fraction with numerator a, over 4, and denominator 4, ), ² = 6 · (a, over 16, ), ²</strong>, which is not <strong>6 · (a, over 4, ), ²</strong>. Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is <strong>6 · (the fraction 4 a, over 4, ), ² = 6 a, ²</strong>, which is not <strong>6 · (a, over 4, ), ²</strong>. Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface area of the cube is <strong>6 · (the fraction 6 a, over 4, ), ² = 6 · (the fraction 3 a, over 2, ), ²</strong>, which is not <strong>6 · (a, over 4, ), ²</strong>.",
    hasFigure: false,
  },
  {
    id: "6d9e01a2",
    type: "spr",
    questionHtml:
      "<strong>f (x) = 4 x² − 50 x + 126</strong><br>The given equation defines the function <strong>f</strong>. For what value of <strong>x</strong> does <strong>f (x)</strong> reach its minimum?",
    choices: [],
    correctAnswer: "25/4, 6.25",
    explanation:
      "The correct answer is <strong>(25) / (4)</strong>. The given equation can be rewritten in the form <strong>f (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants. When <strong>a > 0</strong>, <strong>h</strong> is the value of <strong>x</strong> for which <strong>f (x)</strong> reaches its minimum. The given equation can be rewritten as <strong>f (x) = 4 (x² − (50) / (4) x) + 126</strong>, which is equivalent to <strong>f (x) = 4 (x² − (50) / (4) x + ((50) / (8))² − ((50) / (8))²) + 126</strong>. This equation can be rewritten as <strong>f (x) = 4 ((x − (50) / (8))² − ((50) / (8))²) + 126</strong>, or <strong>f (x) = 4 (x − (50) / (8))² − 4 ((50) / (8))² + 126</strong>, which is equivalent to <strong>f (x) = 4 (x − (25) / (4))² − (121) / (4)</strong>. Therefore, <strong>h = (25) / (4)</strong>, so the value of <strong>x</strong> for which <strong>f (x)</strong> reaches its minimum is <strong>(25) / (4)</strong>. Note that 25/4 and 6.25 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "6f5540a5",
    type: "mcq",
    questionHtml:
      "Kao measured the temperature of a cup of hot chocolate placed in a room with a constant temperature of 70 degrees Fahrenheit (°F). The temperature of the hot chocolate was 185°F at 6:00 p.m. when it started cooling. The temperature of the hot chocolate was 156°F at 6:05 p.m. and 135°F at 6:10 p.m. The hot chocolate’s temperature continued to decrease. Of the following functions, which best models the temperature <strong>T of m</strong>, in degrees Fahrenheit, of Kao’s hot chocolate m minutes after it started cooling?",
    choices: [
      {
        label: "A",
        text: "<strong>T of m = 185 · 1 . 2 5 to the m power</strong>",
      },
      {
        label: "B",
        text: "<strong>T of m = 185 · 0 . 8 5 to the m power</strong>",
      },
      {
        label: "C",
        text: "<strong>T of m = (185 − 70, ) · 0 . 7 5 raised to the m over 5 power</strong>",
      },
      {
        label: "D",
        text: "<strong>T of m = 70 + 115 · 0 . 7 5 raised to the m over 5 power</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The hot chocolate cools from 185°F over time, never going lower than the room temperature, 70°F. Since the base of the exponent in this function, 0.75, is less than 1, <strong>T of m</strong> decreases as time increases. Using the function, the temperature, in °F, at 6:00 p.m. can be estimated as <strong>T of 0</strong> and is equal to <strong>70 + 115 · (0 . 7 5, ), raised to the power zero fifths = 185</strong>. The temperature, in °F, at 6:05 p.m. can be estimated as <strong>T of 5</strong> and is equal to <strong>70 + 115 · (0 . 7 5, ) raised to the power five fifths</strong>, which is approximately 156°F. Finally, the temperature, in °F, at 6:10 p.m. can be estimated as <strong>T of 10</strong> and is equal to <strong>70 + 115 · (0 . 7 5, ), raised to the power ten fifths</strong>, which is approximately 135°F. Since these three given values of m and their corresponding values for <strong>T of m</strong> can be verified using the function <strong>T of m = 70 + 115 · (0 . 7 5, ), raised to the power the fraction m over 5</strong>, this is the best function out of the given choices to model the temperature of Kao’s hot chocolate after m minutes.Choice A is incorrect because the base of the exponent,<strong>1 . 2 5</strong>, results in the value of <strong>T of m</strong> increasing over time rather than decreasing. Choice B is incorrect because when m is large enough, <strong>T of m</strong> becomes less than 70. Choice C is incorrect because the maximum value of <strong>T of m</strong> at 6:00 p.m. is 115°F, not 185°F.",
    hasFigure: false,
  },
  {
    id: "70753f99",
    type: "mcq",
    questionHtml:
      "The function f is defined by <strong>f of x = (x + 3, ) · (x + 1, )</strong>. The graph of f in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of f ?",
    choices: [
      { label: "A", text: "<strong>−4 < x, which < −3</strong>" },
      { label: "B", text: "<strong>−3 < x, which < 1</strong>" },
      { label: "C", text: "<strong>1 < x, which < 3</strong>" },
      { label: "D", text: "<strong>3 < x, which < 4</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The graph of a quadratic function in the xy-plane is a parabola. The axis of symmetry of the parabola passes through the vertex of the parabola. Therefore, the vertex of the parabola and the midpoint of the segment between the two x-intercepts of the graph have the same x-coordinate. Since <strong>f of − 3 = f of − 1, which = 0</strong>, the x-coordinate of the vertex is <strong>the fraction with numerator − 3 + −1, and denominator 2 = −2</strong>. Of the shown intervals, only the interval in choice B contains –2. Choices A, C, and D are incorrect and may result from either calculation errors or misidentification of the graph’s x-intercepts.",
    hasFigure: false,
  },
  {
    id: "7902bed0",
    type: "mcq",
    questionHtml:
      "A machine launches a softball from ground level. The softball reaches a maximum height of <strong>51.84</strong> meters above the ground at <strong>1.8</strong> seconds and hits the ground at <strong>3.6</strong> seconds. Which equation represents the height above ground <strong>h</strong>, in meters, of the softball <strong>t</strong> seconds after it is launched?",
    choices: [
      { label: "A", text: "<strong>h = − t² + 3.6</strong>" },
      { label: "B", text: "<strong>h = − t² + 51.84</strong>" },
      { label: "C", text: "<strong>h = − 16 (t − 1.8)² − 3.6</strong>" },
      { label: "D", text: "<strong>h = − 16 (t − 1.8)² + 51.84</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. An equation representing the height above ground <strong>h</strong>, in meters, of a softball <strong>t</strong> seconds after it is launched by a machine from ground level can be written in the form <strong>h = − a (t − b)² + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are positive constants. In this equation, <strong>b</strong> represents the time, in seconds, at which the softball reaches its maximum height of <strong>c</strong> meters above the ground. It's given that this softball reaches a maximum height of <strong>51.84</strong> meters above the ground at <strong>1.8</strong> seconds; therefore, <strong>b = 1.8</strong> and <strong>c = 51.84</strong>. Substituting <strong>1.8</strong> for <strong>b</strong> and <strong>51.84</strong> for <strong>c</strong> in the equation <strong>h = − a (t − b)² + c</strong> yields <strong>h = − a (t − 1.8)² + 51.84</strong>. It's also given that this softball hits the ground at <strong>3.6</strong> seconds; therefore, <strong>h = 0</strong> when <strong>t = 3.6</strong>. Substituting <strong>0</strong> for <strong>h</strong> and <strong>3.6</strong> for <strong>t</strong> in the equation <strong>h = − a (t − 1.8)² + 51.84</strong> yields <strong>0 = − a (3.6 − 1.8)² + 51.84</strong>, which is equivalent to <strong>0 = − a (1.8)² + 51.84</strong>, or <strong>0 = − 3.24 a + 51.84</strong>. Adding <strong>3.24 a</strong> to both sides of this equation yields <strong>3.24 a = 51.84</strong>. Dividing both sides of this equation by <strong>3.24</strong> yields <strong>a = 16</strong>. Substituting <strong>16</strong> for <strong>a</strong> in the equation <strong>h = − a (t − 1.8)² + 51.84</strong> yields <strong>h = − 16 (t − 1.8)² + 51.84</strong>. Therefore, <strong>h = − 16 (t − 1.8)² + 51.84</strong> represents the height above ground <strong>h</strong>, in meters, of this softball <strong>t</strong> seconds after it is launched.<br>Choice A is incorrect. This equation represents a situation where the maximum height is <strong>3.6</strong> meters above the ground at <strong>0</strong> seconds, not <strong>51.84</strong> meters above the ground at <strong>1.8</strong> seconds.<br>Choice B is incorrect. This equation represents a situation where the maximum height is <strong>51.84</strong> meters above the ground at <strong>0</strong> seconds, not <strong>1.8</strong> seconds.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "7eed640d",
    type: "mcq",
    questionHtml:
      "The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If <strong>y = h of x</strong> is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph?",
    choices: [
      { label: "A", text: "The initial height of the projectile" },
      { label: "B", text: "The maximum height of the projectile" },
      {
        label: "C",
        text: "The time at which the projectile reaches its maximum height",
      },
      { label: "D", text: "The time at which the projectile hits the ground" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The positive x-intercept of the graph of <strong>y = h of x</strong> is a point <strong>with coordinates x, y</strong> for which <strong>y = 0</strong>. Since <strong>y = h of x</strong> models the height above the ground, in feet, of the projectile, a y-value of 0 must correspond to the height of the projectile when it is 0 feet above ground or, in other words, when the projectile is on the ground. Since x represents the time since the projectile was launched, it follows that the positive x-intercept, <strong>x, 0</strong>, represents the time at which the projectile hits the ground.Choice A is incorrect and may result from misidentifying the y-intercept as a positive x-intercept. Choice B is incorrect and may result from misidentifying the y-value of the vertex of the graph of the function as an x-intercept. Choice C is incorrect and may result from misidentifying the x-value of the vertex of the graph of the function as an x-intercept.",
    hasFigure: false,
  },
  {
    id: "841ef26c",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 4 x² + 64 x + 262</strong><br>The function <strong>g</strong> is defined by <strong>g (x) = f (x + 5)</strong>. For what value of <strong>x</strong> does <strong>g (x)</strong> reach its minimum?",
    choices: [
      { label: "A", text: "<strong>−13</strong>" },
      { label: "B", text: "<strong>−8</strong>" },
      { label: "C", text: "<strong>−5</strong>" },
      { label: "D", text: "<strong>−3</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>g (x) = f (x + 5)</strong>. Since <strong>f (x) = 4 x² + 64 x + 262</strong>, it follows that <strong>f (x + 5) = 4 (x + 5)² + 64 (x + 5) + 262</strong>. Expanding the quantity <strong>(x + 5)²</strong> in this equation yields <strong>f (x + 5) = 4 (x² + 10 x + 25) + 64 (x + 5) + 262</strong>. Distributing the <strong>4</strong> and the <strong>64</strong> yields <strong>f (x + 5) = 4 x² + 40 x + 100 + 64 x + 320 + 262</strong>. Combining like terms yields <strong>f (x + 5) = 4 x² + 104 x + 682</strong>. Therefore, <strong>g (x) = 4 x² + 104 x + 682</strong>. For a quadratic function defined by an equation of the form <strong>g (x) = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants and <strong>a</strong> is positive, <strong>g (x)</strong> reaches its minimum, <strong>k</strong>, when the value of <strong>x</strong> is <strong>h</strong>. The equation <strong>g (x) = 4 x² + 104 x + 682</strong> can be rewritten in this form by completing the square. This equation is equivalent to <strong>g (x) = 4 (x² + 26) + 682</strong>, or <strong>g (x) = 4 (x² + 26 x + 169 − 169) + 682</strong>. This equation can be rewritten as <strong>g (x) = 4 ((x + 13)² − 169) + 682</strong>, or <strong>g (x) = 4 (x + 13)² − 4 (169) + 682</strong>, which is equivalent to <strong>g (x) = 4 (x + 13)² + 6</strong>. This equation is in the form <strong>g (x) = a (x − h)² + k</strong>, where <strong>a = 4</strong>, <strong>h = −13</strong>, and <strong>k = 6</strong>. Therefore, <strong>g (x)</strong> reaches its minimum when the value of <strong>x</strong> is <strong>−13</strong>.<br>Choice B is incorrect. This is the value of <strong>x</strong> for which <strong>f (x)</strong>, rather than <strong>g (x)</strong>, reaches its minimum.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the value of <strong>x</strong> for which <strong>f (x − 5)</strong>, rather than <strong>f (x + 5)</strong>, reaches its minimum.",
    hasFigure: false,
  },
  {
    id: "8490cc45",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = (−8) (2)^(x) + 22</strong>. What is the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>(0, 14)</strong>" },
      { label: "B", text: "<strong>(0, 2)</strong>" },
      { label: "C", text: "<strong>(0, 22)</strong>" },
      { label: "D", text: "<strong>(0 −8)</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane occurs at the point on the graph where <strong>x = 0</strong>. In other words, when <strong>x = 0</strong>, the corresponding value of <strong>f (x)</strong> is the y-coordinate of the y-intercept. Substituting <strong>0</strong> for <strong>x</strong> in the given equation yields <strong>f (0) = (−8) (2)⁰ + 22</strong>, which is equivalent to <strong>f (0) = (−8) (1) + 22</strong>, or <strong>f (0) = 14</strong>. Thus, when <strong>x = 0</strong>, the corresponding value of <strong>f (x)</strong> is <strong>14</strong>. Therefore, the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane is <strong>(0, 14)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This could be the y-intercept for <strong>f (x) = (−8) (2)^(x)</strong>, not <strong>f (x) = (−8) (2)^(x) + 22</strong>.",
    hasFigure: false,
  },
  {
    id: "84dd43f8",
    type: "spr",
    questionHtml:
      "For the function <strong>f</strong>, <strong>f (0) = 86</strong>, and for each increase in <strong>x</strong> by <strong>1</strong>, the value of <strong>f (x)</strong> decreases by <strong>80 % sign</strong>. What is the value of <strong>f (2)</strong>?",
    choices: [],
    correctAnswer: "3.44, 86/25",
    explanation:
      "The correct answer is <strong>3.44</strong>. It’s given that <strong>f (0) = 86</strong> and that for each increase in <strong>x</strong> by <strong>1</strong>, the value of <strong>f (x)</strong> decreases by <strong>80 % sign</strong>. Because the output of the function decreases by a constant percentage for each <strong>1</strong>-unit increase in the value of <strong>x</strong>, this relationship can be represented by an exponential function of the form <strong>f (x) = a (b)^(x)</strong>, where <strong>a</strong> represents the initial value of the function and <strong>b</strong> represents the rate of decay,<br>expressed as a decimal. Because <strong>f (0) = 86</strong>, the value of <strong>a</strong> must be <strong>86</strong>. Because the value of <strong>f (x)</strong> decreases by <strong>80 % sign</strong> for each <strong>1</strong>-unit increase in <strong>x</strong>, the value of <strong>b</strong> must be <strong>(1 − 0 . 80)</strong>, or <strong>0 . 2</strong>. Therefore, the function <strong>f</strong> can be defined by <strong>f (x) = 86 (0 . 2)^(x)</strong>. Substituting <strong>2</strong> for <strong>x</strong> in this function yields <strong>f (2) = 86 (0 . 2)²</strong>, which is equivalent to <strong>f (2) = 86 (0 . 04)</strong>, or <strong>f (2) = 3 . 44</strong>. Either <strong>3.44</strong> or <strong>86 ÷ 25</strong> may be entered as the correct answer.<br>Alternate approach: It’s given that <strong>f (0) = 86</strong> and that for each increase in <strong>x</strong> by <strong>1</strong>, the value of <strong>f (x)</strong> decreases by <strong>80 % sign</strong>. Therefore, when <strong>x = 1</strong>, the value of <strong>f (x)</strong> is <strong>(100 − 80) % sign</strong>, or <strong>20 % sign</strong>, of <strong>86</strong>, which can be expressed as <strong>(0 . 20) (86)</strong>. Since <strong>(0 . 20) (86) = 17 . 2</strong>, the value of <strong>f (1)</strong> is <strong>17.2</strong>. Similarly, when <strong>x = 2</strong>, the value of <strong>f (x)</strong> is <strong>20 % sign</strong> of <strong>17.2</strong>, which can be expressed as <strong>(0 . 20) (17 . 2)</strong>. Since <strong>(0 . 20) (17 . 2) = 3 . 44</strong>, the value of <strong>f (2)</strong> is <strong>3.44</strong>. Either <strong>3.44</strong> or <strong>86 ÷ 25</strong> may be entered as the correct answer.",
    hasFigure: false,
  },
  {
    id: "84e8cc72",
    type: "mcq",
    questionHtml:
      "A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object has an initial height of <strong>10</strong> feet above the ground and reaches its maximum height of <strong>1, 034</strong> feet above the ground <strong>8</strong> seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground <strong>10</strong> seconds after being launched?",
    choices: [
      { label: "A", text: "<strong>234</strong>" },
      { label: "B", text: "<strong>778</strong>" },
      { label: "C", text: "<strong>970</strong>" },
      { label: "D", text: "<strong>1, 014</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that a quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. This quadratic function can be defined by an equation of the form <strong>f (x) = a (x − h)² + k</strong>, where <strong>f (x)</strong> is the height of the object <strong>x</strong> seconds after it was launched, and <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are constants such that the function reaches its maximum value, <strong>k</strong>, when <strong>x = h</strong>. Since the model indicates the object reaches its maximum height of <strong>1, 034</strong> feet above the ground <strong>8</strong> seconds after being launched, <strong>f (x)</strong> reaches its maximum value, <strong>1, 034</strong>, when <strong>x = 8</strong>. Therefore, <strong>k = 1, 034</strong> and <strong>h = 8</strong>. Substituting <strong>8</strong> for <strong>h</strong> and <strong>1, 034</strong> for <strong>k</strong> in the function <strong>f (x) = a (x − h)² + k</strong> yields <strong>f (x) = a (x − 8)² + 1, 034</strong>. Since the model indicates the object has an initial height of <strong>10</strong> feet above the ground, the value of <strong>f (x)</strong> is <strong>10</strong> when <strong>x = 0</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>10</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a (x − 8)² + 1, 034</strong> yields <strong>10 = a (0 − 8)² + 1, 034</strong>, or <strong>10 = 64 a + 1, 034</strong>. Subtracting <strong>1, 034</strong> from both sides of this equation yields <strong>64 a = −1, 024</strong>. Dividing both sides of this equation by <strong>64</strong> yields <strong>a = −16</strong>. Therefore, the model can be represented by the equation <strong>f (x) = − 16 (x − 8)² + 1, 034</strong>. Substituting <strong>10</strong> for <strong>x</strong> in this equation yields <strong>f (10) = − 16 (10 − 8)² + 1, 034</strong>, or <strong>f (10) = 970</strong>. Therefore, based on the model, <strong>10</strong> seconds after being launched, the height of the object above the ground is <strong>970</strong> feet.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "91e7ea5e",
    type: "mcq",
    questionHtml:
      "The quadratic function h is defined as shown. In the xy-plane, the graph of <strong>y = h of x</strong> intersects the x-axis at the points <strong>with coordinates 0, 0</strong>and <strong>t, 0</strong>, where t is a constant. What is the value of t ?",
    choices: [
      { label: "A", text: "1" },
      { label: "B", text: "2" },
      { label: "C", text: "4" },
      { label: "D", text: "8" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the graph of <strong>y = h of x</strong> intersects the x-axis at <strong>0, 0</strong> and <strong>t, 0</strong>, where t is a constant. Since this graph intersects the x-axis when <strong>y = 0</strong> or when <strong>h of x = 0</strong>, it follows that <strong>h of 0 = 0</strong> and <strong>h of t = 0</strong>. If <strong>h of t = 0</strong>, then <strong>0 = 2 · (t − 4, ), ² − 32</strong>. Adding 32 to both sides of this equation yields <strong>32 = 2 · (t − 4, ), ²</strong>. Dividing both sides of this equation by 2 yields <strong>16 = (t − 4, ), ²</strong>. Taking the square root of both sides of this equation yields <strong>4 = t − 4</strong>. Adding 4 to both sides of this equation yields <strong>8 = t</strong>. Therefore, the value of t is 8.Choices A, B, and C are incorrect and may result from calculation errors.",
    hasFigure: false,
  },
  {
    id: "92f812bb",
    type: "mcq",
    questionHtml:
      "In the xy-plane, a parabola has vertex <strong>(9 −14)</strong> and intersects the x-axis at two points. If the equation of the parabola is written in the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, which of the following could be the value of <strong>a + b + c</strong>?",
    choices: [
      { label: "A", text: "<strong>−23</strong>" },
      { label: "B", text: "<strong>−19</strong>" },
      { label: "C", text: "<strong>−14</strong>" },
      { label: "D", text: "<strong>−12</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The equation of a parabola in the xy-plane can be written in the form <strong>y = a (x − h)² + k</strong>, where <strong>a</strong> is a constant and <strong>(h, k)</strong> is the vertex of the parabola. If <strong>a</strong> is positive, the parabola will open upward, and if <strong>a</strong> is negative, the parabola will open downward. It’s given that the parabola has vertex <strong>(9 −14)</strong>. Substituting <strong>9</strong> for <strong>h</strong> and <strong>−14</strong> for <strong>k</strong> in the equation <strong>y = a (x − h)² + k</strong> gives <strong>y = a (x − 9)² − 14</strong>, which can be rewritten as <strong>y = a (x − 9) (x − 9) − 14</strong>, or <strong>y = a (x² − 18 x + 81) − 14</strong>. Distributing the factor of <strong>a</strong> on the right-hand side of this equation yields <strong>y = a x² − 18 a x + 81 a − 14</strong>. Therefore, the equation of the parabola, <strong>y = a x² − 18 a x + 81 a − 14</strong>, can be written in the form <strong>y = a x² + b x + c</strong>, where <strong>a = a</strong>, <strong>b = − 18 a</strong>, and <strong>c = 81 a − 14</strong>. Substituting <strong>− 18 a</strong> for <strong>b</strong> and <strong>81 a − 14</strong> for <strong>c</strong> in the expression <strong>a + b + c</strong> yields <strong>(a) + (−18 a) + (81 a − 14)</strong>, or <strong>64 a − 14</strong>. Since the vertex of the parabola, <strong>(9 −14)</strong>, is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant <strong>a</strong> must have a positive value. Setting the expression <strong>64 a − 14</strong> equal to the value in choice D yields <strong>64 a − 14 = −12</strong>. Adding <strong>14</strong> to both sides of this equation yields <strong>64 a = 2</strong>. Dividing both sides of this equation by <strong>64</strong> yields <strong>a = two sixty fourths</strong>, which is a positive value. Therefore, if the equation of the parabola is written in the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, the value of <strong>a + b + c</strong> could be<strong>−12</strong>.<br>Choice A is incorrect. If the equation of a parabola with a vertex at <strong>(9 −14)</strong> is written in the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants and <strong>a + b + c = −23</strong>, then the value of <strong>a</strong> will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points. <br>Choice B is incorrect. If the equation of a parabola with a vertex at <strong>(9 −14)</strong> is written in the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants and <strong>a + b + c = −19</strong>, then the value of <strong>a</strong> will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.<br>Choice C is incorrect. If the equation of a parabola with a vertex at <strong>(9 −14)</strong> is written in the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants and <strong>a + b + c = −14</strong>, then the value of <strong>a</strong> will be <strong>0</strong>, which is inconsistent with the equation of a parabola.",
    hasFigure: false,
  },
  {
    id: "9654add7",
    type: "mcq",
    questionHtml:
      "<strong>f of x = −500, x² + 25, 000 x</strong><br> The revenue <strong>f of x</strong>, in dollars, that a company receives from sales of a product is given by the function f above, where x is the unit price, in dollars, of the product. The graph of <strong>y = f of x</strong> in the xy-plane intersects the x-axis at 0 and a. What does a represent?",
    choices: [
      {
        label: "A",
        text: "The revenue, in dollars, when the unit price of the product is $0",
      },
      {
        label: "B",
        text: "The unit price, in dollars, of the product that will result in maximum revenue",
      },
      {
        label: "C",
        text: "The unit price, in dollars, of the product that will result in a revenue of $0",
      },
      {
        label: "D",
        text: "The maximum revenue, in dollars, that the company can make",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. By definition, the y-value when a function intersects the x-axis is 0. It’s given that the graph of the function intersects the x-axis at 0 and a, that x is the unit price, in dollars, of a product, and that <strong>f of x</strong>, where <strong>y = f of x</strong>, is the revenue, in dollars, that a company receives from the sales of the product. Since the value of a occurs when <strong>y = 0</strong>, a is the unit price, in dollars, of the product that will result in a revenue of $0.Choice A is incorrect. The revenue, in dollars, when the unit price of the product is $0 is represented by <strong>f of x</strong>, when <strong>x = 0</strong>. Choice B is incorrect. The unit price, in dollars, of the product that will result in maximum revenue is represented by the x-coordinate of the maximum of f. Choice D is incorrect. The maximum revenue, in dollars, that the company can make is represented by the y-coordinate of the maximum of f.",
    hasFigure: false,
  },
  {
    id: "9afe2370",
    type: "mcq",
    questionHtml:
      "If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?",
    choices: [
      { label: "A", text: "<strong>r < −1</strong>" },
      { label: "B", text: "<strong>−1 < r, which < 0</strong>" },
      { label: "C", text: "<strong>0 < r, which < 1</strong>" },
      { label: "D", text: "<strong>r > 1</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The term (1 + r) represents a percent change. Since the population is decreasing, the percent change must be between 0% and 100%. When the percent change is expressed as a decimal rather than as a percent, the percentage change must be between 0 and 1. Because (1 + r) represents percent change, this can be expressed as 0 r r Choice A is incorrect. If r P would be a negative value, which is not possible. Choices C and D are incorrect. For any value of r > 0, 1 + r > 1, and the exponential function models growth for positive values of the exponent. This contradicts the given information that the population is decreasing.",
    hasFigure: false,
  },
  {
    id: "9f2ecade",
    type: "spr",
    questionHtml:
      "The function h is defined above, where a, b, and c are integer constants. If the zeros of the function are <strong>−5</strong> 6, and 7, what is the value of c ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 210. Since <strong>−5</strong>, 6, and 7 are zeros of the function, the function can be rewritten as <strong>h of x = (x + 5, ) · (x − 6, ) · (x − 7, )</strong>. Expanding the function yields <strong>h of x = x³ − 8, x² − 23 x + 210</strong>. Thus, <strong>a = −8</strong>, <strong>b = −23</strong>, and <strong>c = 210</strong>. Therefore, the value of c is 210.",
    hasFigure: false,
  },
  {
    id: "a45ffacb",
    type: "spr",
    questionHtml:
      "Function <strong>f</strong> is defined by <strong>f (x) = − a^(x) + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. In the xy-plane, the graph of <strong>y = f (x) − 15</strong> has a y-intercept at <strong>(0 −(99) / (7))</strong>. The product of <strong>a</strong> and <strong>b</strong> is <strong>(65) / (7)</strong>. What is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: "5",
    explanation:
      "The correct answer is <strong>5</strong>. It’s given that <strong>f (x) = − a^(x) + b</strong>. Substituting <strong>− a^(x + b)</strong> for <strong>f (x)</strong> in the equation <strong>y = f (x) − 15</strong> yields <strong>y = − a^(x) + b − 15</strong>. It’s given that the y-intercept of the graph of <strong>y = f (x) − 15</strong> is <strong>(0 −(99) / (7))</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>−(99) / (7)</strong> for <strong>y</strong> in the equation <strong>y = − a^(x) + b − 15</strong> yields <strong>−(99) / (7) = − a⁰ + b − 15</strong>, which is equivalent to <strong>−(99) / (7) = −1 + b − 15</strong>, or <strong>−(99) / (7) = b − 16</strong>. Adding <strong>16</strong> to both sides of this equation yields <strong>(13) / (7) = b</strong>. It’s given that the product of <strong>a</strong> and <strong>b</strong> is <strong>(65) / (7)</strong>, or <strong>a b = (65) / (7)</strong>. Substituting <strong>(13) / (7)</strong>  for <strong>b</strong> in this equation yields <strong>(a) ((13) / (7)) = (65) / (7)</strong>. Dividing both sides of this equation by <strong>(13) / (7)</strong> yields <strong>a = 5</strong>.",
    hasFigure: false,
  },
  {
    id: "a7711fe8",
    type: "mcq",
    questionHtml:
      "What is the minimum value of the function f defined by <strong>f of x = (x − 2, ), ² − 4</strong> ?",
    choices: [
      { label: "A", text: "<strong>−4</strong>" },
      { label: "B", text: "<strong>−2</strong>" },
      { label: "C", text: "<strong>2</strong>" },
      { label: "D", text: "<strong>4</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given quadratic function f is in vertex form, <strong>f of x = (x − h, ), ² + k</strong>, where <strong>h, k</strong> is the vertex of the graph of <strong>y = f of x</strong> in the xy-plane. Therefore, the vertex of the graph of <strong>y = f of x</strong> is <strong>2 −4</strong>. In addition, the y-coordinate of the vertex represents either the minimum or maximum value of a quadratic function, depending on whether the graph of the function opens upward or downward. Since the leading coefficient of f (the coefficient of the term <strong>(x − 2, ), ²</strong>) is 1, which is positive, the graph of <strong>y = f of x</strong> opens upward. It follows that at <strong>x = 2</strong>, the minimum value of the function f is <strong>−4</strong>.Choice B is incorrect and may result from making a sign error and from using the x-coordinate of the vertex. Choice C is incorrect and may result from using the x-coordinate of the vertex. Choice D is incorrect and may result from making a sign error.",
    hasFigure: false,
  },
  {
    id: "a8ae0d22",
    type: "mcq",
    questionHtml:
      "Two variables, <strong>x</strong> and <strong>y</strong>, are related such that for each increase of <strong>1</strong> in the value of <strong>x</strong>, the value of <strong>y</strong> increases by a factor of <strong>4</strong>. When <strong>x = 0</strong>, <strong>y = 200</strong>. Which equation represents this relationship?",
    choices: [
      { label: "A", text: "<strong>y = 4 (x)²⁰⁰</strong>" },
      { label: "B", text: "<strong>y = 4 (200)^(x)</strong>" },
      { label: "C", text: "<strong>y = 200 (x)⁴</strong>" },
      { label: "D", text: "<strong>y = 200 (4)^(x)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since the value of <strong>y</strong> increases by a constant factor, <strong>4</strong>, for each increase of <strong>1</strong> in the value of <strong>x</strong>, the relationship between <strong>x</strong> and <strong>y</strong> is exponential. An exponential relationship between <strong>x</strong> and <strong>y</strong> can be represented by an equation of the form <strong>y = a (b)^(x)</strong>, where <strong>a</strong> is the value of <strong>x</strong> when <strong>y = 0</strong> and <strong>y</strong> increases by a factor of <strong>b</strong> for each increase of <strong>1</strong> in the value of <strong>x</strong>. Since <strong>y = 200</strong> when <strong>x = 0</strong>, <strong>a = 200</strong>. Since <strong>y</strong> increases by a factor of <strong>4</strong> for each increase of <strong>1</strong> in the value of <strong>x</strong>, <strong>b = 4</strong>. Substituting <strong>200</strong> for <strong>a</strong> and <strong>4</strong> for <strong>b</strong> in the equation <strong>y = a (b)^(x)</strong> yields <strong>y = 200 (4)^(x)</strong>. Thus, the equation <strong>y = 200 (4)^(x)</strong> represents the relationship between <strong>x</strong> and <strong>y</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect. This equation represents a relationship where for each increase of <strong>1</strong> in the value of <strong>x</strong>, the value of <strong>y</strong> increases by a factor of <strong>200</strong>, not <strong>4</strong>, and when <strong>x = 0</strong>, <strong>y</strong> is equal to <strong>4</strong>, not <strong>200</strong>.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "a9084ca4",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 9, 000 (0.66)^(x)</strong><br>The given function <strong>f</strong> models the number of advertisements a company sent to its clients each year, where <strong>x</strong> represents the number of years since <strong>1997</strong>, and <strong>0 < or = x < or = 5</strong>. If <strong>y = f (x)</strong> is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?",
    choices: [
      {
        label: "A",
        text: "The minimum estimated number of advertisements the company sent to its clients during the <strong>5</strong> years was <strong>1, 708</strong>.",
      },
      {
        label: "B",
        text: "The minimum estimated number of advertisements the company sent to its clients during the <strong>5</strong> years was <strong>9, 000</strong>.",
      },
      {
        label: "C",
        text: "The estimated number of advertisements the company sent to its clients in <strong>1997</strong> was <strong>1, 708</strong>.",
      },
      {
        label: "D",
        text: "The estimated number of advertisements the company sent to its clients in <strong>1997</strong> was <strong>9, 000</strong>.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The y-intercept of a graph in the xy-plane is the point where <strong>x = 0</strong>. For the given function <strong>f</strong>, the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane can be found by substituting <strong>0</strong> for <strong>x</strong> in the equation <strong>y = 9, 000 (0.66)^(x)</strong>, which gives <strong>y = 9, 000 (0.66)⁰</strong>. This is equivalent to <strong>y = 9, 000 (1)</strong>, or <strong>y = 9, 000</strong>. Therefore, the y-intercept of the graph of <strong>y = f (x)</strong> is <strong>(0, 9, 000)</strong>. It’s given that the function <strong>f</strong> models the number of advertisements a company sent to its clients each year. Therefore, <strong>f (x)</strong> represents the estimated number of advertisements the company sent to its clients each year. It's also given that <strong>x</strong> represents the number of years since <strong>1997</strong>. Therefore, <strong>x = 0</strong> represents <strong>0</strong> years since <strong>1997</strong>, or <strong>1997</strong>. Thus, the best interpretation of the y-intercept of the graph of <strong>y = f (x)</strong> is that the estimated number of advertisements the company sent to its clients in <strong>1997</strong> was <strong>9, 000</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b73ee6cf",
    type: "mcq",
    questionHtml:
      "The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ?",
    choices: [
      {
        label: "A",
        text: "<strong>50, 000 = 60, 000 · (0 . 0 3, ), to the power t</strong>",
      },
      {
        label: "B",
        text: "<strong>50, 000 = 60, 000 · (3, ), to the power t</strong>",
      },
      {
        label: "C",
        text: "<strong>60, 000 = 50, 000 · (0 . 0 3, ), to the power t</strong>",
      },
      {
        label: "D",
        text: "<strong>60, 000 = 50, 000 · (1 . 0 3, ), to the power t</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Stating that the population will increase each year by 3% from the previous year is equivalent to saying that the population each year will be 103% of the population the year before. Since the initial population is 50,000, the population after t years is given by 50,000(1.03)t. It follows that the equation 60,000 = 50,000(1.03)t can be used to estimate the number of years it will take for the population to reach 60,000.Choice A is incorrect. This equation models how long it will take the population to decrease from 60,000 to 50,000, which is impossible given the growth factor. Choice B is incorrect and may result from misinterpreting a 3% growth as growth by a factor of 3. Additionally, this equation attempts to model how long it will take the population to decrease from 60,000 to 50,000. Choice C is incorrect and may result from misunderstanding how to model percent growth by multiplying the initial amount by a factor greater than 1.",
    hasFigure: false,
  },
  {
    id: "b7c74b73",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 5, 470 (0.64)^((x) / (12))</strong><br>The function <strong>f</strong> gives the value, in dollars, of a certain piece of equipment after <strong>x</strong> months of use. If the value of the equipment decreases each year by <strong>p % sign</strong> of its value the preceding year, what is the value of <strong>p</strong>?",
    choices: [
      { label: "A", text: "<strong>4</strong>" },
      { label: "B", text: "<strong>5</strong>" },
      { label: "C", text: "<strong>36</strong>" },
      { label: "D", text: "<strong>64</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. For a function of the form <strong>f (x) = a (r)^((x) / (k))</strong>, where <strong>a</strong>, <strong>r</strong>, and <strong>k</strong> are constants and <strong>r < 1</strong>, the value of <strong>f (x)</strong> decreases by <strong>100 (1 − r) % sign</strong> for every increase of <strong>x</strong> by <strong>k</strong>. In the given function, <strong>a = 5, 470</strong>, <strong>r = 0.64</strong>, and <strong>k = 12</strong>. Therefore, for the given function, the value of <strong>f (x)</strong> decreases by <strong>100 (1 − 0.64) % sign</strong>, or <strong>36 % sign</strong>, for every increase of <strong>x</strong> by <strong>12</strong>. Since <strong>f (x)</strong> represents the value, in dollars, of the equipment after <strong>x</strong> months of use, it follows that the value of the equipment decreases every <strong>12</strong> months by <strong>36 % sign</strong> of its value the preceding <strong>12</strong> months. Since there are <strong>12</strong> months in a year, the value of the equipment decreases each year by <strong>36 % sign</strong> of its value the preceding year. Thus, the value of <strong>p</strong> is <strong>36</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "b8f13a3a",
    type: "spr",
    questionHtml:
      "Function <strong>f</strong> is defined by <strong>f (x) = − a^(x) + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. In the xy-plane, the graph of <strong>y = f (x) − 12</strong> has a y-intercept at <strong>(0 −(75) / (7))</strong>. The product of <strong>a</strong> and <strong>b</strong> is <strong>(320) / (7)</strong>. What is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: "20",
    explanation:
      "The correct answer is <strong>20</strong>. It’s given that <strong>f (x) = − a^(x) + b</strong>. Substituting <strong>− a^(x + b)</strong> for <strong>f (x)</strong> in the equation <strong>y = f (x) − 12</strong> yields <strong>y = − a^(x) + b − 12</strong>. It’s given that the y-intercept of the graph of <strong>y = f (x) − 12</strong> is <strong>(0 −(75) / (7))</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>−(75) / (7)</strong> for <strong>y</strong> in the equation <strong>y = − a^(x) + b − 12</strong> yields <strong>−(75) / (7) = − a⁰ + b − 12</strong>, which is equivalent to <strong>−(75) / (7) = −1 + b − 12</strong>, or <strong>−(75) / (7) = b − 13</strong>. Adding <strong>13</strong> to both sides of this equation yields <strong>(16) / (7) = b</strong>. It’s given that the product of <strong>a</strong> and <strong>b</strong> is <strong>(320) / (7)</strong>, or <strong>a b = (320) / (7)</strong>. Substituting <strong>(16) / (7)</strong> for <strong>b</strong> in this equation yields <strong>(a) ((16) / (7)) = (320) / (7)</strong>. Dividing both sides of this equation by <strong>(16) / (7)</strong> yields <strong>a = 20</strong>.",
    hasFigure: false,
  },
  {
    id: "bba18ecb",
    type: "mcq",
    questionHtml:
      "When the quadratic function <strong>f</strong> is graphed in the xy-plane, where <strong>y = f (x)</strong>, its vertex is <strong>(−3, 6)</strong>. One of the x-intercepts of this graph is <strong>(−(17) / (4), 0)</strong>. What is the other x-intercept of the graph?",
    choices: [
      { label: "A", text: "<strong>(−(29) / (4), 0)</strong>" },
      { label: "B", text: "<strong>(−seven fourths, 0)</strong>" },
      { label: "C", text: "<strong>(five fourths, 0)</strong>" },
      { label: "D", text: "<strong>((17) / (4), 0)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the x-coordinate of the vertex of the graph of <strong>y = f (x)</strong> is the x-coordinate of the midpoint of its two x-intercepts. The midpoint of two points with x-coordinates <strong>x 1</strong> and <strong>x 2</strong> has x-coordinate <strong>x Subscript m</strong>, where <strong>x Subscript m Baseline = (x 1 + x 2) / (2)</strong>. It′s given that the vertex is <strong>(−3, 6)</strong> and one of the x-intercepts is <strong>(−(17) / (4), 0)</strong>. Substituting <strong>−3</strong> for <strong>x Subscript m</strong> and <strong>−(17) / (4)</strong> for <strong>x 1</strong> in the equation <strong>x Subscript m Baseline = (x 1 + x 2) / (2)</strong> yields <strong>−3 = Start(−StartFraction 17) / (4) + x 2 OverOver 2 EndEndFraction</strong>. Multiplying each side of this equation by <strong>2</strong> yields <strong>−6 = −(17) / (4) + x 2</strong>. Adding <strong>(17) / (4)</strong> to each side of this equation yields <strong>−seven fourths = x 2</strong>. Therefore, the other x-intercept is <strong>(−seven fourths, 0)</strong>.  <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "cd358b89",
    type: "mcq",
    questionHtml:
      "Function <strong>f</strong> is defined by <strong>f (x) = (x + 6) (x + 5) (x + 1)</strong>. Function <strong>g</strong> is defined by <strong>g (x) = f (x − 1)</strong>. The graph of <strong>y = g (x)</strong> in the xy-plane has x-intercepts at <strong>(a, 0)</strong>, <strong>(b, 0)</strong>, and <strong>(c, 0)</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are distinct constants. What is the value of <strong>a + b + c</strong>?",
    choices: [
      { label: "A", text: "<strong>−15</strong>" },
      { label: "B", text: "<strong>−9</strong>" },
      { label: "C", text: "<strong>11</strong>" },
      { label: "D", text: "<strong>15</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that <strong>g (x) = f (x − 1)</strong>. Since <strong>f (x) = (x + 6) (x + 5) (x + 1)</strong>, it follows that <strong>f (x − 1) = (x − 1 + 6) (x − 1 + 5) (x − 1 + 1)</strong>. Combining like terms yields <strong>f (x − 1) = (x + 5) (x + 4) (x)</strong>. Therefore, <strong>g (x) = x (x + 5) (x + 4)</strong>. The x-intercepts of a graph in the xy-plane are the points where <strong>y = 0</strong>. The x-coordinates of the x-intercepts of the graph of <strong>y = g (x)</strong> in the xy-plane can be found by solving the equation <strong>0 = x (x + 5) (x + 4)</strong>. Applying the zero product property to this equation yields three equations: <strong>x = 0</strong>, <strong>x + 5 = 0</strong>, and <strong>x + 4 = 0</strong>. Solving each of these equations for <strong>x</strong> yields <strong>x = 0</strong>, <strong>x = −5</strong>, and <strong>x = −4</strong>, respectively. Therefore, the x-intercepts of the graph of <strong>y = g (x)</strong> are <strong>(0, 0)</strong>, <strong>(−5, 0)</strong>, and <strong>(−4, 0)</strong>. It follows that the values of <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are <strong>0</strong>, <strong>−5</strong>, and <strong>−4</strong>. Thus, the value of <strong>a + b + c</strong> is <strong>0 + (−5) + (−4)</strong>, which is equal to <strong>−9</strong>.<br>Choice A is incorrect. This is the value of <strong>a + b + c</strong> if <strong>g (x) = f (x + 1)</strong>.<br>Choice C is incorrect. This is the value of <strong>a + b + c − 1</strong> if <strong>g (x) = (x − 6) (x − 5) (x − 1)</strong>.<br>Choice D is incorrect. This is the value of <strong>a + b + c</strong> if <strong>f (x) = (x − 6) (x − 5) (x − 1)</strong>.",
    hasFigure: false,
  },
  {
    id: "ce579859",
    type: "mcq",
    questionHtml:
      "A model estimates that at the end of each year from <strong>2015</strong> to <strong>2020</strong>, the number of squirrels in a population was <strong>150 % sign</strong> more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of <strong>2016</strong>, there were <strong>180</strong> squirrels in the population. Which of the following equations represents this model, where <strong>n</strong> is the estimated number of squirrels in the population <strong>t</strong> years after the end of <strong>2015</strong> and <strong>t < or = 5</strong>?",
    choices: [
      { label: "A", text: "<strong>n = 72 (1.5)^(t)</strong>" },
      { label: "B", text: "<strong>n = 72 (2.5)^(t)</strong>" },
      { label: "C", text: "<strong>n = 180 (1.5)^(t)</strong>" },
      { label: "D", text: "<strong>n = 180 (2.5)^(t)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, <strong>150 % sign</strong>, each year, the model can be represented by an exponential equation of the form <strong>n = a (1 + (p) / (100))^(t)</strong>, where <strong>a</strong> is the estimated number of squirrels in the population at the end of <strong>2015</strong>, and the model estimates that at the end of each year, the number is <strong>p % sign</strong> more than the number at the end of the previous year. Since the model estimates that at the end of each year, the number was <strong>150 % sign</strong> more than the number at the end of the previous year, <strong>p = 150</strong>. Substituting <strong>150</strong> for <strong>p</strong> in the equation <strong>n = a (1 + (p) / (100))^(t)</strong> yields <strong>n = a (1 + (150) / (100))^(t)</strong>, which is equivalent to <strong>n = a (1 + 1.5)^(t)</strong>, or <strong>n = a (2.5)^(t)</strong>. It’s given that the estimated number of squirrels at the end of <strong>2016</strong> was <strong>180</strong>. This means that when <strong>t = 1</strong>, <strong>n = 180</strong>. Substituting <strong>1</strong> for <strong>t</strong> and <strong>180</strong> for <strong>n</strong> in the equation <strong>n = a (2.5)^(t)</strong> yields <strong>180 = a (2.5)¹</strong>, or <strong>180 = 2.5 a</strong>. Dividing each side of this equation by <strong>2.5</strong> yields <strong>72 = a</strong>. Substituting <strong>72</strong> for <strong>a</strong> in the equation <strong>n = a (2.5)^(t)</strong> yields <strong>n = 72 (2.5)^(t)</strong>.<br>Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was <strong>150 % sign</strong> of, not <strong>150 % sign</strong> more than, the estimated number at the end of the previous year.<br>Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was <strong>150 % sign</strong> of, not <strong>150 % sign</strong> more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of <strong>2015</strong>, not the end of <strong>2016</strong>, was <strong>180</strong>.<br>Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of <strong>2015</strong>, not the end of <strong>2016</strong>, was <strong>180</strong>.",
    hasFigure: false,
  },
  {
    id: "d135f4bf",
    type: "spr",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = (x − 6) (x − 2) (x + 6)</strong>. In the xy-plane, the graph of <strong>y = g (x)</strong> is the result of translating the graph of <strong>y = f (x)</strong> up <strong>4</strong> units. What is the value of <strong>g (0)</strong>?",
    choices: [],
    correctAnswer: "76",
    explanation:
      "The correct answer is <strong>76</strong>. It's given that the graph of <strong>y = g (x)</strong> is the result of translating the graph of <strong>y = f (x)</strong> up <strong>4</strong> units in the xy-plane. It follows that the graph of <strong>y = g (x)</strong> is the same as the graph of <strong>y = f (x) + 4</strong>. Substituting <strong>g (x)</strong> for <strong>y</strong> in the equation <strong>y = f (x) + 4</strong> yields <strong>g (x) = f (x) + 4</strong>. It's given that <strong>f (x) = (x − 6) (x − 2) (x + 6)</strong>. Substituting <strong>(x − 6) (x − 2) (x + 6)</strong> for <strong>f (x)</strong> in the equation <strong>g (x) = f (x) + 4</strong> yields <strong>g (x) = (x − 6) (x − 2) (x + 6) + 4</strong>. Substituting <strong>0</strong> for <strong>x</strong> in this equation yields <strong>g (0) = (0 − 6) (0 − 2) (0 + 6) + 4</strong>, or <strong>g (0) = 76</strong>. Thus, the value of <strong>g (0)</strong> is <strong>76</strong>.",
    hasFigure: false,
  },
  {
    id: "de39858a",
    type: "mcq",
    questionHtml:
      "The function <em>(expression)</em> is defined by <em>(expression)</em>, where <em>(expression)</em> and <em>(expression)</em> are positive constants. The graph of <em>(expression)</em> in the <em>(expression)</em>-plane passes through the points (<em>(expression)</em>, <em>(expression)</em>) and (<em>(expression)</em>, <em>(expression)</em>). What is the value of <em>(expression)</em>?",
    choices: [
      { label: "A", text: "<em>(expression)</em>" },
      { label: "B", text: "<em>(expression)</em>" },
      { label: "C", text: "<em>(expression)</em>" },
      { label: "D", text: "<em>(expression)</em>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the function <strong>h</strong> is defined by <strong>h (x) = a^(x) + b</strong> and that the graph of <strong>y = h (x)</strong> in the xy-plane passes through the points <strong>(0, 10)</strong> and <strong>(−2, (325) / (36))</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>10</strong> for <strong>h (x)</strong> in the equation <strong>h (x) = a^(x) + b</strong> yields <strong>10 = a⁰ + b</strong>, or <strong>10 = 1 + b</strong>. Subtracting <strong>1</strong> from both sides of this equation yields <strong>9 = b</strong>. Substituting <strong>−2</strong> for <strong>x</strong> and <strong>(325) / (36)</strong> for <strong>h (x)</strong> in the equation <strong>h (x) = a^(x) + 9</strong> yields <strong>(325) / (36) = a^(−2) + 9</strong>. Subtracting <strong>9</strong> from both sides of this equation yields <strong>one thirty sixth = a^(−2)</strong>, which can be rewritten as <strong>a² = 36</strong>. Taking the square root of both sides of this equation yields <strong>a = 6</strong> and <strong>a = −6</strong>, but because it’s given that <strong>a</strong> is a positive constant, <strong>a</strong> must equal <strong>6</strong>. Because the value of <strong>a</strong> is <strong>6</strong> and the value of <strong>b</strong> is <strong>9</strong>, the value of <strong>a b</strong> is <strong>(6) (9)</strong>, or <strong>54</strong>.<br><br>Choice A is incorrect and may result from finding the value of <strong>a^(−2) b</strong> rather than the value of <strong>a b</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from correctly finding the value of <strong>a</strong> as <strong>6</strong>, but multiplying it by the y-value in the first ordered pair rather than by the value of <strong>b</strong>.",
    hasFigure: false,
  },
  {
    id: "ee857afb",
    type: "spr",
    questionHtml:
      "<strong>y = x² − 14 x + 22</strong><br>The given equation relates the variables <strong>x</strong> and <strong>y</strong>. For what value of <strong>x</strong> does the value of <strong>y</strong> reach its minimum?",
    choices: [],
    correctAnswer: "7",
    explanation:
      "The correct answer is <strong>7</strong>. When an equation is of the form <strong>y = a x² + b x + c</strong>, where <strong>a</strong>, <strong>b</strong>, and <strong>c</strong> are constants, the value of <strong>y</strong> reaches its minimum when <strong>x = − (b) / (2 a)</strong>. Since the given equation is of the form <strong>y = a x² + b x + c</strong>, it follows that <strong>a = 1</strong>, <strong>b = −14</strong>, and <strong>c = 22</strong>. Therefore, the value of <strong>y</strong> reaches its minimum when <strong>x = − ((−14)) / (2 (1))</strong>, or <strong>x = 7</strong>.",
    hasFigure: false,
  },
  {
    id: "ef926848",
    type: "mcq",
    questionHtml:
      "Square P has a side length of <strong>x</strong> inches. Square Q has a perimeter that is <strong>176</strong> inches greater than the perimeter of square P. The function <strong>f</strong> gives the area of square Q, in square inches. Which of the following defines <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = (x + 44)²</strong>" },
      { label: "B", text: "<strong>f (x) = (x + 176)²</strong>" },
      { label: "C", text: "<strong>f (x) = (176 x + 44)²</strong>" },
      { label: "D", text: "<strong>f (x) = (176 x + 176)²</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Let <strong>x</strong> represent the side length, in inches, of square P. It follows that the perimeter of square P is <strong>4 x</strong> inches. It's given that square Q has a perimeter that is <strong>176</strong> inches greater than the perimeter of square P. Thus, the perimeter of square Q is <strong>176</strong> inches greater than <strong>4 x</strong> inches, or <strong>4 x + 176</strong> inches. Since the perimeter of a square is <strong>4</strong> times the side length of the square, each side length of Q is <strong>(4 x + 176) / (4)</strong>, or <strong>x + 44</strong> inches. Since the area of a square is calculated by multiplying the length of two sides, the area of square Q is <strong>(x + 44) (x + 44)</strong>, or <strong>(x + 44)²</strong> square inches. It follows that function <strong>f</strong> is defined by <strong>f (x) = (x + 44)²</strong>. <br>Choice B is incorrect. This function represents a square with side lengths <strong>(x + 176)</strong> inches.<br>Choice C is incorrect. This function represents a square with side lengths <strong>(176 x + 44)</strong> inches.<br>Choice D is incorrect. This function represents a square with side lengths <strong>(176 x + 176)</strong> inches.",
    hasFigure: false,
  },
  {
    id: "f2d60b99",
    type: "mcq",
    questionHtml:
      "The function <strong>f (x) = one ninth (x − 7)² + 3</strong> gives a metal ball's height above the ground <strong>f (x)</strong>, in inches, <strong>x</strong> seconds after it started moving on a track, where <strong>0 < or = x < or = 10</strong>. Which of the following is the best interpretation of the vertex of the graph of <strong>y = f (x)</strong> in the xy-plane?",
    choices: [
      {
        label: "A",
        text: "The metal ball's minimum height was <strong>3</strong> inches above the ground.",
      },
      {
        label: "B",
        text: "The metal ball's minimum height was <strong>7</strong> inches above the ground.",
      },
      {
        label: "C",
        text: "The metal ball's height was <strong>3</strong> inches above the ground when it started moving.",
      },
      {
        label: "D",
        text: "The metal ball's height was <strong>7</strong> inches above the ground when it started moving.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The graph of a quadratic equation in the form <strong>y = a (x − h)² + k</strong>, where <strong>a</strong>, <strong>h</strong>, and <strong>k</strong> are positive constants, is a parabola that opens upward with vertex <strong>(h, k)</strong>. The given function <strong>f (x) = one ninth (x − 7)² + 3</strong> is in the form <strong>y = a (x − h)² + k</strong>, where <strong>y = f (x)</strong>, <strong>a = one ninth</strong>, <strong>h = 7</strong>, and <strong>k = 3</strong>. Therefore, the graph of <strong>y = f (x)</strong> is a parabola that opens upward with vertex <strong>(7, 3)</strong>. Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the y-coordinate of the vertex of the graph of <strong>y = f (x)</strong> is the minimum value of <strong>f (x)</strong>. Therefore, the minimum value of <strong>f (x)</strong> is <strong>3</strong>. It’s given that <strong>f (x) = one ninth (x − 7)² + 3</strong> represents the metal ball’s height above the ground, in inches, <strong>x</strong> seconds after it started moving on a track. Therefore, the best interpretation of the vertex of the graph of <strong>y = f (x)</strong> is that the metal ball’s minimum height was <strong>3</strong> inches above the ground.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];
