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

export const LINEAR_FUNC_EASY: PracticeQuestion[] = [
  {
    id: "06fc1726",
    type: "mcq",
    questionHtml:
      "If f is the function defined by <strong>f of x = the fraction with numerator 2 x − 1, and denominator 3</strong>, what is the value of <strong>f of 5</strong> ?",
    choices: [
      { label: "A", text: "<strong>4 thirds</strong>" },
      { label: "B", text: "<strong>7 thirds</strong>" },
      { label: "C", text: "3" },
      { label: "D", text: "9" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. If <strong>f of x = the fraction with numerator 2 x − 1, and denominator 3</strong>, then <strong>f of 5 = the fraction with numerator 2 · 5 − 1, and denominator 3, which = the fraction with numerator 10 − 1, and denominator 3, which = the fraction 9 over 3, which = 3</strong>.Choice A is incorrect and may result from not multiplying x by 2 in the numerator. Choice B is incorrect and may result from dividing 2x by 3 and then subtracting 1. Choice D is incorrect and may result from evaluating only the numerator 2x – 1.",
    hasFigure: false,
  },
  {
    id: "0b332f00",
    type: "spr",
    questionHtml:
      "The function <strong>g</strong> is defined by <strong>g (x) = 6 x</strong>. For what value of <strong>x</strong> is <strong>g (x) = 54</strong>?",
    choices: [],
    correctAnswer: "9",
    explanation:
      "The correct answer is <strong>9</strong>. It’s given that <strong>g (x) = 6 x</strong>. Substituting <strong>54</strong> for <strong>g (x)</strong> in the given function yields <strong>54 = 6 x</strong>. Dividing both sides of this equation by <strong>6</strong> yields <strong>x = 9</strong>. Therefore, the value of <strong>x</strong> when <strong>g (x) = 54</strong> is <strong>9</strong>.",
    hasFigure: false,
  },
  {
    id: "0d391910",
    type: "spr",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 4 x</strong>. For what value of <strong>x</strong> does <strong>f (x) = 8</strong>?",
    choices: [],
    correctAnswer: "2",
    explanation:
      "The correct answer is <strong>2</strong>. Substituting <strong>8</strong> for <strong>f (x)</strong> in the given equation yields <strong>8 = 4 x</strong>. Dividing the left- and right-hand sides of this equation by <strong>4</strong> yields <strong>x = 2</strong>. Therefore, the value of <strong>x</strong> is <strong>2</strong> when <strong>f (x) = 8</strong>.",
    hasFigure: false,
  },
  {
    id: "0ea7ef01",
    type: "mcq",
    questionHtml:
      "Oxygen gas is placed inside a tank with a constant volume. The graph shows the estimated temperature <strong>y</strong>, in kelvins, of the oxygen gas when its pressure is <strong>x</strong> atmospheres.<br>The line slants sharply up from left to right.<br>The line begins at the approximate point (1.6 comma 190).<br>The line passes through the following approximate points:<br><br>(3 comma 350)<br>(6 comma 700)<br>(8 comma 933)<br><br>What is the estimated temperature, in kelvins, of the oxygen gas when its pressure is <strong>6</strong> atmospheres?",
    choices: [
      { label: "A", text: "<strong>6</strong>" },
      { label: "B", text: "<strong>60</strong>" },
      { label: "C", text: "<strong>700</strong>" },
      { label: "D", text: "<strong>760</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. For the graph shown, the x-axis represents pressure, in atmospheres, and the y-axis represents temperature, in kelvins. Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is <strong>6</strong> atmospheres is represented by the y-coordinate of the point on the graph that has an x-coordinate of <strong>6</strong>. The point on the graph with an x-coordinate of <strong>6</strong> has a y-coordinate of approximately <strong>700</strong>. Therefore, the estimated temperature, in kelvins, of the oxygen gas when its pressure is <strong>6</strong> atmospheres is <strong>700</strong>.<br>Choice A is incorrect. This is the pressure, in atmospheres, not the estimated temperature, in kelvins, of the oxygen gas when its pressure is <strong>6</strong> atmospheres.<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/0ea7ef01_svg1.svg",
  },
  {
    id: "0eae6be1",
    type: "mcq",
    questionHtml:
      "The number <strong>y</strong> is <strong>84</strong> less than the number <strong>x</strong>. Which equation represents the relationship between <strong>x</strong> and <strong>y</strong>?",
    choices: [
      { label: "A", text: "<strong>y = x + 84</strong>" },
      { label: "B", text: "<strong>y = one eighty fourth x</strong>" },
      { label: "C", text: "<strong>y = 84 x</strong>" },
      { label: "D", text: "<strong>y = x − 84</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the number <strong>y</strong> is <strong>84</strong> less than the number <strong>x</strong>. A number that's <strong>84</strong> less than the number <strong>x</strong> is equivalent to <strong>84</strong> subtracted from the number <strong>x</strong>, or <strong>x − 84</strong>. Therefore, the equation <strong>y = x − 84</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 and may result from conceptual errors.<br>Choice C is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "12983c1e",
    type: "mcq",
    questionHtml:
      "Some values of the linear function f are shown in the table above. Which of the following defines f ?",
    choices: [
      { label: "A", text: "<strong>f of x = 2 x + 3</strong>" },
      { label: "B", text: "<strong>f of x = 3 x + 2</strong>" },
      { label: "C", text: "<strong>f of x = 4 x + 1</strong>" },
      { label: "D", text: "<strong>f of x = 5 x</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Because f is a linear function of x, the equation <strong>f of x = m x + b</strong>, where m and b are constants, can be used to define the relationship between x and f (x). In this equation, m represents the increase in the value of f (x) for every increase in the value of x by 1. From the table, it can be determined that the value of f (x) increases by 8 for every increase in the value of x by 2. In other words, for the function f the value of m is <strong>eight halves</strong>, or 4. The value of b can be found by substituting the values of x and f (x) from any row of the table and the value of m into the equation <strong>f of x = m x + b</strong> and solving for b. For example, using <strong>x = 1</strong>, <strong>f of x = 5</strong>, and <strong>m = 4</strong> yields <strong>5 = 4 · 1 + b</strong>. Solving for b yields <strong>b = 1</strong>. Therefore, the equation defining the function f can be written in the form <strong>f of x = 4 x + 1</strong>.Choices A, B, and D are incorrect. Any equation defining the linear function f must give values of f (x) for corresponding values of x, as shown in each row of the table. According to the table, if <strong>x = 3</strong>, <strong>f of x = 13</strong>. However, substituting <strong>x = 3</strong> into the equation given in choice A gives <strong>f of 3 = 2 · 3 + 3</strong>, or <strong>f of 3 = 9</strong>, not 13. Similarly, substituting <strong>x = 3</strong> into the equation given in choice B gives <strong>f of 3 = 3 · 3 + 2</strong>, or <strong>f of 3 = 11</strong>, not 13.<br>Lastly, substituting <strong>x = 3</strong> into the equation given in choice D gives <strong>f of 3 = 5 · 3</strong>, or <strong>f of 3 = 15</strong>, not 13. Therefore, the equations in choices A, B, and D cannot define f.",
    hasFigure: true,
    figureUrl: "/practice-images/12983c1e_img1.png",
  },
  {
    id: "13294295",
    type: "mcq",
    questionHtml:
      "The graph shown models the number of candy bars a certain machine wraps with a label in <strong>x</strong> seconds.<br>The line slants sharply up from left to right.<br>The line passes through the following points:<br><br>(0 comma 0)<br>(2 comma 80)<br>(4 comma 160)<br>(6 comma 240)<br><br>According to the graph, what is the estimated number of candy bars the machine wraps with a label per second?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>40</strong>" },
      { label: "C", text: "<strong>78</strong>" },
      { label: "D", text: "<strong>80</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. For the graph shown, the x-axis represents time, in seconds, and the y-axis represents the number of candy bars wrapped. The slope of a line in the xy-plane is the change in <strong>y</strong> for each <strong>1</strong>-unit increase in <strong>x</strong>. It follows that the slope of the graph shown represents the estimated number of candy bars the machine wraps with a label per second. The slope, <strong>m</strong>, of a line in the xy-plane can be found using any two points, <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, on the line and the slope formula <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. The graph shown passes through the points <strong>(0, 0)</strong> and <strong>(2, 80)</strong>. Substituting these points for <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong>, respectively, in the slope formula yields <strong>m = (80 − 0) / (2 − 0)</strong>, which is equivalent to <strong>m = (80) / (2)</strong>, or <strong>m = 40</strong>. Therefore, the estimated number of candy bars the machine wraps with a label per second is <strong>40</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/13294295_svg1.svg",
  },
  {
    id: "13909d78",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by the equation <strong>f (x) = 100 x + 2</strong>. What is the value of <strong>f (x)</strong> when <strong>x = 9</strong>?",
    choices: [
      { label: "A", text: "<strong>111</strong>" },
      { label: "B", text: "<strong>118</strong>" },
      { label: "C", text: "<strong>900</strong>" },
      { label: "D", text: "<strong>902</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Substituting <strong>9</strong> for <strong>x</strong> in the given equation yields <strong>f (9) = 100 (9) + 2</strong>, or <strong>f (9) = 902</strong>. Therefore, the value of <strong>f (x)</strong> when <strong>x = 9</strong> is <strong>902</strong>.<br>Choice A is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 1.09</strong>.<br>Choice B is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 1.16</strong>.<br>Choice C is incorrect. This is the value of <strong>f (x)</strong> when <strong>x = 8.98</strong>.",
    hasFigure: false,
  },
  {
    id: "1ecaa9c0",
    type: "mcq",
    questionHtml:
      "Robert rented a truck to transport materials he purchased from a hardware store. He was charged an initial fee of $20.00 plus an additional $0.70 per mile driven. If the truck was driven 38 miles, what was the total amount Robert was charged?",
    choices: [
      { label: "A", text: "$46.60" },
      { label: "B", text: "$52.90" },
      { label: "C", text: "$66.90" },
      { label: "D", text: "$86.50" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that Robert was charged an initial fee of $20.00 to rent the truck plus an additional $0.70 per mile driven. Let m represent the number of miles the truck was driven. Since the rental charge is $0.70 per mile driven, <strong>0 . 7 0 m</strong> represents the amount Robert was charged for m miles driven. Let c equal the total amount, in dollars, Robert was charged to rent the truck. The total amount can be represented by the equation <strong>c = 20 . 0 0 + 0 . 7 0 m</strong>. It’s given that the truck was driven 38 miles, thus <strong>m = 38</strong>. Substituting 38 into the equation gives <strong>c = 20 . 0 0 + 0 . 7 0 · 38</strong>. Multiplying <strong>0 . 7 0 · 38</strong> gives <strong>c = 20 . 0 0 + 26 . 6 0</strong>. Adding these values gives <strong>c = 46 . 6 0</strong>, so the total amount Robert was charged is $46.60.Choices B, C, and D are incorrect and may result from setting up the equation incorrectly or from making calculation errors.",
    hasFigure: false,
  },
  {
    id: "255996a6",
    type: "mcq",
    questionHtml:
      "In the equation above, T represents Brittany’s total take-home pay, in dollars, for her first week of work, where h represents the number of hours she worked that week and 1,000 represents a sign-on bonus. If Brittany’s total take-home pay was $1,576, for how many hours was Brittany paid for her first week of work?",
    choices: [
      { label: "A", text: "16" },
      { label: "B", text: "32" },
      { label: "C", text: "55" },
      { label: "D", text: "88" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since Brittany’s total take-home pay was $1,576, the value 1,576 can be substituted for T in the given equation <strong>T = 1, 000 + 18 h</strong> to give <strong>1, 576 = 1, 000 + 18 h</strong>. Subtracting 1,000 from both sides of this equation gives <strong>576 = 18 h</strong>. Dividing both sides of this equation by 18 gives <strong>32 = h</strong>. Therefore, Brittany was paid for 32 hours for her first week of work.Choice A is incorrect. This is half the number of hours Brittany was paid for. Choice C is incorrect and may result from dividing 1,000 by 18. Choice D is incorrect and may result from dividing 1,576 by 18.",
    hasFigure: false,
  },
  {
    id: "2eef7e61",
    type: "mcq",
    questionHtml:
      "The graph of the function f is a line in the xy-plane. If the line has slope <strong>three-fourths</strong> and <strong>f of 0 = 3</strong>, which of the following defines f?",
    choices: [
      { label: "A", text: "<strong>f of x = three fourths x − 3</strong>" },
      { label: "B", text: "<strong>f of x = three fourths x + 3</strong>" },
      { label: "C", text: "<strong>f of x = 4 x − 3</strong>" },
      { label: "D", text: "<strong>f of x = 4 x + 3</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The equation for the function f in the xy-plane can be represented by <strong>f of x = m x + b</strong>, where m is the slope and b is the y-coordinate of the y-intercept. Since it’s given that the line has a slope of <strong>three fourths</strong>, it follows that <strong>m = three fourths</strong> in <strong>f of x = m x + b</strong>, which yields <strong>y = three fourths x + b</strong>. It’s given that <strong>f of 0 = 3</strong>. This implies that the graph of the function f in the xy-plane passes through the point <strong>with coordinates 0, 3</strong>. Thus, the y-coordinate of the y-intercept of the graph is 3, so <strong>b = 3</strong> in <strong>f of x = three fourths x + b</strong>, which yields <strong>f of x = three fourths x + 3</strong>. Therefore, the equation <strong>f of x = three fourths x + 3</strong> defines the function f. Choice A is incorrect and may result from a sign error for the y-intercept. Choice C is incorrect and may result from using the denominator of the given slope as m in <strong>f of x = m x + b</strong>, in addition to a sign error for the y-intercept. Choice D is incorrect and may result from using the denominator of the given slope as m in <strong>f of x = m x + b</strong>.",
    hasFigure: false,
  },
  {
    id: "3462d850",
    type: "mcq",
    questionHtml: "Which of the following does 45 represent in the equation?",
    choices: [
      { label: "A", text: "Marisol took 45 trips from City A to City B." },
      {
        label: "B",
        text: "The distance between City A and City B is 45 miles.",
      },
      {
        label: "C",
        text: "Marisol drove at an average speed of about 45 miles per hour.",
      },
      {
        label: "D",
        text: "It took Marisol 45 hours to drive from City A to City B.",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that d is the distance, in miles, Marisol traveled after driving for t hours. Therefore, 45 represents the distance in miles traveled per hour, which is the speed she drove in miles per hour.Choice A is incorrect and may result from misidentifying speed as the number of trips. Choice B is incorrect and may result from misidentifying speed as the total distance. Choice D is incorrect and may result from misidentifying the speed as the time, in hours.",
    hasFigure: false,
  },
  {
    id: "3f5375d9",
    type: "mcq",
    questionHtml:
      "According to the graph, what is the cost for each additional mile traveled, in dollars, of a cab ride?",
    choices: [
      { label: "A", text: "$2.00" },
      { label: "B", text: "$2.60" },
      { label: "C", text: "$3.00" },
      { label: "D", text: "$5.00" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The cost of each additional mile traveled is represented by the slope of the given line. The slope of the line can be calculated by identifying two points on the line and then calculating the ratio of the change in y to the change in x between the two points. Using the points <strong>with coordinates 1, 5</strong> and <strong>2, 7</strong>, the slope is equal to <strong>the fraction with numerator 7 − 5, and denominator 2 − 1, end fraction</strong>, or 2. Therefore, the cost for each additional mile traveled of the cab ride is $2.00.Choice B is incorrect and may result from calculating the slope of the line that passes through the points <strong>with coordinates 5, 13</strong> and <strong>0, 0</strong>. However, <strong>0, 0</strong> does not lie on the line shown. Choice C is incorrect. This is the y-coordinate of the y-intercept of the graph and represents the flat fee for a cab ride before the charge for any miles traveled is added. Choice D is incorrect. This value represents the total cost of a 1-mile cab ride.",
    hasFigure: true,
    figureUrl: "/practice-images/3f5375d9_img1.png",
  },
  {
    id: "447fa970",
    type: "spr",
    questionHtml:
      "The function <strong>f</strong> is defined by the equation <strong>f (x) = 7 x + 2</strong>. What is the value of <strong>f (x)</strong> when <strong>x = 4</strong>?",
    choices: [],
    correctAnswer: "30",
    explanation:
      "The correct answer is <strong>30</strong>. The value of <strong>f (x)</strong> when <strong>x = 4</strong> can be found by substituting <strong>4</strong> for <strong>x</strong> in the given equation <strong>f (x) = 7 x + 2</strong>. This yields <strong>f (4) = 7 (4) + 2</strong>, or <strong>f (4) = 30</strong>. Therefore, when <strong>x = 4</strong>, the value of <strong>f (x)</strong> is <strong>30</strong>.",
    hasFigure: false,
  },
  {
    id: "5ad6bc97",
    type: "spr",
    questionHtml:
      "<strong>f (x) = 7 x + 1</strong><br>The function gives the total number of people on a company retreat with <strong>x</strong> managers. What is the total number of people on a company retreat with <strong>7</strong> managers?",
    choices: [],
    correctAnswer: "50",
    explanation:
      "The correct answer is <strong>50</strong>. It's given that the function <strong>f</strong> gives the total number of people on a company retreat with <strong>x</strong> managers. It's also given that <strong>7</strong> managers are on the company retreat. Substituting <strong>7</strong> for <strong>x</strong> in the given function yields <strong>f (7) = 7 (7) + 1</strong>, or <strong>f (7) = 50</strong>. Therefore, there are a total of <strong>50</strong> people on a company retreat with <strong>7</strong> managers.",
    hasFigure: false,
  },
  {
    id: "6863c7ce",
    type: "mcq",
    questionHtml:
      "<strong>d = 16 t</strong><br>The given equation represents the distance <strong>d</strong>, in inches, where <strong>t</strong> represents the number of seconds since an object started moving. Which of the following is the best interpretation of <strong>16</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "The object moved a total of <strong>16</strong> inches.",
      },
      {
        label: "B",
        text: "The object moved a total of <strong>16 t</strong> inches.",
      },
      {
        label: "C",
        text: "The object is moving at a rate of <strong>16</strong> inches per second.",
      },
      {
        label: "D",
        text: "The object is moving at a rate of <strong>one sixteenth</strong> inches per second.",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that in the equation <strong>d = 16 t</strong>, <strong>d</strong> represents the distance, in inches, and <strong>t</strong> represents the number of seconds since an object started moving. In this equation, <strong>t</strong> is being multiplied by <strong>16</strong>. This means that the object’s distance increases by <strong>16</strong> inches each second. Therefore, the best interpretation of <strong>16</strong> in this context is that the object is moving at a rate of <strong>16</strong> inches per second.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect. This is the best interpretation of <strong>16 t</strong>, rather than <strong>16</strong>, in this context.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "6efcc0a3",
    type: "mcq",
    questionHtml:
      "In the linear function <strong>h</strong>, <strong>h (0) = 41</strong> and <strong>h (1) = 40</strong>. Which equation defines <strong>h</strong>?",
    choices: [
      { label: "A", text: "<strong>h (x) = −x + 41</strong>" },
      { label: "B", text: "<strong>h (x) = −x</strong>" },
      { label: "C", text: "<strong>h (x) = − 41 x</strong>" },
      { label: "D", text: "<strong>h (x) = −41</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. An equation defining a linear function can be written in the form <strong>h (x) = a x + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. It’s given that <strong>h (0) = 41</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>41</strong> for <strong>h (x)</strong> in the equation <strong>h (x) = a x + b</strong> yields <strong>41 = a (0) + b</strong>, or <strong>b = 41</strong>. Substituting <strong>41</strong> for <strong>b</strong> in the equation <strong>h (x) = a x + b</strong> yields <strong>h (x) = a x + 41</strong>. It’s also given that <strong>h (1) = 40</strong>. Substituting <strong>1</strong> for <strong>x</strong> and <strong>40</strong> for <strong>h (x)</strong> in the equation <strong>h (x) = a x + 41</strong> yields <strong>40 = a (1) + 41</strong>, or <strong>40 = a + 41</strong>. Subtracting <strong>41</strong> from the left- and right-hand sides of this equation yields <strong>−1 = a</strong>. Substituting <strong>−1</strong> for <strong>a</strong> in the equation <strong>h (x) = a x + 41</strong> yields <strong>h (x) = − 1 x + 41</strong>, or <strong>h (x) = −x + 41</strong>.<br>Choice B is incorrect. Substituting <strong>0</strong> for <strong>x</strong> and <strong>41</strong> for <strong>h (x)</strong> in this equation yields <strong>41 = −0</strong>, which isn't a true statement.<br>Choice C is incorrect. Substituting <strong>0</strong> for <strong>x</strong> and <strong>41</strong> for <strong>h (x)</strong> in this equation yields <strong>41 = − 41 (0)</strong>, or <strong>41 = 0</strong>, which isn't a true statement.<br>Choice D is incorrect. Substituting <strong>41</strong> for <strong>h (x)</strong> in this equation yields <strong>41 = −41</strong>, which isn't a true statement.",
    hasFigure: false,
  },
  {
    id: "70d9516e",
    type: "mcq",
    questionHtml:
      "A bus is traveling at a constant speed along a straight portion of road. The equation <strong>d = 30 t</strong> gives the distance <strong>d</strong>, in feet from a road marker, that the bus will be <strong>t</strong> seconds after passing the marker. How many feet from the marker will the bus be <strong>2</strong> seconds after passing the marker?",
    choices: [
      { label: "A", text: "<strong>30</strong>" },
      { label: "B", text: "<strong>32</strong>" },
      { label: "C", text: "<strong>60</strong>" },
      { label: "D", text: "<strong>90</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that <strong>t</strong> represents the number of seconds after the bus passes the marker. Substituting <strong>2</strong> for <strong>t</strong> in the given equation <strong>d = 30 t</strong> yields <strong>d = 30 (2)</strong>, or <strong>d = 60</strong>. Therefore, the bus will be <strong>60</strong> feet from the marker <strong>2</strong> seconds after passing it.<br>Choice A is incorrect. This is the distance, in feet, the bus will be from the marker <strong>1</strong> second, not <strong>2</strong> seconds, after passing it.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the distance, in feet, the bus will be from the marker <strong>3</strong> seconds, not <strong>2</strong> seconds, after passing it.",
    hasFigure: false,
  },
  {
    id: "720e51ac",
    type: "mcq",
    questionHtml:
      "The cost <strong>y</strong>, in dollars, for a manufacturer to make <strong>x</strong> rings is represented by the line shown.<br>The line slants gradually up from left to right.<br>The line passes through the following points:<br><br>(0 comma 100)<br>(60 comma 175)<br>(100 comma 225)<br><br>What is the cost, in dollars, for the manufacturer to make <strong>60</strong> rings?",
    choices: [
      { label: "A", text: "<strong>100</strong>" },
      { label: "B", text: "<strong>125</strong>" },
      { label: "C", text: "<strong>175</strong>" },
      { label: "D", text: "<strong>225</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The line shown represents the cost <strong>y</strong>, in dollars, for a manufacturer to make <strong>x</strong> rings. For the line shown, the x-axis represents the number of rings made by the manufacturer and the y-axis represents the cost, in dollars. Therefore, the cost, in dollars, for the manufacturer to make <strong>60</strong> rings is represented by the y-coordinate of the point on the line that has an x-coordinate of <strong>60</strong>. The point on the line with an x-coordinate of <strong>60</strong> has a y-coordinate of <strong>175</strong>. Therefore, the cost, in dollars, for the manufacturer to make <strong>60</strong> rings is <strong>175</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the cost, in dollars, for the manufacturer to make <strong>20</strong> rings.<br>Choice D is incorrect. This is the cost, in dollars, for the manufacturer to make <strong>100</strong> rings.",
    hasFigure: true,
    figureUrl: "/practice-images/720e51ac_svg1.svg",
  },
  {
    id: "73b5f330",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 5 x + 8</strong>. For what value of <strong>x</strong> does <strong>f (x) = 58</strong>?",
    choices: [
      { label: "A", text: "<strong>10</strong>" },
      { label: "B", text: "<strong>13</strong>" },
      { label: "C", text: "<strong>50</strong>" },
      { label: "D", text: "<strong>298</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the function <strong>f</strong> is defined by <strong>f (x) = 5 x + 8</strong>. Substituting <strong>58</strong> for <strong>f (x)</strong> in this equation yields <strong>58 = 5 x + 8</strong>. Subtracting <strong>8</strong> from both sides of this equation yields <strong>50 = 5 x</strong>. Dividing both sides of this equation by <strong>5</strong> yields <strong>10 = x</strong>. Therefore, the value of <strong>x</strong> when <strong>f (x) = 58</strong> is <strong>10</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 is the value of <strong>f (58)</strong>, not the value of <strong>x</strong> when <strong>f (x) = 58</strong>.",
    hasFigure: false,
  },
  {
    id: "776cfa7c",
    type: "mcq",
    questionHtml:
      "Hana deposited a fixed amount into her bank account each month. The function <strong>f (t) = 100 + 25 t</strong> gives the amount, in dollars, in Hana's bank account after <strong>t</strong> monthly deposits. What is the best interpretation of <strong>25</strong> in this context?",
    choices: [
      {
        label: "A",
        text: "With each monthly deposit, the amount in Hana's bank account increased by <strong>dollar sign 25</strong>.",
      },
      {
        label: "B",
        text: "Before Hana made any monthly deposits, the amount in her bank account was <strong>dollar sign 25</strong>.",
      },
      {
        label: "C",
        text: "After <strong>1</strong> monthly deposit, the amount in Hana's bank account was <strong>dollar sign 25</strong>.",
      },
      {
        label: "D",
        text: "Hana made a total of <strong>25</strong> monthly deposits.",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>t</strong> represents the number of monthly deposits. In the given function <strong>f (t) = 100 + 25 t</strong>, the coefficient of <strong>t</strong> is <strong>25</strong>. This means that for every increase in the value of <strong>t</strong> by <strong>1</strong>, the value of <strong>f (t)</strong> increases by <strong>25</strong>. It follows that with each monthly deposit, the amount in Hana's bank account increased by <strong>dollar sign 25</strong>.<br>Choice B is incorrect. Before Hana made any monthly deposits, the amount in her bank account was <strong>dollar sign 100</strong>. <br>Choice C is incorrect. After <strong>1</strong> monthly deposit, the amount in Hana's bank account was <strong>dollar sign 125</strong>. <br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: false,
  },
  {
    id: "81390d6c",
    type: "mcq",
    questionHtml:
      "The function <strong>h</strong> is defined by <strong>h (x) = x + 200</strong>. What is the value of <strong>h (50)</strong>?",
    choices: [
      { label: "A", text: "<strong>200</strong>" },
      { label: "B", text: "<strong>250</strong>" },
      { label: "C", text: "<strong>10, 000</strong>" },
      { label: "D", text: "<strong>50, 200</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Substituting <strong>50</strong> for <strong>x</strong> in the given function yields <strong>h (50) = 50 + 200</strong>, or <strong>h (50) = 250</strong>. Therefore, the value of <strong>h (50)</strong> is <strong>250</strong>.<br>Choice A is incorrect. This is the value of <strong>h (0)</strong>.<br>Choice C is incorrect. This is the value of <strong>h (9, 800)</strong>.<br>Choice D is incorrect. This is the value of <strong>h (50, 000)</strong>.",
    hasFigure: false,
  },
  {
    id: "84664a7c",
    type: "mcq",
    questionHtml:
      "The front of a roller-coaster car is at the bottom of a hill and is 15 feet above the ground. If the front of the roller-coaster car rises at a constant rate of 8 feet per second, which of the following equations gives the height h, in feet, of the front of the roller-coaster car s seconds after it starts up the hill?",
    choices: [
      { label: "A", text: "<strong>h = 8 s + 15</strong>" },
      {
        label: "B",
        text: "<strong>h = 15 s + the fraction 335 over 8</strong>",
      },
      {
        label: "C",
        text: "<strong>h = 8 s + the fraction 335 over 15</strong>",
      },
      { label: "D", text: "<strong>h = 15 s + 8</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the front of the roller-coaster car starts rising when it’s 15 feet above the ground. This initial height of 15 feet can be represented by a constant term, 15, in an equation. Each second, the front of the roller-coaster car rises 8 feet, which can be represented by 8s. Thus, the equation <strong>h = 8 s + 15</strong> gives the height, in feet, of the front of the roller-coaster car s seconds after it starts up the hill.Choices B and C are incorrect and may result from conceptual errors in creating a linear equation. Choice D is incorrect and may result from switching the rate at which the roller-coaster car rises with its initial height.",
    hasFigure: false,
  },
  {
    id: "8643d906",
    type: "mcq",
    questionHtml:
      "The population of snow leopards in a certain area can be modeled by the function P defined above, where <strong>P of t</strong> is the population t years after 1990. Of the following, which is the best interpretation of the equation <strong>P of 30 = 550</strong> ?",
    choices: [
      {
        label: "A",
        text: "The snow leopard population in this area is predicted to be 30 in the year 2020.",
      },
      {
        label: "B",
        text: "The snow leopard population in this area is predicted to be 30 in the year 2030.",
      },
      {
        label: "C",
        text: "The snow leopard population in this area is predicted to be 550 in the year 2020.",
      },
      {
        label: "D",
        text: "The snow leopard population in this area is predicted to be 550 in the year 2030.",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that <strong>P of t</strong> represents the population of snow leopards t years after 1990. <strong>P of 30 = 550</strong> corresponds to <strong>t = 30</strong> and <strong>P of t = 550</strong>. It follows that <strong>t = 30</strong> corresponds to 30 years after 1990, or 2020. Thus, the best interpretation of <strong>P of 30 = 550</strong> is that the snow leopard population in this area is predicted to be 550 in the year 2020.Choices A and B are incorrect and may result from reversing the interpretations of t and <strong>P of t</strong>. Choice D is incorrect and may result from determining that 30 years after 1990 is 2030, not 2020.",
    hasFigure: false,
  },
  {
    id: "88e13c8c",
    type: "mcq",
    questionHtml:
      "The total cost <strong>f (x)</strong>, in dollars, to lease a car for <strong>36</strong> months from a particular car dealership is given by <strong>f (x) = 36 x + 1, 000</strong>, where <strong>x</strong> is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is <strong>dollar sign 400</strong>?",
    choices: [
      { label: "A", text: "<strong>dollar sign 13, 400</strong>" },
      { label: "B", text: "<strong>dollar sign 13, 000</strong>" },
      { label: "C", text: "<strong>dollar sign 15, 400</strong>" },
      { label: "D", text: "<strong>dollar sign 37, 400</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>f (x)</strong> is the total cost, in dollars, to lease a car from this dealership with a monthly payment of <strong>x</strong> dollars. Therefore, the total cost, in dollars, to lease the car when the monthly payment is <strong>dollar sign 400</strong> is represented by the value of <strong>f (x)</strong> when <strong>x = 400</strong>. Substituting <strong>400</strong> for <strong>x</strong> in the equation <strong>f (x) = 36 x + 1, 000</strong> yields <strong>f (400) = 36 (400) + 1, 000</strong>, or <strong>f (400) = 15, 400</strong>. Thus, when the monthly payment is <strong>dollar sign 400</strong>, the total cost to lease a car is <strong>dollar sign 15, 400</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: "a1696f3e",
    type: "mcq",
    questionHtml:
      "The function [] is defined as <strong>g of x = 5 x + a</strong>, where a is a constant. If <strong>g of 4 = 31</strong>, what is the value of a ?",
    choices: [
      { label: "A", text: "[]" },
      { label: "B", text: "[]" },
      { label: "C", text: "[]" },
      { label: "D", text: "<strong>−23</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Substituting 4 for x in g(x) = 5x + a gives g(4) = 5(4) + a. Since g(4) = 31, the equation g(4) = 5(4) + a simplifies to 31 = 20 + a. It follows that a = 11.Choices A, B, and D are incorrect and may result from arithmetic errors.",
    hasFigure: false,
  },
  {
    id: "a396ed75",
    type: "mcq",
    questionHtml:
      "For a training program, Juan rides his bike at an average rate of <strong>5.7</strong> minutes per mile. Which function <strong>m</strong> models the number of minutes it will take Juan to ride <strong>x</strong> miles at this rate?",
    choices: [
      { label: "A", text: "<strong>m (x) = (x) / (5.7)</strong>" },
      { label: "B", text: "<strong>m (x) = x + 5.7</strong>" },
      { label: "C", text: "<strong>m (x) = x − 5.7</strong>" },
      { label: "D", text: "<strong>m (x) = 5.7 x</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It′s given that Juan rides his bike at an average rate of <strong>5.7</strong> minutes per mile. The number of minutes it will take Juan to ride <strong>x</strong> miles can be determined by multiplying his average rate by the number of miles, <strong>x</strong>, which yields <strong>5.7 x</strong>. Therefore, the function <strong>m (x) = 5.7 x</strong> models the number of minutes it will take Juan to ride <strong>x</strong> miles.<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: "a4d6fbec",
    type: "spr",
    questionHtml:
      "If <strong>y = 5 x + 10</strong>, what is the value of <strong>y</strong> when <strong>x = 8</strong>?",
    choices: [],
    correctAnswer: "50",
    explanation:
      "The correct answer is <strong>50</strong>. Substituting <strong>8</strong> for <strong>x</strong> in the given equation yields <strong>y = 5 (8) + 10</strong>, or <strong>y = 50</strong>. Therefore, the value of <strong>y</strong> is <strong>50</strong> when <strong>x = 8</strong>.",
    hasFigure: false,
  },
  {
    id: "a73a5c22",
    type: "mcq",
    questionHtml:
      "The function <strong>g</strong> is defined by <strong>g (x) = 10 x + 8</strong>. What is the value of <strong>g (x)</strong> when <strong>x = 8</strong>?",
    choices: [
      { label: "A", text: "<strong>0</strong>" },
      { label: "B", text: "<strong>8</strong>" },
      { label: "C", text: "<strong>10</strong>" },
      { label: "D", text: "<strong>88</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The value of <strong>g (x)</strong> when <strong>x = 8</strong> can be found by substituting <strong>8</strong> for <strong>x</strong> in the given equation <strong>g (x) = 10 x + 8</strong>. This yields <strong>g (8) = 10 (8) + 8</strong>, or <strong>g (8) = 88</strong>. Therefore, when <strong>x = 8</strong>, the value of <strong>g (x)</strong> is <strong>88</strong>.<br>Choice A is incorrect. This is the value of <strong>x</strong> when <strong>g (x) = 8</strong>, rather than the value of <strong>g (x)</strong> when <strong>x = 8</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.",
    hasFigure: false,
  },
  {
    id: "a8e6bd75",
    type: "mcq",
    questionHtml:
      "Which of the following is the graph of the equation <strong>y = 2 x − 5</strong> in the xy-plane?",
    choices: [
      {
        label: "A",
        text: "<strong>The figure presents the graph of a line in the xy-plane with the origin labeled O. The numbers − five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the x-axis at − 5 and the y-axis at 2 . 5, ending in quadrant 1</strong>",
      },
      {
        label: "B",
        text: "<strong>The figure presents the graph of a line in the xy-plane with the origin labeled O. The numbers − five through five are indicated on each axis. The line begins in quadrant 2 and moves downward and to the right. It crosses the x-axis at − 2 . 5 and the y-axis at − 5, ending in quadrant 4</strong>",
      },
      {
        label: "C",
        text: "<strong>The figure presents the graph of a line in the xy-plane with the origin labeled O. The numbers − five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the x-axis at − 2 . 5 and the y-axis at 5, ending in quadrant 1</strong>",
      },
      {
        label: "D",
        text: "<strong>The figure presents the graph of a line in the xy-plane with the origin labeled O. The numbers − five through five are indicated on each axis. The line begins in quadrant 3 and moves upward and to the right. It crosses the y-axis at − 5 and the x-axis at 2 . 5, ending in quadrant 1</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. In the xy-plane, the graph of the equation <strong>y = m x + b</strong>, where m and b are constants, is a line with slope m and y-intercept <strong>with coordinates 0, b</strong>. Therefore, the graph of <strong>y = 2 x − 5</strong> in the xy-plane is a line with slope 2 and a y-intercept <strong>with coordinates 0 −5</strong>. Having a slope of 2 means that for each increase in x by 1, the value of y increases by 2. Only the graph in choice D has a slope of 2 and crosses the y-axis at <strong>0 −5</strong>. Therefore, the graph shown in choice D must be the correct answer.Choices A, B, and C are incorrect. The graph of <strong>y = 2 x − 5</strong> in the xy-plane is a line with slope 2 and a y-intercept at <strong>0 −5</strong>. The graph in choice A crosses the y-axis at the point <strong>with coordinates 0, 2 . 5</strong>, not <strong>0 −5</strong>, and it has a slope of <strong>one-half</strong>, not 2. The graph in choice B crosses the y-axis at <strong>0 −5</strong>; however, the slope of this line is <strong>−2</strong>, not 2. The graph in choice C has a slope of 2; however, the graph crosses the y-axis at <strong>0, 5</strong>, not <strong>0 −5</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/a8e6bd75_img2.png",
  },
  {
    id: "a9039591",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>0</strong><br><strong>29</strong><br><br><strong>1</strong><br><strong>32</strong><br><br><strong>2</strong><br><strong>35</strong><br><br>For the linear 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 (x)</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 3 x + 29</strong>" },
      { label: "B", text: "<strong>f (x) = 29 x + 32</strong>" },
      { label: "C", text: "<strong>f (x) = 35 x + 29</strong>" },
      { label: "D", text: "<strong>f (x) = 32 x + 35</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. An equation that defines a linear function <strong>f</strong> can be written in the form <strong>f (x) = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants. It's given in the table that when <strong>x = 0</strong>, <strong>f (x) = 29</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>29</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = m x + b</strong> yields <strong>29 = m (0) + b</strong>, or <strong>29 = b</strong>. Substituting <strong>29</strong> for <strong>b</strong> in the equation <strong>f (x) = m x + b</strong> yields <strong>f (x) = m x + 29</strong>. It's also given in the table that when <strong>x = 1</strong>, <strong>f (x) = 32</strong>. Substituting <strong>1</strong> for <strong>x</strong> and <strong>32</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = m x + 29</strong> yields <strong>32 = m (1) + 29</strong>, or <strong>32 = m + 29</strong>. Subtracting <strong>29</strong> from both sides of this equation yields <strong>3 = m</strong>. Substituting <strong>3</strong> for <strong>m</strong> in the equation <strong>f (x) = m x + 29</strong> yields <strong>f (x) = 3 x + 29</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: "aad7e1b9",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = one tenth x − 2</strong>. What is the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>(−2, 0)</strong>" },
      { label: "B", text: "<strong>(0 −2)</strong>" },
      { label: "C", text: "<strong>(0, one tenth)</strong>" },
      { label: "D", text: "<strong>(one tenth, 0)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The y-intercept of the graph of a function in the xy-plane is the point on the graph where <strong>x = 0</strong>. It′s given that <strong>f (x) = one tenth x − 2</strong>. Substituting <strong>0</strong> for <strong>x</strong> in this equation yields <strong>f (0) = one tenth (0) − 2</strong>, or <strong>f (0) = −2</strong>. Since it′s given that <strong>y = f (x)</strong>, it follows that <strong>y = −2</strong> when <strong>x = 0</strong>. Therefore, the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane 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: false,
  },
  {
    id: "b8cbe394",
    type: "mcq",
    questionHtml:
      "Sean rents a tent at a cost of <strong>dollar sign 11</strong> per day plus a onetime insurance fee of <strong>dollar sign 10</strong>. Which equation represents the total cost <strong>c</strong>, in dollars, to rent the tent with insurance for <strong>d</strong> days?",
    choices: [
      { label: "A", text: "<strong>c = 11 (d + 10)</strong>" },
      { label: "B", text: "<strong>c = 10 (d + 11)</strong>" },
      { label: "C", text: "<strong>c = 11 d + 10</strong>" },
      { label: "D", text: "<strong>c = 10 d + 11</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the cost of renting a tent is <strong>dollar sign 11</strong> per day for <strong>d</strong> days. Multiplying the rental cost by the number of days yields <strong>dollar sign 11 d</strong>, which represents the cost of renting the tent for <strong>d</strong> days before the insurance is added. Adding the onetime insurance fee of <strong>dollar sign 10</strong> to the rental cost of <strong>dollar sign 11 d</strong> gives the total cost <strong>c</strong>, in dollars, which can be represented by the equation <strong>c = 11 d + 10</strong>.<br>Choice A is incorrect. This equation represents the total cost to rent the tent if the insurance fee was charged every day.<br>Choice B is incorrect. This equation represents the total cost to rent the tent if the daily fee was <strong>dollar sign (d + 11)</strong> for <strong>10</strong> days.<br>Choice D is incorrect. This equation represents the total cost to rent the tent if the daily fee was <strong>dollar sign 10</strong> and the onetime fee was <strong>dollar sign 11</strong>.",
    hasFigure: false,
  },
  {
    id: "bf36c815",
    type: "mcq",
    questionHtml:
      "The function g is defined by <strong>g of x = −x + 8</strong>. What is the value of <strong>g of 0</strong>?",
    choices: [
      { label: "A", text: "<strong>−8</strong>" },
      { label: "B", text: "<strong>0</strong>" },
      { label: "C", text: "<strong>4</strong>" },
      { label: "D", text: "<strong>8</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The value of <strong>g of 0</strong> is found by substituting 0 for x in the function g. This yields <strong>g of 0 = −0 + 8</strong>, which can be rewritten as <strong>g of 0 = 8</strong>.Choice A is incorrect and may result from misinterpreting the equation as <strong>g of x = x + −8</strong> instead of <strong>g of x = −x + −8</strong>. Choice B is incorrect. This is the value of x, not <strong>g of x</strong>. Choice C is incorrect and may result from calculation errors.",
    hasFigure: false,
  },
  {
    id: "bf883fde",
    type: "mcq",
    questionHtml:
      "For the function <strong>f</strong>, the graph of <strong>y = f (x)</strong> in the xy-plane has a slope of <strong>3</strong> and passes through the point <strong>(0 −8)</strong>. Which equation defines <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 3 x</strong>" },
      { label: "B", text: "<strong>f (x) = 3 x − 8</strong>" },
      { label: "C", text: "<strong>f (x) = 3 x + 5</strong>" },
      { label: "D", text: "<strong>f (x) = 3 x + 11</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. An equation defining a linear function can be written in the form <strong>f (x) = m x + b</strong>, where <strong>m</strong> and <strong>b</strong> are constants, <strong>m</strong> is the slope of the graph of <strong>y = f (x)</strong> in the xy-plane, and <strong>(0, b)</strong> is the y-intercept of the graph. It's given that for the function <strong>f</strong>, the graph of <strong>y = f (x)</strong> in the xy-plane has a slope of <strong>3</strong>. Therefore, <strong>m = 3</strong>. It's also given that this graph passes through the point <strong>(0 −8)</strong>. Therefore, the y-intercept of the graph is <strong>(0 −8)</strong>, and it follows that <strong>b = −8</strong>. Substituting <strong>3</strong> for <strong>m</strong> and <strong>−8</strong> for <strong>b</strong> in the equation <strong>f (x) = m x + b</strong> yields <strong>f (x) = 3 x − 8</strong>. Thus, the equation that defines <strong>f</strong> is <strong>f (x) = 3 x − 8</strong>.<br>Choice A is incorrect. For this function, the graph of <strong>y = f (x)</strong> in the xy-plane passes through the point <strong>(0, 0)</strong>, not <strong>(0 −8)</strong>.<br>Choice C is incorrect. For this function, the graph of <strong>y = f (x)</strong> in the xy-plane passes through the point <strong>(0, 5)</strong>, not <strong>(0 −8)</strong>.<br>Choice D is incorrect. For this function, the graph of <strong>y = f (x)</strong> in the xy-plane passes through the point <strong>(0, 11)</strong>, not <strong>(0 −8)</strong>.",
    hasFigure: false,
  },
  {
    id: "c4d49134",
    type: "mcq",
    questionHtml:
      "<strong>s = 40 + 3 t</strong><br>The equation gives the speed <strong>s</strong>, in miles per hour, of a certain car <strong>t</strong> seconds after it began to accelerate. What is the speed, in miles per hour, of the car <strong>5</strong> seconds after it began to accelerate?",
    choices: [
      { label: "A", text: "<strong>40</strong>" },
      { label: "B", text: "<strong>43</strong>" },
      { label: "C", text: "<strong>45</strong>" },
      { label: "D", text: "<strong>55</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. In the given equation, <strong>s</strong> is the speed, in miles per hour, of a certain car <strong>t</strong> seconds after it began to accelerate. Therefore, the speed of the car, in miles per hour, <strong>5</strong> seconds after it began to accelerate can be found by substituting <strong>5</strong> for <strong>t</strong> in the given equation, which yields <strong>s = 40 + 3 (5)</strong>, or <strong>s = 55</strong>. Thus, the speed of the car <strong>5</strong> seconds after it began to accelerate is <strong>55</strong> miles per hour.<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: "cee5b352",
    type: "mcq",
    questionHtml:
      "The length, <strong>y</strong>, of a white whale was <strong>162 centimeters (cm)</strong> when it was born and increased an average of <strong>4.8 cm</strong> per month for the first <strong>12</strong> months after it was born. Which equation best represents this situation, where <strong>x</strong> is the number of months after the whale was born and <strong>y</strong> is the length, in <strong>cm</strong>, of the whale?",
    choices: [
      { label: "A", text: "<strong>y = 162 x</strong>" },
      { label: "B", text: "<strong>y = 162 x + 162</strong>" },
      { label: "C", text: "<strong>y = 4.8 x + 4.8</strong>" },
      { label: "D", text: "<strong>y = 4.8 x + 162</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the length of the whale was <strong>162 cm</strong> when it was born and that its length increased an average of <strong>4.8 cm</strong> per month for the first <strong>12</strong> months after it was born. Since <strong>x</strong> represents the number of months after the whale was born, the total increase in the whale's length, in <strong>cm</strong>, is <strong>4.8</strong> times <strong>x</strong>, or <strong>4.8 x</strong>. The length of the whale <strong>y</strong>, in <strong>cm</strong>, can be found by adding the whale's length at birth, <strong>162 cm</strong>, to the total increase in length, <strong>4.8 x cm</strong>. Therefore, the equation that best represents this situation is <strong>y = 4.8 x + 162</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: false,
  },
  {
    id: "de6fe450",
    type: "mcq",
    questionHtml:
      "On January 1, 2015, a city’s minimum hourly wage was $9.25. It will increase by $0.50 on the first day of the year for the next 5 years. Which of the following functions best models the minimum hourly wage, in dollars, x years after January 1, 2015, where <strong>x = the following five values: 1, 2, 3, 4, 5</strong> ?",
    choices: [
      { label: "A", text: "<strong>f of x = 9 . 2 5 − 0 . 5 0 x</strong>" },
      { label: "B", text: "<strong>f of x = 9 . 2 5 x − 0 . 5 0</strong>" },
      { label: "C", text: "<strong>f of x = 9 . 2 5 + 0 . 5 0 x</strong>" },
      { label: "D", text: "<strong>f of x = 9 . 2 5 x + 0 . 5 0</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the city’s minimum hourly wage will increase by $0.50 on the first day of the year for the 5 years after January 1, 2015. Therefore, the total increase, in dollars, in the minimum hourly wage x years after January 1, 2015, is represented by <strong>0 . 5 0 x</strong>. Since the minimum hourly wage on January 1, 2015, was $9.25, it follows that the minimum hourly wage, in dollars, x years after January 1, 2015, is represented by <strong>9 . 2 5 + 0 . 5 0 x</strong>. Therefore, the function <strong>f of x = 9 . 2 5 + 0 . 5 0 x</strong> best models this situation.Choices A, B, and D are incorrect. In choice A, the function models a situation where the minimum hourly wage is $9.25 on January 1, 2015, but decreases by $0.50 on the first day of the year for the next 5 years. The functions in choices B and D both model a situation where the minimum hourly wage is increasing by $9.25 on the first day of the year for the 5 years after January 1, 2015.",
    hasFigure: false,
  },
  {
    id: "f79fffba",
    type: "mcq",
    questionHtml:
      "The function <strong>h</strong> is defined by <strong>h (x) = 3 x − 7</strong>. What is the value of <strong>h (−2)</strong>?",
    choices: [
      { label: "A", text: "<strong>−13</strong>" },
      { label: "B", text: "<strong>−10</strong>" },
      { label: "C", text: "<strong>10</strong>" },
      { label: "D", text: "<strong>13</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The value of <strong>h (−2)</strong> can be found by substituting <strong>−2</strong> for <strong>x</strong> in the equation defining <strong>h</strong>. Substituting <strong>−2</strong> for <strong>x</strong> in <strong>h (x) = 3 x − 7</strong> yields <strong>h (−2) = 3 (−2) − 7</strong>, or <strong>h (−2) = −13</strong>. Therefore, the value of <strong>h (−2)</strong> is <strong>−13</strong>.<br>Choice B is incorrect. This is the value of <strong>h (−1)</strong>, not <strong>h (−2)</strong>.<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,
  },
];

export const LINEAR_FUNC_MEDIUM: PracticeQuestion[] = [
  {
    id: "042aa429",
    type: "mcq",
    questionHtml:
      "If <em>(expression)</em> and <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. The value of <strong>f (2)</strong> can be found by substituting <strong>2</strong> for <strong>x</strong> in the given equation <strong>f (x) = x + 7</strong>, which yields <strong>f (2) = 2 + 7</strong>, or <strong>f (2) = 9</strong>. The value of <strong>g (2)</strong> can be found by substituting <strong>2</strong> for <strong>x</strong> in the given equation <strong>g (x) = 7 x</strong>, which yields <strong>g (2) = 7 (2)</strong>, or <strong>g (2) = 14</strong>. The value of the expression <strong>4 f (2) − g (2)</strong> can be found by substituting the corresponding values into the expression, which gives <strong>4 (9) − 14</strong>. This expression is equivalent to <strong>36 − 14</strong>, or <strong>22</strong>.<br>Choice A is incorrect. This is the value of <strong>f (2) − g (2)</strong>, not <strong>4 f (2) − g (2)</strong>.<br>Choice B is incorrect and may result from calculating <strong>4 f (2)</strong> as <strong>4 (2) + 7</strong>, rather than<br><strong>4 (2 + 7)</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "0cadb20e",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = (x + 15) / (5)</strong>, and <strong>f (a) = 10</strong>, where <strong>a</strong> is a constant. What is the value of <strong>a</strong>?",
    choices: [
      { label: "A", text: "<strong>5</strong>" },
      { label: "B", text: "<strong>10</strong>" },
      { label: "C", text: "<strong>35</strong>" },
      { label: "D", text: "<strong>65</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that <strong>f (x) = (x + 15) / (5)</strong> and <strong>f (a) = 10</strong>, where <strong>a</strong> is a constant. Therefore, for the given function <strong>f</strong>, when <strong>x = a</strong>, <strong>f (x) = 10</strong>. Substituting <strong>a</strong> for <strong>x</strong> and <strong>10</strong> for <strong>f (x)</strong> in the given function <strong>f</strong> yields <strong>10 = (a + 15) / (5)</strong>. Multiplying both sides of this equation by <strong>5</strong> yields <strong>50 = a + 15</strong>. Subtracting <strong>15</strong> from both sides of this equation yields <strong>35 = a</strong>. Therefore, the value of <strong>a</strong> is <strong>35</strong>.<br>Choice A is incorrect. This is the value of <strong>a</strong> if <strong>f (a) = 4</strong>.<br>Choice B is incorrect. This is the value of <strong>a</strong> if <strong>f (a) = 5</strong>.<br>Choice D is incorrect. This is the value of <strong>a</strong> if <strong>f (a) = 16</strong>.",
    hasFigure: false,
  },
  {
    id: "113b938e",
    type: "mcq",
    questionHtml:
      "The equation above represents the speed y, in feet per second, of Sheila’s bicycle x seconds after she applied the brakes at the end of a ride. If the equation is graphed in the xy-plane, which of the following is the best interpretation of the x-coordinate of the line’s x-intercept in the context of the problem?",
    choices: [
      {
        label: "A",
        text: "The speed of Sheila’s bicycle, in feet per second, before Sheila applied the brakes",
      },
      {
        label: "B",
        text: "The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes",
      },
      {
        label: "C",
        text: "The number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop",
      },
      {
        label: "D",
        text: "The number of feet Sheila’s bicycle traveled from the time she began applying the brakes until the bicycle came to a complete stop",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that for each point <strong>with coordinates x, y</strong> on the graph of the given equation, the x-coordinate represents the number of seconds after Sheila applied the brakes, and the y-coordinate represents the speed of Sheila’s bicycle at that moment in time. For the graph of the equation, the y-coordinate of the x-intercept is 0. Therefore, the x-coordinate of the x-intercept of the graph of the given equation represents the number of seconds it took from the time Sheila began applying the brakes until the bicycle came to a complete stop.Choice A is incorrect. The speed of Sheila’s bicycle before she applied the brakes is represented by the y-coordinate of the y-intercept of the graph of the given equation, not the x-coordinate of the x-intercept. Choice B is incorrect. The number of feet per second the speed of Sheila’s bicycle decreased each second after Sheila applied the brakes is represented by the slope of the graph of the given equation, not the x-coordinate of the x-intercept. Choice D is incorrect and may result from misinterpreting x as the distance, in feet, traveled after applying the brakes, rather than the time, in seconds, after applying the brakes.",
    hasFigure: false,
  },
  {
    id: "11e1ab81",
    type: "mcq",
    questionHtml:
      "The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after the product is placed on the belt. Which of the following equations correctly relates d and m ?",
    choices: [
      { label: "A", text: "<strong>d = 2 m</strong>" },
      { label: "B", text: "<strong>d = one-half m</strong>" },
      { label: "C", text: "<strong>d = m + 2</strong>" },
      { label: "D", text: "<strong>d = 2 m + 2</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The line passes through the origin. Therefore, this is a relationship of the form <strong>d = k m</strong>, where k is a constant representing the slope of the graph. To find the value of k, choose a point <strong>with coordinates m, d</strong> on the graph of the line other than the origin and substitute the values of m and d into the equation. For example, if the point <strong>with coordinates 2, 4</strong> is chosen, then <strong>4 = k · 2</strong>, and <strong>k = 2</strong>. Therefore, the equation of the line is <strong>d = 2 m</strong>.Choice B is incorrect and may result from calculating the slope of the line as the change in time over the change in distance traveled instead of the change in distance traveled over the change in time. Choices C and D are incorrect because each of these equations represents a line with a d-intercept of 2. However, the graph shows a line with a d-intercept of 0.",
    hasFigure: true,
    figureUrl: "/practice-images/11e1ab81_img1.png",
  },
  {
    id: "1bc11c4e",
    type: "mcq",
    questionHtml:
      "<strong>g (m) = −0.05 m + 12.1</strong><br>The given function <strong>g</strong> models the number of gallons of gasoline that remains from a full gas tank in a car after driving <strong>m</strong> miles. According to the model, about how many gallons of gasoline are used to drive each mile?",
    choices: [
      { label: "A", text: "<strong>0.05</strong>" },
      { label: "B", text: "<strong>12.1</strong>" },
      { label: "C", text: "<strong>20</strong>" },
      { label: "D", text: "<strong>242.0</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the function <strong>g</strong> models the number of gallons that remain from a full gas tank in a car after driving <strong>m</strong> miles. In the given function <strong>g (m) = − 0.05 m + 12.1</strong>, the coefficient of <strong>m</strong> is <strong>−0.05</strong>. This means that for every increase in the value of <strong>m</strong> by <strong>1</strong>, the value of <strong>g (m)</strong> decreases by <strong>0.05</strong>. It follows that for each mile driven, there is a decrease of <strong>0.05</strong> gallons of gasoline. Therefore, <strong>0.05</strong> gallons of gasoline are used to drive each mile.<br>Choice B is incorrect and represents the number of gallons of gasoline in a full gas tank.<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: "23dedddd",
    type: "mcq",
    questionHtml:
      "In the linear function <strong>f</strong>, <strong>f (0) = 8</strong> and <strong>f (1) = 12</strong>. Which equation defines <strong>f</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 12 x + 8</strong>" },
      { label: "B", text: "<strong>f (x) = 4 x</strong>" },
      { label: "C", text: "<strong>f (x) = 4 x + 12</strong>" },
      { label: "D", text: "<strong>f (x) = 4 x + 8</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since <strong>f</strong> is a linear function, it can be defined by an equation of the form <strong>f (x) = a x + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. It's given that <strong>f (0) = 8</strong>. Substituting <strong>0</strong> for <strong>x</strong> and <strong>8</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a x + b</strong> yields <strong>8 = a (0) + b</strong>, or <strong>8 = b</strong>. Substituting <strong>8</strong> for <strong>b</strong> in the equation <strong>f (x) = a x + b</strong> yields <strong>f (x) = a x + 8</strong>. It's given that <strong>f (1) = 12</strong>. Substituting <strong>1</strong> for <strong>x</strong> and <strong>12</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = a x + 8</strong> yields <strong>12 = a (1) + 8</strong>, or <strong>12 = a + 8</strong>. Subtracting <strong>8</strong> from both sides of this equation yields <strong>a = 4</strong>. Substituting <strong>4</strong> for <strong>a</strong> in the equation <strong>f (x) = a x + 8</strong> yields <strong>f (x) = 4 x + 8</strong>. Therefore, an equation that defines <strong>f</strong> is <strong>f (x) = 4 x + 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: false,
  },
  {
    id: "271f7e3f",
    type: "mcq",
    questionHtml:
      "For the function f defined above, what is the value of <strong>f of 9 − f of 1</strong> ?",
    choices: [
      { label: "A", text: "<strong>1</strong>" },
      { label: "B", text: "<strong>2</strong>" },
      { label: "C", text: "<strong>one fourth</strong>" },
      { label: "D", text: "<strong>nine fourths</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The value of <strong>f of 9 − f of 1</strong> can be calculated by finding the values of <strong>f of 9</strong> and <strong>f of 1</strong>. The value of <strong>f of 9</strong> can be found by substituting 9 for x in the given function: <strong>f of 9 = the fraction with numerator, (9 + 7, ), and denominator 4</strong>. This equation can be rewritten as <strong>f of 9 = 16 over 4</strong>, or 4. Then, the value of <strong>f of 1</strong> can be found by substituting 1 for x in the given function: <strong>f of 1 = the fraction with numerator, (1 + 7, ), and denominator 4</strong>. This equation can be rewritten as <strong>f of 1 = 8 over 4</strong>, or 2. Therefore, <strong>f of 9 − f of 1 = 4 − 2</strong>, which is equivalent to 2.Choices A, C, and D are incorrect and may result from incorrectly substituting values of x in the given function or making computational errors.",
    hasFigure: false,
  },
  {
    id: "3122fc7b",
    type: "spr",
    questionHtml:
      "A linear model estimates the population of a city from <strong>1991</strong> to <strong>2015</strong>. The model estimates the population was <strong>57</strong> thousand in <strong>1991</strong>, <strong>224</strong> thousand in <strong>2011</strong>, and <strong>x</strong> thousand in <strong>2015</strong>. To the nearest whole number, what is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "257",
    explanation:
      "The correct answer is <strong>257</strong>. It’s given that a linear model estimates the population of a city from <strong>1991</strong> to <strong>2015</strong>. Since the population can be estimated using a linear model, it follows that there is a constant rate of change for the model. It’s also given that the model estimates the population was <strong>57</strong> thousand in <strong>1991</strong>, <strong>224</strong> thousand in <strong>2011</strong>, and <strong>x</strong> thousand in <strong>2015</strong>. The change in the population between <strong>2011</strong> and <strong>1991</strong> is <strong>224 − 57</strong>, or <strong>167</strong>, thousand. The change in the number of years between <strong>2011</strong> and <strong>1991</strong> is <strong>2011 − 1991</strong>, or <strong>20</strong>, years. Dividing <strong>167</strong> by <strong>20</strong> gives <strong>167 ÷ 20</strong>, or <strong>8.35</strong>, thousand per year. Thus, the change in population per year from <strong>1991</strong> to <strong>2015</strong> estimated by the model is <strong>8.35</strong> thousand. The change in the number of years between <strong>2015</strong> and <strong>2011</strong> is <strong>2015 − 2011</strong>, or <strong>4</strong>, years. Multiplying the change in population per year by the change in number of years yields the increase in population from <strong>2011</strong> to <strong>2015</strong> estimated by the model: <strong>(8.35) (4)</strong>, or <strong>33.4</strong>, thousand. Adding the change in population from <strong>2011</strong> to <strong>2015</strong> estimated by the model to the estimated population in <strong>2011</strong> yields the estimated population in <strong>2015</strong>. Thus, the estimated population in <strong>2015</strong> is <strong>33.4 + 224</strong>, or <strong>257.4</strong>, thousand. Therefore to the nearest whole number, the value of <strong>x</strong> is <strong>257</strong>.",
    hasFigure: false,
  },
  {
    id: "3ce92ce8",
    type: "spr",
    questionHtml:
      "<strong>f (x) = 2 x + 3</strong><br>For the given function <strong>f</strong>, the graph of <strong>y = f (x)</strong> in the xy-plane is parallel to line <strong>j</strong>. What is the slope of line <strong>j</strong>?",
    choices: [],
    correctAnswer: "2",
    explanation:
      "The correct answer is <strong>2</strong>. It’s given that function <strong>f</strong> is defined by <strong>f (x) = 2 x + 3</strong>. Therefore, the equation representing the graph of <strong>y = f (x)</strong> in the xy-plane is <strong>y = 2 x + 3</strong>, and the graph is a line. For a linear equation in the form <strong>y = m x + b</strong>, <strong>m</strong> represents the slope of the line. Since the value of <strong>m</strong> in the equation <strong>y = 2 x + 3</strong> is <strong>2</strong>, the slope of the line defined by function <strong>f</strong> is <strong>2</strong>. It's given that line <strong>j</strong> is parallel to the line defined by function <strong>f</strong>. The slopes of parallel lines are equal. Therefore, the slope of line <strong>j</strong> is also <strong>2</strong>.",
    hasFigure: false,
  },
  {
    id: "3e9eaffc",
    type: "mcq",
    questionHtml:
      "Caleb used juice to make popsicles. The function <strong>f (x) = − 5 x + 30</strong> approximates the volume, in fluid ounces, of juice Caleb had remaining after making <strong>x</strong> popsicles. Which statement is the best interpretation of the y-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane in this context?",
    choices: [
      {
        label: "A",
        text: "Caleb used approximately <strong>5</strong> fluid ounces of juice for each popsicle.",
      },
      {
        label: "B",
        text: "Caleb had approximately <strong>5</strong> fluid ounces of juice when he began to make the popsicles.",
      },
      {
        label: "C",
        text: "Caleb had approximately <strong>30</strong> fluid ounces of juice when he began to make the popsicles.",
      },
      {
        label: "D",
        text: "Caleb used approximately <strong>30</strong> fluid ounces of juice for each popsicle.",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. An equation that defines a linear function <strong>f</strong> can be written in the form <strong>f (x) = m x + b</strong>, where <strong>m</strong> represents the slope and <strong>b</strong> represents the y-intercept, <strong>(0, b)</strong>, of the line of <strong>y = f (x)</strong> in the xy-plane. The function <strong>f (x) = − 5 x + 30</strong> is linear. Therefore, the graph of the given function <strong>y = f (x)</strong> in the xy-plane has a y-intercept of <strong>(0, 30)</strong>. It’s given that <strong>f (x)</strong> gives the approximate volume, in fluid ounces, of juice Caleb had remaining after making <strong>x</strong> popsicles. It follows that the y-intercept of <strong>(0, 30)</strong> means that Caleb had approximately <strong>30</strong> fluid ounces of juice remaining after making <strong>0</strong> popsicles. In other words, Caleb had approximately <strong>30</strong> fluid ounces of juice when he began to make the popsicles.<br>Choice A is incorrect. This is an interpretation of the slope, rather than the y-intercept, of the graph of <strong>y = f (x)</strong> in the xy-plane.<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: "41fdc0b8",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents a 2-column table, with two rows of data, titled “Population of Greenleaf, Idaho.” The heading of the first column is “Year.” The heading of the second column is “Population.” The 2 rows of data are as follows. Row 1. Year, 2000. Population, 862. Row 2. Year, 2010. Population, 846</strong>The table above shows the population of Greenleaf, Idaho, for the years 2000 and 2010. If the relationship between population and year is linear, which of the following functions P models the population of Greenleaf t years after 2000?",
    choices: [
      { label: "A", text: "<strong>P of t = 862 − 1 . 6 · t</strong>" },
      { label: "B", text: "<strong>P of t = 862 − 16 · t</strong>" },
      {
        label: "C",
        text: "<strong>P of t = 862 + 16 · (t − 2, 000, )</strong>",
      },
      {
        label: "D",
        text: "<strong>P of t = 862 − 1 . 6 · (t − 2, 000, )</strong>",
      },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It is given that the relationship between population and year is linear; therefore, the function that models the population t years after 2000 is of the form <strong>P of t = m, t + b</strong>, where m is the slope and b is the population when <strong>t = 0</strong>. In the year 2000, <strong>t = 0</strong>. Therefore, <strong>b = 862</strong>. The slope is given by <strong>m = the fraction with numerator P of 10 − P of 0, and denominator 10 − 0, end fraction, which = the fraction with numerator 846 − 862, and denominator 10 − 0, end fraction, which = −16 over 10, which = −1 . 6</strong>. Therefore, <strong>P of t = −1 . 6, t + 862</strong>, which is equivalent to the equation in choice A.Choice B is incorrect and may be the result of incorrectly calculating the slope as just the change in the value of P. Choice C is incorrect and may be the result of the same error as in choice B, in addition to incorrectly using t to represent the year, instead of the number of years after 2000. Choice D is incorrect and may be the result of incorrectly using t to represent the year instead of the number of years after 2000.",
    hasFigure: true,
    figureUrl: "/practice-images/41fdc0b8_img1.png",
  },
  {
    id: "441558e7",
    type: "mcq",
    questionHtml:
      "Scientists collected fallen acorns that each housed a colony of the ant species P. ohioensis and analyzed each colony's structure. For any of these colonies, if the colony has <strong>x</strong> worker ants, the equation <strong>y = 0.67 x + 2.6</strong>, where <strong>20 < or = x < or = 110</strong>, gives the predicted number of larvae, <strong>y</strong>, in the colony. If one of these colonies has <strong>58</strong> worker ants, which of the following is closest to the predicted number of larvae in the colony?",
    choices: [
      { label: "A", text: "<strong>41</strong>" },
      { label: "B", text: "<strong>61</strong>" },
      { label: "C", text: "<strong>83</strong>" },
      { label: "D", text: "<strong>190</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that the equation <strong>y = 0.67 x + 2.6</strong>, where <strong>20 < or = x < or = 110</strong>, gives the predicted number of larvae, <strong>y</strong>, in a colony of ants if the colony has <strong>x</strong> worker ants. If one of these colonies has <strong>58</strong> worker ants, the predicted number of larvae in that colony can be found by substituting <strong>58</strong> for <strong>x</strong> in the given equation. Substituting <strong>58</strong> for <strong>x</strong> in the given equation yields <strong>y = 0.67 (58) + 2.6</strong>, or <strong>y = 41.46</strong>. Of the given choices, <strong>41</strong> is closest to the predicted number of larvae in the colony.<br>Choice B is incorrect. This is closest to the predicted number of larvae in a colony with <strong>87</strong> worker ants.<br>Choice C is incorrect. This is closest to the number of worker ants for which the predicted number of larvae in a colony is <strong>58</strong>.<br>Choice D is incorrect. This is closest to the predicted number of larvae in a colony with <strong>280</strong> worker ants.",
    hasFigure: false,
  },
  {
    id: "4fe4fd7c",
    type: "spr",
    questionHtml:
      "A company’s total cost <strong>c of x</strong>, in dollars, to produce x shirts is given by the function above, where m is a constant and <strong>x > 0</strong>. The total cost to produce 100 shirts is $800. What is the total cost, in dollars, to produce 1000 shirts? (Disregard the $ sign when gridding your answer.)",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 3500. The given information includes a cost, $800, to produce 100 shirts. Substituting <strong>c of x = 800</strong> and <strong>x = 100</strong> into the given equation yields <strong>800 = m · 100 + 500</strong>. Subtracting 500 from both sides of the equation yields <strong>300 = m · 100</strong>. Dividing both sides of this equation by 100 yields <strong>3 = m</strong>. Substituting the value of m into the given equation yields <strong>c of x = 3 x + 500</strong>. Substituting 1000 for x in this equation and solving for <strong>c of x</strong> gives the cost of 1000 shirts: <strong>3 · 1000 + 500</strong>, or 3500.",
    hasFigure: false,
  },
  {
    id: "548a4929",
    type: "mcq",
    questionHtml:
      "The function <strong>h</strong> is defined by <strong>h (x) = 4 x + 28</strong>. The graph of <strong>y = h (x)</strong> in the xy-plane has an x-intercept at <strong>(a, 0)</strong> and a y-intercept at <strong>(0, b)</strong>, where <strong>a</strong> and <strong>b</strong> are constants. What is the value of <strong>a + b</strong>?",
    choices: [
      { label: "A", text: "<strong>21</strong>" },
      { label: "B", text: "<strong>28</strong>" },
      { label: "C", text: "<strong>32</strong>" },
      { label: "D", text: "<strong>35</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where <strong>y = 0</strong>. It's given that function <strong>h</strong> is defined by <strong>h (x) = 4 x + 28</strong>. Therefore, the equation representing the graph of <strong>y = h (x)</strong> is <strong>y = 4 x + 28</strong>. Substituting <strong>0</strong> for <strong>y</strong> in the equation <strong>y = 4 x + 28</strong> yields <strong>0 = 4 x + 28</strong>. Subtracting <strong>28</strong> from both sides of this equation yields <strong>−28 = 4 x</strong>. Dividing both sides of this equation by <strong>4</strong> yields <strong>−7 = x</strong>. Therefore, the x-intercept of the graph of <strong>y = h (x)</strong> in the xy-plane is <strong>(−7, 0)</strong>. It's given that the x-intercept of the graph of <strong>y = h (x)</strong> is <strong>(a, 0)</strong>. Therefore, <strong>a = −7</strong>. The y-intercept of a graph 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>y = 4 x + 28</strong> yields <strong>y = 4 (0) + 28</strong>, or <strong>y = 28</strong>. Therefore, the y-intercept of the graph of <strong>y = h (x)</strong> in the xy-plane is <strong>(0, 28)</strong>. It's given that the y-intercept of the graph of <strong>y = h (x)</strong> is <strong>(0, b)</strong>. Therefore, <strong>b = 28</strong>. If <strong>a = −7</strong> and <strong>b = 28</strong>, then the value of <strong>a + b</strong> is <strong>−7 + 28</strong>, or <strong>21</strong>.<br>Choice B is incorrect. This is the value of <strong>b</strong>, not <strong>a + b</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the value of <strong>−a + b</strong>, not <strong>a + b</strong>.",
    hasFigure: false,
  },
  {
    id: "56dc8045",
    type: "mcq",
    questionHtml:
      "<strong>w (t) = 300 − 4 t</strong><br>The function <strong>w</strong> models the volume of liquid, in milliliters, in a container <strong>t</strong> seconds after it begins draining from a hole at the bottom. According to the model, what is the predicted volume, in milliliters, draining from the container each second?",
    choices: [
      { label: "A", text: "<strong>300</strong>" },
      { label: "B", text: "<strong>296</strong>" },
      { label: "C", text: "<strong>75</strong>" },
      { label: "D", text: "<strong>4</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the function <strong>w</strong> models the volume of liquid, in milliliters, in a container <strong>t</strong> seconds after it begins draining from a hole at the bottom. The given function <strong>w (t) = 300 − 4 t</strong> can be rewritten as <strong>w (t) = − 4 t + 300</strong>. Thus, for each increase of <strong>t</strong> by <strong>1</strong>, the value of <strong>w (t)</strong> decreases by <strong>4 (1)</strong>, or <strong>4</strong>. Therefore, the predicted volume, in milliliters, draining from the container each second is <strong>4</strong> milliliters.<br>Choice A is incorrect. This is the amount of liquid, in milliliters, in the container before the liquid begins draining.<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: "5cf1bbc9",
    type: "mcq",
    questionHtml:
      "The graph of the function <em>(expression)</em>, where <em>(expression)</em>, gives the total cost <em>(expression)</em>, in dollars, for a certain video game system and <em>(expression)</em> games. What is the best interpretation of the slope of the graph in this context?",
    choices: [
      { label: "A", text: "Each game costs <em>(expression)</em>." },
      {
        label: "B",
        text: "The video game system costs <em>(expression)</em>.",
      },
      {
        label: "C",
        text: "The video game system costs <em>(expression)</em>.",
      },
      { label: "D", text: "Each game costs <em>(expression)</em>." },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The given graph is a line, and the slope of a line is defined as the change in the value of <strong>y</strong> for each increase in the value of <strong>x</strong> by <strong>1</strong>. It’s given that <strong>y</strong> represents the total cost, in dollars, and that <strong>x</strong> represents the number of games. Therefore, the change in the value of <strong>y</strong> for each increase in the value of <strong>x</strong> by <strong>1</strong> represents the change in total cost, in dollars, for each increase in the number of games by <strong>1</strong>. In other words, the slope represents the cost, in dollars, per game. The graph shows that when the value of <strong>x</strong> increases from <strong>0</strong> to <strong>1</strong>, the value of <strong>y</strong> increases from <strong>100</strong> to <strong>125</strong>. It follows that the slope is <strong>25</strong>, or the cost per game is <strong>dollar sign 25</strong>. Thus, the best interpretation of the slope of the graph is that each game costs <strong>dollar sign 25</strong>.<br>Choice B is incorrect. This is an interpretation of the y-intercept of the graph rather than the slope of the graph.<br>Choice C is incorrect. The slope of the graph is the cost per game, not the cost of the video game system.<br>Choice D is incorrect. Each game costs <strong>dollar sign 25</strong>, not <strong>dollar sign 100</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/5cf1bbc9_svg1.svg",
  },
  {
    id: "620fe971",
    type: "mcq",
    questionHtml:
      "The graph of this equation in the xy-plane is a line. What is the best interpretation of the x-intercept in this context?",
    choices: [
      {
        label: "A",
        text: "The team will have moved all the cargo in about 4.8 hours.",
      },
      {
        label: "B",
        text: "The team has been moving about 4.8 tons of cargo per hour.",
      },
      {
        label: "C",
        text: "The team has been moving about 25 tons of cargo per hour.",
      },
      { label: "D", text: "The team started with 120 tons of cargo to move." },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The x-intercept of the line with equation y = 120 – 25x can be found by substituting 0 for y and finding the value of x. When y = 0, x = 4.8, so the x-intercept is at (4.8, 0). Since y represents the number of tons of cargo remaining to be moved x hours after the team started working, it follows that the x-intercept refers to the team having no cargo remaining to be moved after 4.8 hours. In other words, the team will have moved all of the cargo after about 4.8 hours.Choice B is incorrect and may result from incorrectly interpreting the value 4.8. Choices C and D are incorrect and may result from misunderstanding the x-intercept. These statements are accurate but not directly relevant to the x-intercept.",
    hasFigure: false,
  },
  {
    id: "7e1bff94",
    type: "mcq",
    questionHtml:
      "The table gives the number of hours, <strong>h</strong>, of labor and a plumber’s total charge <strong>f (h)</strong>, in dollars, for two different jobs.<br><br><strong>h</strong><br><strong>f (h)</strong><br><br><strong>1</strong><br><strong>155</strong><br><br><strong>3</strong><br><strong>285</strong><br><br>There is a linear relationship between <strong>h</strong> and <strong>f (h)</strong>. Which equation represents this relationship?",
    choices: [
      { label: "A", text: "<strong>f (h) = 25 h + 130</strong>" },
      { label: "B", text: "<strong>f (h) = 130 h + 25</strong>" },
      { label: "C", text: "<strong>f (h) = 65 h + 90</strong>" },
      { label: "D", text: "<strong>f (h) = 90 h + 65</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, <strong>h</strong>, and the plumber's total charge <strong>f (h)</strong>, in dollars. It follows that the relationship can be represented by an equation of the form <strong>f (h) = m h + b</strong>, where <strong>m</strong> is the rate of change of the function <strong>f</strong> and <strong>b</strong> is a constant. The rate of change of <strong>f</strong> can be calculated by dividing the difference in two values of <strong>f (h)</strong> by the difference in the corresponding values of <strong>h</strong>. Based on the values given in the table, the rate of change of <strong>f</strong> is <strong>(285 − 155) / (3 − 1)</strong>, or <strong>65</strong>. Substituting <strong>65</strong> for <strong>m</strong> in the equation <strong>f (h) = m h + b</strong> yields <strong>f (h) = 65 h + b</strong>. The value of <strong>b</strong> can be found by substituting any value of <strong>h</strong> and its corresponding value of <strong>f (h)</strong> for <strong>h</strong> and <strong>f (h)</strong>, respectively, in this equation. Substituting <strong>1</strong> for <strong>h</strong> and <strong>155</strong> for <strong>f (h)</strong> yields <strong>155 = 65 (1) + b</strong>, or <strong>155 = 65 + b</strong>. Subtracting <strong>65</strong> from both sides of this equation yields <strong>90 = b</strong>. Substituting <strong>90</strong> for <strong>b</strong> in the equation <strong>f (h) = 65 h + b</strong> yields <strong>f (h) = 65 h + 90</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: "7e3f8363",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the graph of the linear function <strong>f</strong> contains the points <strong>(0, 3)</strong> and <strong>(7, 31)</strong>. Which equation defines <strong>f</strong>, where <strong>y = f (x)</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 28 x + 34</strong>" },
      { label: "B", text: "<strong>f (x) = 3 x + 38</strong>" },
      { label: "C", text: "<strong>f (x) = 4 x + 3</strong>" },
      { label: "D", text: "<strong>f (x) = 7 x + 3</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. In the xy-plane, an equation of the graph of a linear function can be written in the form <strong>f (x) = m x + b</strong>, where <strong>m</strong> represents the slope and <strong>(0, b)</strong> represents the y-intercept of the graph of <strong>y = f (x)</strong>. It’s given that the graph of the linear function <strong>f</strong>, where <strong>y = f (x)</strong>, in the xy-plane contains the point <strong>(0, 3)</strong>. Thus, <strong>b = 3</strong>. The slope of the graph of a line containing any two points <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong> can be found using the slope formula, <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Since it’s given that the graph of the linear function <strong>f</strong> contains the points <strong>(0, 3)</strong> and <strong>(7, 31)</strong>, it follows that the slope of the graph of the line containing these points is <strong>m = (31 − 3) / (7 − 0)</strong>, or <strong>m = 4</strong>. Substituting <strong>4</strong> for <strong>m</strong> and <strong>3</strong> for <strong>b</strong> in <strong>f (x) = m x + b</strong> yields <strong>f (x) = 4 x + 3</strong>.<br>Choice A is incorrect. This function represents a graph with a slope of <strong>28</strong> and a y-intercept of <strong>(0, 34)</strong>.<br>Choice B is incorrect. This function represents a graph with a slope of <strong>3</strong> and a y-intercept of <strong>(0, 38)</strong>.<br>Choice D is incorrect. This function represents a graph with a slope of <strong>7</strong> and a y-intercept of <strong>(0, 3)</strong>.",
    hasFigure: false,
  },
  {
    id: "868fc236",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents a 3-column table, with 3 rows of data, titled “Energy per Gram of Typical Macronutrients.” The heading for column 1 is “Macronutrient, ” the heading for column 2 is “Food calories, ” and the heading for column 3 is “Kilojoules.” The 3 rows of data are as follows. Row 1.Macronutrient, Protein; Food calories, 4 . zero; Kilojoules, 16 . 7. Row 2. Macronutrient, Fat; Food calories, 9 . zero; Kilojoules, 37 . 7. Row 3. Macronutrient, Carbohydrate; Food calories, 4 . zero; Kilojoules, 16 . 7.</strong>The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If x food calories is equivalent to k kilojoules, of the following, which best represents the relationship between x and k ?",
    choices: [
      { label: "A", text: "<strong>k = zero . 2 4 x</strong>" },
      { label: "B", text: "<strong>k = 4 . 2 x</strong>" },
      { label: "C", text: "<strong>x = 4 . 2 k</strong>" },
      { label: "D", text: "<strong>x k = 4 . 2</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The relationship between x food calories and k kilojoules can be modeled as a proportional relationship. Let <strong>the ordered pair x sub 1, k sub 1</strong> and <strong>the ordered pair x sub 2, k sub 2</strong> represent the values in the first two rows in the table: <strong>the ordered pair 4 . 0, 16 . 7</strong> and <strong>the ordered pair 9 . 0, 37 . 7</strong>. The rate of change, or <strong>the fraction with numerator, k sub 2 − k sub 1, and denominator x sub 2 − x sub 1, end fraction</strong>, is <strong>21 over 5 = 4 . 2</strong>; therefore, the equation that best represents the relationship between x and k is <strong>k = 4 . 2 x</strong>.Choice A is incorrect and may be the result of calculating the rate of change using <strong>the fraction with numerator, x sub 2 − x sub 1, and denominator k sub 2 − k sub 1, end fraction</strong>. Choice C is incorrect because the number of kilojoules is greater than the number of food calories. Choice D is incorrect and may be the result of an error when setting up the equation.",
    hasFigure: true,
    figureUrl: "/practice-images/868fc236_img1.png",
  },
  {
    id: "8a6de407",
    type: "spr",
    questionHtml:
      "The function f is defined by <strong>f of x = m x + b</strong>, where m and b are constants. If <strong>f of 0 = 18</strong> and <strong>f of 1 = 20</strong>, what is the value of m ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 2. The slope-intercept form of an equation for a line is <strong>y = m x + b</strong>, where m is the slope and b is the y-coordinate of the y-intercept. Two ordered pairs, <strong>x sub 1, y sub 1</strong> and <strong>x sub 2, y sub 2</strong>, can be used to compute the slope using the formula <strong>m = the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction</strong>. It’s given that <strong>f of 0 = 18</strong> and <strong>f of 1 = 20</strong>; therefore, the two ordered pairs for this line are <strong>0, 18</strong> and <strong>1, 20</strong>. Substituting these values for <strong>x sub 1, y sub 1</strong> and <strong>x sub 2, y sub 2</strong> gives <strong>the fraction with numerator 20 − 18, and denominator 1 − 0, end fraction = the fraction 2 over 1</strong>, or 2.",
    hasFigure: false,
  },
  {
    id: "a5834ea4",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 39</strong><br>For the given linear function <strong>f</strong>, which table gives three values of <strong>x</strong> and their corresponding values of <strong>f (x)</strong>?",
    choices: [
      {
        label: "A",
        text: "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>0</strong><br><br><strong>2</strong><br><strong>0</strong>",
      },
      {
        label: "B",
        text: "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>0</strong><br><strong>39</strong><br><br><strong>1</strong><br><strong>39</strong><br><br><strong>2</strong><br><strong>39</strong>",
      },
      {
        label: "C",
        text: "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>0</strong><br><strong>0</strong><br><br><strong>1</strong><br><strong>39</strong><br><br><strong>2</strong><br><strong>78</strong>",
      },
      {
        label: "D",
        text: "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>0</strong><br><strong>39</strong><br><br><strong>1</strong><br><strong>0</strong><br><br><strong>2</strong><br><strong>−39</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. For the given linear function <strong>f</strong>, <strong>f (x)</strong> must equal <strong>39</strong> for all values of <strong>x</strong>. Of the given choices, only choice B gives three values of <strong>x</strong> and their corresponding values of <strong>f (x)</strong> for the given linear function <strong>f</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: "a775af14",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the graph of the linear function <strong>f</strong> contains the points <strong>(0, 2)</strong> and <strong>(8, 34)</strong>. Which equation defines <strong>f</strong>, where <strong>y = f (x)</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 2 x + 42</strong>" },
      { label: "B", text: "<strong>f (x) = 32 x + 36</strong>" },
      { label: "C", text: "<strong>f (x) = 4 x + 2</strong>" },
      { label: "D", text: "<strong>f (x) = 8 x + 2</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. In the xy-plane, the graph of a linear function can be written in the form <strong>f (x) = m x + b</strong>, where <strong>m</strong> represents the slope and <strong>(0, b)</strong> represents the y-intercept of the graph of <strong>y = f (x)</strong>. It’s given that the graph of the linear function <strong>f</strong>, where <strong>y = f (x)</strong>, in the xy-plane contains the point <strong>(0, 2)</strong>. Thus, <strong>b = 2</strong>. The slope of the graph of a line containing any two points <strong>(x 1, y 1)</strong> and <strong>(x 2, y 2)</strong> can be found using the slope formula, <strong>m = (y 2 − y 1) / (x 2 − x 1)</strong>. Since it’s given that the graph of the linear function <strong>f</strong> contains the points <strong>(0, 2)</strong> and <strong>(8, 34)</strong>, it follows that the slope of the graph of the line containing these points is <strong>m = (34 − 2) / (8 − 0)</strong>, or  <strong>m = 4</strong>. Substituting <strong>4</strong> for <strong>m</strong> and <strong>2</strong> for <strong>b</strong> in <strong>f (x) = m x + b</strong> yields <strong>f (x) = 4 x + 2</strong>.<br>Choice A is incorrect. This function represents a graph with a slope of <strong>2</strong> and a y-intercept of <strong>(0, 42)</strong>.<br>Choice B is incorrect. This function represents a graph with a slope of <strong>32</strong> and a y-intercept of <strong>(0, 36)</strong>.<br>Choice D is incorrect. This function represents a graph with a slope of <strong>8</strong> and a y-intercept of <strong>(0, 2)</strong>.",
    hasFigure: false,
  },
  {
    id: "c01f4a95",
    type: "spr",
    questionHtml:
      "<strong>j (x) = m x + 144</strong><br>For the linear function <strong>j</strong>, <strong>m</strong> is a constant and <strong>j (12) = 18</strong>. What is the value of <strong>j (10)</strong>?",
    choices: [],
    correctAnswer: "39",
    explanation:
      "The correct answer is <strong>39</strong>. It’s given that for the linear function <strong>j</strong>, <strong>m</strong> is a constant and <strong>j (12) = 18</strong>. Substituting <strong>12</strong> for <strong>x</strong> and <strong>18</strong> for <strong>j (x)</strong> in the given equation yields <strong>18 = m (12) + 144</strong>. Subtracting <strong>144</strong> from both sides of this equation yields <strong>−126 = m (12)</strong>. Dividing both sides of this equation by <strong>12</strong> yields <strong>−10.5 = m</strong>. Substituting <strong>−10.5</strong> for <strong>m</strong> in the given equation, <strong>j (x) = m x + 144</strong>, yields <strong>j (x) = − 10.5 x + 144</strong>. Substituting <strong>10</strong> for <strong>x</strong> in this equation yields <strong>j (10) = (−10.5) (10) + 144</strong>, or <strong>j (10) = 39</strong>. Therefore, the value of <strong>j (10)</strong> is <strong>39</strong>.",
    hasFigure: false,
  },
  {
    id: "c1bd5301",
    type: "spr",
    questionHtml:
      "A model predicts that a certain animal weighed <strong>241</strong> pounds when it was born and that the animal gained <strong>3</strong> pounds per day in its first year of life. This model is defined by an equation in the form <strong>f (x) = a + b x</strong>, where <strong>f (x)</strong> is the predicted weight, in pounds, of the animal <strong>x</strong> days after it was born, and <strong>a</strong> and <strong>b</strong> are constants. What is the value of <strong>a</strong>?",
    choices: [],
    correctAnswer: "241",
    explanation:
      "The correct answer is <strong>241</strong>. For a certain animal, it's given that a model predicts the animal weighed <strong>241</strong> pounds when it was born and gained <strong>3</strong> pounds per day in its first year of life. It's also given that this model is defined by an equation in the form <strong>f (x) = a + b x</strong>, where <strong>f (x)</strong> is the predicted weight, in pounds, of the animal <strong>x</strong> days after it was born, and <strong>a</strong> and <strong>b</strong> are constants. It follows that <strong>a</strong> represents the predicted weight, in pounds, of the animal when it was born and <strong>b</strong> represents the predicted rate of weight gain, in pounds per day, in its first year of life. Thus, the value of <strong>a</strong> is <strong>241</strong>.",
    hasFigure: false,
  },
  {
    id: "c22b5f25",
    type: "mcq",
    questionHtml:
      "In the xy-plane, the points <strong>with coordinates − 2, 3</strong> and <strong>4 −5</strong> lie on the graph of which of the following linear functions?",
    choices: [
      { label: "A", text: "<strong>f of x = x + 5</strong>" },
      { label: "B", text: "<strong>f of x = one half x + 4</strong>" },
      {
        label: "C",
        text: "<strong>f of x = negative, four thirds x + one third</strong>",
      },
      {
        label: "D",
        text: "<strong>f of x = negative, three halves x + 1</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. A linear function can be written in the form <strong>f of x = m x + b</strong>, where m is the slope and b is the y-coordinate of the y-intercept of the line. The slope of the graph can be found using the formula <strong>m = the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction</strong>. Substituting the values of the given points into this formula yields <strong>m = the fraction with numerator − 5 − 3, and denominator 4 − −2, end fraction</strong> or <strong>m = −8 over 6</strong>, which simplifies to <strong>m = −four thirds</strong>. Only choice C shows an equation with this slope.<br><br>Choices A, B, and D are incorrect and may result from computation errors or misinterpreting the given information.",
    hasFigure: false,
  },
  {
    id: "c651cc56",
    type: "mcq",
    questionHtml:
      "Some values of the linear function f are shown in the table above. What is the value of <strong>f of 3</strong>?",
    choices: [
      { label: "A", text: "6" },
      { label: "B", text: "7" },
      { label: "C", text: "8" },
      { label: "D", text: "9" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A linear function has a constant rate of change, and any two rows of the table shown can be used to calculate this rate. From the first row to the second, the value of x is increased by 2 and the value of <strong>f of x</strong> is increased by <strong>6 = 4 − −2</strong>. So the values of <strong>f of x</strong> increase by 3 for every increase by 1 in the value of x. Since <strong>f of 2 = 4</strong>, it follows that <strong>f of, (2 + 1, ) = 4 + 3, which = 7</strong>. Therefore, <strong>f of 3 = 7</strong>.Choice A is incorrect. This is the third x-value in the table, not <strong>f of 3</strong>. Choices C and D are incorrect and may result from errors when calculating the function’s rate of change.",
    hasFigure: true,
    figureUrl: "/practice-images/c651cc56_img1.png",
  },
  {
    id: "c8fb6bcb",
    type: "mcq",
    questionHtml:
      "<strong>f (x) = 2 x + 244</strong><br>The given function <strong>f</strong> represents the perimeter, in <strong>centimeters (cm)</strong>, of a rectangle with a length of <strong>x cm</strong> and a fixed width. What is the width, in <strong>cm</strong>, of the rectangle?",
    choices: [
      { label: "A", text: "<strong>2</strong>" },
      { label: "B", text: "<strong>122</strong>" },
      { label: "C", text: "<strong>244</strong>" },
      { label: "D", text: "<strong>488</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that <strong>f (x) = 2 x + 244</strong> represents the perimeter, in <strong>centimeters (cm)</strong>, of a rectangle with a length of <strong>x</strong> <strong>c m</strong> and a fixed width. If <strong>w</strong> represents a fixed width, in <strong>cm</strong>, then the perimeter, in <strong>cm</strong>, of a rectangle with a length of <strong>x</strong> <strong>cm</strong> and a fixed width of <strong>w</strong> <strong>cm</strong> can be given by the function <strong>f (x) = 2 x + 2 w</strong>. Therefore, <strong>2 x + 2 w = 2 x + 244</strong>. Subtracting <strong>2 x</strong> from both sides of this equation yields <strong>2 w = 244</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>w = 122</strong>. Therefore, the width, in <strong>cm</strong>, of the rectangle is <strong>122</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: "dae126d7",
    type: "mcq",
    questionHtml:
      "The boiling point of water at sea level is 212 degrees Fahrenheit <strong>° Fahrenheit</strong>. For every 550 feet above sea level, the boiling point of water is lowered by about <strong>1 ° Fahrenheit</strong>. Which of the following equations can be used to find the boiling point B of water, in <strong>° Fahrenheit</strong>, x feet above sea level?",
    choices: [
      {
        label: "A",
        text: "<strong>B = 550 + the fraction x over 212</strong>",
      },
      {
        label: "B",
        text: "<strong>B = 550 − the fraction x over 212</strong>",
      },
      {
        label: "C",
        text: "<strong>B = 212 + the fraction x over 550</strong>",
      },
      {
        label: "D",
        text: "<strong>B = 212 − the fraction x over 550</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that the boiling point of water at sea level is 212°F and that for every 550 feet above sea level, the boiling point of water is lowered by about 1°F. Therefore, the change in the boiling point of water x feet above sea level is represented by the expression <strong>the − of the fraction x over 550</strong>. Adding this expression to the boiling point of water at sea level gives the equation for the boiling point B of water, in °F, x feet above sea level: <strong>B = the − of the fraction x over 550, end fraction + 212</strong>, or <strong>B = 212 − the fraction x over 550</strong>.Choices A and B are incorrect and may result from using the boiling point of water at sea level as the rate of change and the rate of change as the initial boiling point of water at sea level. Choice C is incorrect and may result from representing the change in the boiling point of water as an increase rather than a decrease.",
    hasFigure: false,
  },
  {
    id: "e470e19d",
    type: "mcq",
    questionHtml:
      "The function <strong>f</strong> is defined by <strong>f (x) = 7 x − 84</strong>. What is the x-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>(−12, 0)</strong>" },
      { label: "B", text: "<strong>(−7, 0)</strong>" },
      { label: "C", text: "<strong>(7, 0)</strong>" },
      { label: "D", text: "<strong>(12, 0)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The given function <strong>f</strong> is a linear function. Therefore, the graph of <strong>y = f (x)</strong> in the xy-plane has one x-intercept at the point <strong>(k, 0)</strong>, where <strong>k</strong> is a constant. Substituting <strong>0</strong> for <strong>f (x)</strong> and <strong>k</strong> for <strong>x</strong> in the given function yields <strong>0 = 7 k − 84</strong>. Adding <strong>84</strong> to both sides of this equation yields <strong>84 = 7 k</strong>. Dividing both sides of this equation by <strong>7</strong> yields <strong>12 = k</strong>. Therefore, the x-intercept of the graph of <strong>y = f (x)</strong> in the xy-plane is <strong>(12, 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 C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "e62cfe5f",
    type: "spr",
    questionHtml:
      "According to a model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to four times the body weight of the bee, in grams. According to the model, what would be the head width, in millimeters, of a worker bumblebee that has a body weight of 0.5 grams?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 2.6. According to the model, the head width, in millimeters, of a worker bumblebee can be estimated by adding 0.6 to 4 times the body weight, in grams, of the bee. Let x represent the body weight, in grams, of a worker bumblebee and let y represent the head width, in millimeters. Translating the verbal description of the model into an equation yields <strong>y = 0 . 6 + 4 x</strong>. Substituting 0.5 grams for x in this equation yields <strong>y = 0 . 6 + 4 · 0 . 5</strong>, or <strong>y = 2 . 6</strong>. Therefore, a worker bumblebee with a body weight of 0.5 grams has an estimated head width of 2.6 millimeters. Note that 2.6 and 13/5 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "f0773a55",
    type: "mcq",
    questionHtml:
      "The line begins on the y axis above the x axis.<br>The line slants sharply up from left to right.<br><br>The graph represents the total charge, in dollars, by an electrician for <strong>x</strong> hours of work. The electrician charges a onetime fee plus an hourly rate. What is the best interpretation of the slope of the graph?",
    choices: [
      { label: "A", text: "The electrician’s hourly rate" },
      { label: "B", text: "The electrician’s onetime fee" },
      { label: "C", text: "The maximum amount that the electrician charges" },
      { label: "D", text: "The total amount that the electrician charges" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the electrician charges a onetime fee plus an hourly rate. It's also given that the graph represents the total charge, in dollars, for <strong>x</strong> hours of work. This graph shows a linear relationship in the xy-plane. Thus, the total charge <strong>y</strong>, in dollars, for <strong>x</strong> hours of work can be represented as <strong>y = m x + b</strong>, where <strong>m</strong> is the slope and <strong>(0, b)</strong> is the y-intercept of the graph of the equation in the xy-plane. Since the given graph represents the total charge, in dollars, by an electrician for <strong>x</strong> hours of work, it follows that its slope is <strong>m</strong>, or the electrician’s hourly rate.<br>Choice B is incorrect. The electrician's onetime fee is represented by the y-coordinate of the y-intercept, not the slope, of the graph.<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/f0773a55_svg1.svg",
  },
  {
    id: "f7e39fe9",
    type: "spr",
    questionHtml:
      "<strong>x</strong><br><strong>10</strong><br><strong>15</strong><br><strong>20</strong><br><strong>25</strong><br><br><strong>f (x)</strong><br><strong>82</strong><br><strong>137</strong><br><strong>192</strong><br><strong>247</strong><br><br>The table shows four values of <strong>x</strong> and their corresponding values of <strong>f (x)</strong>. There is a linear relationship between <strong>x</strong> and <strong>f (x)</strong> that is defined by the equation <strong>f (x) = m x − 28</strong>, where <strong>m</strong> is a constant. What is the value of <strong>m</strong>?",
    choices: [],
    correctAnswer: "11",
    explanation:
      "The correct answer is <strong>11</strong>. It's given that <strong>f (x)</strong> is defined by the equation <strong>f (x) = m x − 28</strong>, where <strong>m</strong> is a constant. It's also given in the table that when <strong>x = 10</strong>, <strong>f (x) = 82</strong>. Substituting <strong>10</strong> for <strong>x</strong> and <strong>82</strong> for <strong>f (x)</strong> in the equation <strong>f (x) = m x − 28</strong> yields, <strong>82 = m (10) − 28</strong>. Adding <strong>28</strong> to both sides of this equation yields <strong>110 = 10 m</strong>. Dividing both sides of this equation by <strong>10</strong> yields <strong>11 = m</strong>. Therefore, the value of <strong>m</strong> is <strong>11</strong>.",
    hasFigure: false,
  },
];

export const LINEAR_FUNC_HARD: PracticeQuestion[] = [
  {
    id: "023c0a8d",
    type: "mcq",
    questionHtml:
      "For the function f, if <strong>f of (3 x, ) = x − 6</strong> for all values of x, what is the value of <strong>f of 6</strong> ?",
    choices: [
      { label: "A", text: "<strong>−6</strong>" },
      { label: "B", text: "<strong>−4</strong>" },
      { label: "C", text: "<strong>0</strong>" },
      { label: "D", text: "<strong>2</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that <strong>f of 3 x = x − 6</strong> for all values of x. If <strong>3 x = 6</strong>, then <strong>f of 3 x</strong> will equal <strong>f of 6</strong>. Dividing both sides of <strong>3 x = 6</strong> by 3 gives <strong>x = 2</strong>. Therefore, substituting 2 for x in the given equation yields <strong>f of, (3 · 2, ) = 2 − 6</strong>, which can be rewritten as <strong>f of 6 = −4</strong>.Choice A is incorrect. This is the value of the constant in the given equation for f. Choice C is incorrect and may result from substituting <strong>x = 6</strong>, rather than <strong>x = 2</strong>, into the given equation. Choice D is incorrect. This is the value of x that yields <strong>f of 6</strong> for the left-hand side of the given equation; it’s not the value of <strong>f of 6</strong>.",
    hasFigure: false,
  },
  {
    id: "16889ef3",
    type: "mcq",
    questionHtml:
      "Oil and gas production in a certain area dropped from 4 million barrels in 2000 to 1.9 million barrels in 2013. Assuming that the oil and gas production decreased at a constant rate, which of the following linear functions f best models the production, in millions of barrels, t years after the year 2000?",
    choices: [
      {
        label: "A",
        text: "<strong>f of t = the fraction 21 over 130, end fraction, t + 4</strong>",
      },
      {
        label: "B",
        text: "<strong>f of t = the fraction 19 over 130, end fraction, t + 4</strong>",
      },
      {
        label: "C",
        text: "<strong>f of t = the − fraction 21 over 130, end fraction, t + 4</strong>",
      },
      {
        label: "D",
        text: "<strong>f of t = the − fraction 19 over 130, end fraction, t + 4</strong>",
      },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It is assumed that the oil and gas production decreased at a constant rate. Therefore, the function f that best models the production t years after the year 2000 can be written as a linear function, <strong>f of t = m, t + b</strong>, where m is the rate of change of the oil and gas production and b is the oil and gas production, in millions of barrels, in the year 2000. Since there were 4 million barrels of oil and gas produced in 2000, <strong>b = 4</strong>. The rate of change, m, can be calculated as <strong>the fraction with numerator 4 − 1 . 9, and denominator 0 − 13, end fraction = the − of the fraction 2 . 1 over 13</strong>, which is equivalent to <strong>the − of the fraction 21 over 130</strong>, the rate of change in choice C.Choices A and B are incorrect because each of these functions has a positive rate of change. Since the oil and gas production decreased over time, the rate of change must be negative. Choice D is incorrect. This model may result from misinterpreting 1.9 million barrels as the amount by which the production decreased.",
    hasFigure: false,
  },
  {
    id: "2b15d65f",
    type: "mcq",
    questionHtml:
      "An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?",
    choices: [
      { label: "A", text: "16,250" },
      { label: "B", text: "16,500" },
      { label: "C", text: "16,750" },
      { label: "D", text: "17,500" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Let the economist’s model be the linear function <strong>Q = m P + b</strong>, where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where <strong>y = Q</strong>. Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs <strong>P, Q</strong>, two points that satisfy the function are <strong>40, 20, 000</strong> and <strong>60, 15, 000</strong>. The slope m of the function can be found using the formula <strong>m = the fraction with numerator Q sub 2 − Q sub 1, and denominator P sub 2 − P sub 1, end fraction</strong>. Substituting the given values into this formula yields <strong>m = the fraction with numerator 15, 000 − 20, 000, and denominator 60 − 40, end fraction</strong>, or  <strong>m = −250</strong>. Therefore, <strong>Q = −250 P + b</strong>. The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point <strong>with coordinates 40, 20, 000</strong> yields <strong>20, 000 = −250 · 40 + b</strong>, or <strong>20, 000 = −10, 000 + b</strong>. Adding 10,000 to both sides of this equation yields <strong>b = 30, 000</strong>. Therefore, the linear function the economist used as the model is <strong>Q = −250 P + 30, 000</strong>. Substituting 55 for P yields <strong>Q = −250 · 55 + 30, 000 = 16, 250</strong>. It follows that when the selling price is $55 per unit, the demand is 16,250 units.Choices B, C, and D are incorrect and may result from calculation or conceptual errors.",
    hasFigure: false,
  },
  {
    id: "50f4cb9c",
    type: "mcq",
    questionHtml:
      "<strong>x</strong><br><strong>f (x)</strong><br><br><strong>1</strong><br><strong>−64</strong><br><br><strong>2</strong><br><strong>0</strong><br><br><strong>3</strong><br><strong>64</strong><br><br>For the linear function <strong>f</strong>, the table shows three values of <strong>x</strong> and their corresponding values of <strong>f (x)</strong>. Function <strong>f</strong> is defined by <strong>f (x) = a x + b</strong>, where <strong>a</strong> and <strong>b</strong> are constants. What is the value of <strong>a − b</strong>?",
    choices: [
      { label: "A", text: "<strong>−64</strong>" },
      { label: "B", text: "<strong>62</strong>" },
      { label: "C", text: "<strong>128</strong>" },
      { label: "D", text: "<strong>192</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The table gives that <strong>f (x) = 0</strong> when <strong>x = 2</strong>. Substituting 0 for <strong>f (x)</strong> and <strong>2</strong> for <strong>x</strong> into the equation <strong>f (x) = a x + b</strong>yields <strong>0 = 2 a + b</strong>. Subtracting <strong>2 a</strong> from both sides of this equation yields <strong>b = − 2 a</strong>. The table gives that <strong>f (x) = −64</strong> when <strong>x = 1</strong>. Substituting <strong>− 2 a</strong> for <strong>b</strong>, <strong>−64</strong> for <strong>f (x)</strong>, and <strong>1</strong> for <strong>x</strong> into the equation <strong>f (x) = a x + b</strong> yields <strong>−64 = a (1) + (−2 a)</strong>. Combining like terms yields <strong>−64 = −a</strong>, or <strong>a = 64</strong>. Since <strong>b = − 2 a</strong>, substituting <strong>64</strong> for <strong>a</strong> into this equation gives <strong>b = (−2) (64)</strong>, which yields <strong>b = −128</strong>. Thus, the value of <strong>a − b</strong> can be written as <strong>64 − (−128)</strong>, which is <strong>192</strong>.<br>Choice A is incorrect. This is the value of <strong>a + b</strong>, not <strong>a − b</strong>.<br>Choice B is incorrect. This is the value of <strong>a − 2</strong>, not <strong>a − b</strong>.<br>Choice C is incorrect. This is the value of <strong>2 a</strong>, not <strong>a − b</strong>.",
    hasFigure: false,
  },
  {
    id: "6989c80a",
    type: "mcq",
    questionHtml:
      "<strong>F (x) = nine fifths (x − 273.15) + 32</strong><br>The function <strong>F</strong> gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of <strong>x</strong> kelvins. If a temperature increased by <strong>2.10</strong> kelvins, by how much did the temperature increase, in degrees Fahrenheit?",
    choices: [
      { label: "A", text: "<strong>3.78</strong>" },
      { label: "B", text: "<strong>35.78</strong>" },
      { label: "C", text: "<strong>487.89</strong>" },
      { label: "D", text: "<strong>519.89</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the function <strong>F (x) = nine fifths (x − 273.15) + 32</strong> gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of <strong>x</strong> kelvins. A temperature that increased by <strong>2.10</strong> kelvins means that the value of <strong>x</strong> increased by <strong>2.10</strong> kelvins. It follows that an increase in <strong>x</strong> by <strong>2.10</strong> increases <strong>F (x)</strong> by <strong>nine fifths (2.10)</strong>, or <strong>3.78</strong>. Therefore, if a temperature increased by <strong>2.10</strong> kelvins, the temperature increased by <strong>3.78</strong> degrees Fahrenheit.<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: "78391fcc",
    type: "mcq",
    questionHtml:
      "The table above shows some values of x and their corresponding values <strong>f of x</strong> for the linear function f. What is the x-intercept of the graph of <strong>y = f of x</strong> in the xy-plane?",
    choices: [
      { label: "A", text: "<strong>−3, 0</strong>" },
      { label: "B", text: "<strong>−4, 0</strong>" },
      { label: "C", text: "<strong>−9, 0</strong>" },
      { label: "D", text: "<strong>−12, 0</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The equation of a linear function can be written in the form <strong>y = m x + b</strong>, where <strong>y = f of x</strong>, m is the slope of the graph of <strong>y = f of x</strong>, and b is the y-coordinate of the y-intercept of the graph. The value of m can be found using the slope formula, <strong>m = the fraction with numerator y sub 2 − y sub 1, and denominator x sub 2 − x sub 1, end fraction</strong>. According to the table, the points <strong>with coordinates − 11, 21</strong> and <strong>with coordinates − 10, 18</strong> lie on the graph of <strong>y = f of x</strong>. Using these two points in the slope formula yields<strong>m = the fraction with numerator 18 − 21, and denominator − 10 + 11, end fraction</strong>, or <strong>−3</strong>. Substituting <strong>−3</strong> for m in the slope-intercept form of the equation yields <strong>y = −3 x + b</strong>. The value of b can be found by substituting values from the table and solving; for example, substituting the coordinates of the point <strong>with coordinates − 11, 21</strong> into the equation <strong>y = −3 x + b</strong> gives <strong>21 = −3 · −11 + b</strong>, which yields <strong>b = −12</strong>. This means the function given by the table can be represented by the equation <strong>y = −3 x − 12</strong>. The value of the x-intercept of the graph of <strong>y = f of x</strong> can be determined by finding the value of x when <strong>y = 0</strong>. Substituting <strong>y = 0</strong> into <strong>y = −3 x − 12</strong> yields <strong>0 = −3 x − 12</strong>, or <strong>x = −4</strong>. This corresponds to the point <strong>with coordinates − 4, 0</strong>.Choice A is incorrect and may result from substituting the value of the slope for the x-coordinate of the x-intercept. Choice C is incorrect and may result from a calculation error. Choice D is incorrect and may result from substituting the y-coordinate of the y-intercept for the x-coordinate of the x-intercept.",
    hasFigure: false,
  },
  {
    id: "8c5e6702",
    type: "mcq",
    questionHtml:
      "A window repair specialist charges <strong>dollar sign 220</strong> for the first two hours of repair plus an hourly fee for each additional hour. The total cost for <strong>5</strong> hours of repair is <strong>dollar sign 400</strong>. Which function <strong>f</strong> gives the total cost, in dollars, for <strong>x</strong> hours of repair, where <strong>x > or = 2</strong>?",
    choices: [
      { label: "A", text: "<strong>f (x) = 60 x + 100</strong>" },
      { label: "B", text: "<strong>f (x) = 60 x + 220</strong>" },
      { label: "C", text: "<strong>f (x) = 80 x</strong>" },
      { label: "D", text: "<strong>f (x) = 80 x + 220</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the window repair specialist charges <strong>dollar sign 220</strong> for the first two hours of repair plus an hourly fee for each additional hour. Let <strong>n</strong> represent the hourly fee for each additional hour after the first two hours. Since it’s given that <strong>x</strong> is the number of hours of repair, it follows that the charge generated by the hourly fee after the first two hours can be represented by the expression <strong>n (x − 2)</strong>. Therefore, the total cost, in dollars, for <strong>x</strong> hours of repair is <strong>f (x) = 220 + n (x − 2)</strong>. It’s given that the total cost for <strong>5</strong> hours of repair is <strong>dollar sign 400</strong>. Substituting <strong>5</strong> for <strong>x</strong> and <strong>400</strong> for <strong>f (x)</strong> into the equation <strong>f (x) = 220 + n (x − 2)</strong> yields <strong>400 = 220 + n (5 − 2)</strong>, or <strong>400 = 220 + 3 n</strong>. Subtracting <strong>220</strong> from both sides of this equation yields <strong>180 = 3 n</strong>. Dividing both sides of this equation by <strong>3</strong> yields <strong>n = 60</strong>. Substituting <strong>60</strong> for <strong>n</strong> in the equation <strong>f (x) = 220 + n (x − 2)</strong> yields <strong>f (x) = 220 + 60 (x − 2)</strong>, which is equivalent to <strong>f (x) = 220 + 60 x − 120</strong>, or <strong>f (x) = 60 x + 100</strong>. Therefore, the total cost, in dollars, for <strong>x</strong> hours of repair is <strong>f (x) = 60 x + 100</strong>.<br>Choice B is incorrect. This function represents the total cost, in dollars, for <strong>x</strong> hours of repair where the specialist charges <strong>dollar sign 340</strong>, rather than <strong>dollar sign 220</strong>, for the first two hours of repair.<br>Choice C is incorrect. This function represents the total cost, in dollars, for <strong>x</strong> hours of repair where the specialist charges <strong>dollar sign 160</strong>, rather than <strong>dollar sign 220</strong>, for the first two hours of repair, and an hourly fee of <strong>dollar sign 80</strong>, rather than <strong>dollar sign 60</strong>, after the first two hours.<br>Choice D is incorrect. This function represents the total cost, in dollars, for <strong>x</strong> hours of repair where the specialist charges <strong>dollar sign 380</strong>, rather than <strong>dollar sign 220</strong>, for the first two hours of repair, and an hourly fee of <strong>dollar sign 80</strong>, rather than <strong>dollar sign 60</strong>, after the first two hours.",
    hasFigure: false,
  },
  {
    id: "a04050d8",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents a 3-column table, with 3 rows of data, titled “Energy per Gram of Typical Macronutrients.” The heading for column 1 is “Macronutrient, ” the heading for column 2 is “Food calories, ” and the heading for column 3 is “Kilojoules.” The 3 rows of data are as follows. Row 1.Macronutrient, Protein; Food calories, 4 . zero; Kilojoules, 16 . 7. Row 2. Macronutrient, Fat; Food calories, 9 . zero; Kilojoules, 37 . 7. Row 3. Macronutrient, Carbohydrate; Food calories, 4 . zero; Kilojoules, 16 . 7.</strong>The table above gives the typical amounts of energy per gram, expressed in both food calories and kilojoules, of the three macronutrients in food. If the 180 food calories in a granola bar come entirely from p grams of protein, f grams of fat, and c grams of carbohydrate, which of the following expresses f in terms of p and c ?",
    choices: [
      {
        label: "A",
        text: "<strong>f = 20 + the fraction 4 over 9, end fraction · (p + c, )</strong>",
      },
      {
        label: "B",
        text: "<strong>f = 20 − the fraction 4 over 9, end fraction · (p + c, )</strong>",
      },
      {
        label: "C",
        text: "<strong>f = 20 − the fraction 4 over 9, end fraction · (p − c, )</strong>",
      },
      {
        label: "D",
        text: "<strong>f = 20 + the fraction 9 over 4, end fraction · (p + c, )</strong>",
      },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It is given that there are 4.0 food calories per gram of protein, 9.0 food calories per gram of fat, and 4.0 food calories per gram of carbohydrate. If 180 food calories in a granola bar came from p grams of protein, f grams of fat, and c grams of carbohydrate, then the situation can be represented by the equation <strong>180 = 4 p + 9 f + 4 c</strong>. The equation can then be rewritten in terms of f by subtracting 4p and 4c from both sides of the equation and then dividing both sides of the equation by 9. The result is the equation <strong>f = 20 − four ninths · (p + c, )</strong>.Choices A, C, and D are incorrect and may be the result of not representing the situation with the correct equation or incorrectly rewriting the equation in terms of f.",
    hasFigure: true,
    figureUrl: "/practice-images/a04050d8_img1.png",
  },
  {
    id: "a309803e",
    type: "mcq",
    questionHtml:
      "One gallon of paint will cover <strong>220</strong> square feet of a surface. A room has a total wall area of <strong>w</strong> square feet. Which equation represents the total amount of paint <strong>P</strong>, in gallons, needed to paint the walls of the room twice?",
    choices: [
      { label: "A", text: "<strong>P = (w) / (110)</strong>" },
      { label: "B", text: "<strong>P = 440 w</strong>" },
      { label: "C", text: "<strong>P = (w) / (220)</strong>" },
      { label: "D", text: "<strong>P = 220 w</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It's given that <strong>w</strong> represents the total wall area, in square feet. Since the walls of the room will be painted twice, the amount of paint, in gallons, needs to cover <strong>2 w</strong> square feet. It’s also given that one gallon of paint will cover <strong>220</strong> square feet. Dividing the total area, in square feet, of the surface to be painted by the number of square feet covered by one gallon of paint gives the number of gallons of paint that will be needed. Dividing <strong>2 w</strong> by <strong>220</strong> yields  <strong>(2 w) / (220)</strong>, or <strong>(w) / (110)</strong>. Therefore, the equation that represents the total amount of paint <strong>P</strong>, in gallons, needed to paint the walls of the room twice is <strong>P = (w) / (110)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from finding the amount of paint needed to paint the walls once rather than twice.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "af2ba762",
    type: "mcq",
    questionHtml:
      "The constant 2.74 in this function estimates which of the following?",
    choices: [
      {
        label: "A",
        text: "The average monthly decrease in the price per gallon",
      },
      {
        label: "B",
        text: "The difference in the average price per gallon from September 1, 2014, to December 1, 2014",
      },
      { label: "C", text: "The average price per gallon on September 1, 2014" },
      { label: "D", text: "The average price per gallon on December 1, 2014" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since 2.74 is a constant term, it represents an actual price of gas rather than a measure of change in gas price. To determine what gas price it represents, find x such that F(x) = 2.74, or 2.74 = 2.74 – 0.19(x – 3). Subtracting 2.74 from both sides gives 0 = –0.19(x – 3). Dividing both sides by –0.19 results in 0 = x – 3, or x = 3. Therefore, the average price of gas is $2.74 per gallon 3 months after September 1, 2014, which is December 1, 2014.Choice A is incorrect. Since 2.74 is a constant, not a multiple of x, it cannot represent a rate of change in price. Choice B is incorrect. The difference in the average price from September 1, 2014, to December 1, 2014, is F(3) – F(0) = 2.74 – 0.19(3 – 3) – (2.74 – 0.19(0 – 3)) = 2.74 – (2.74 + 0.57) = –0.57, which is not 2.74. Choice C is incorrect. The average price per gallon on September 1, 2014, is F(0) = 2.74 – 0.19(0 – 3) = 2.74 + 0.57 = 3.31, which is not 2.74.",
    hasFigure: false,
  },
  {
    id: "b3abf40f",
    type: "mcq",
    questionHtml:
      "<strong>F (x) = nine fifths (x − 273.15) + 32</strong><br>The function <strong>F</strong> gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of <strong>x</strong> kelvins. If a temperature increased by <strong>9.10</strong> kelvins, by how much did the temperature increase, in degrees Fahrenheit?",
    choices: [
      { label: "A", text: "<strong>16.38</strong>" },
      { label: "B", text: "<strong>48.38</strong>" },
      { label: "C", text: "<strong>475.29</strong>" },
      { label: "D", text: "<strong>507.29</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. It’s given that the function <strong>F (x) = nine fifths (x − 273.15) + 32</strong> gives the temperature, in degrees Fahrenheit, that corresponds to a temperature of <strong>x</strong> kelvins. A temperature that increased by <strong>9.10</strong> kelvins means that the value of <strong>x</strong> increased by <strong>9.10</strong> kelvins. It follows that an increase in <strong>x</strong> by <strong>9.10</strong> increases <strong>F (x)</strong> by <strong>nine fifths (9.10)</strong>, or <strong>16.38</strong>. Therefore, if a temperature increased by <strong>9.10</strong> kelvins, the temperature increased by <strong>16.38</strong> degrees Fahrenheit. <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: "b988eeec",
    type: "spr",
    questionHtml:
      "The functions <strong>f</strong> and <strong>g</strong> are defined as <strong>f (x) = one fourth x − 9</strong> and <strong>g (x) = three fourths x + 21</strong>. If the function <strong>h</strong> is defined as <strong>h (x) = f (x) + g (x)</strong>, what is the x-coordinate of the x-intercept of the graph of <strong>y = h (x)</strong> in the xy-plane?",
    choices: [],
    correctAnswer: "-12",
    explanation:
      "The correct answer is <strong>−12</strong>. It's given that the functions <strong>f</strong> and <strong>g</strong> are defined as <strong>f (x) = one fourth x − 9</strong> and <strong>g (x) = three fourths x + 21</strong>. If the function <strong>h</strong> is defined as <strong>h (x) = f (x) + g (x)</strong>, then substituting <strong>one fourth x − 9</strong> for <strong>f (x)</strong> and <strong>three fourths x + 21</strong> for <strong>g (x)</strong> in this function yields <strong>h (x) = one fourth x − 9 + three fourths x + 21</strong>. This can be rewritten as <strong>h (x) = four fourths x + 12</strong>, or <strong>h (x) = x + 12</strong>. The x-intercept of a graph in the xy-plane is the point on the graph where <strong>y = 0</strong>. The equation representing the graph of <strong>y = h (x)</strong> is <strong>y = x + 12</strong>. Substituting <strong>0</strong> for <strong>y</strong> in this equation yields <strong>0 = x + 12</strong>. Subtracting <strong>12</strong> from both sides of this equation yields <strong>−12 = x</strong>, or <strong>x = −12</strong>. Therefore, the x-coordinate of the x-intercept of the graph of <strong>y = h (x)</strong> in the xy-plane is <strong>−12</strong>.",
    hasFigure: false,
  },
  {
    id: "bbf9e5ce",
    type: "mcq",
    questionHtml:
      "For groups of <strong>25</strong> or more people, a museum charges <strong>dollar sign 21</strong> per person for the first <strong>25</strong> people and <strong>dollar sign 14</strong> for each additional person. Which function <strong>f</strong> gives the total charge, in dollars, for a tour group with <strong>n</strong> people, where <strong>n > or = 25</strong>?",
    choices: [
      { label: "A", text: "<strong>f (n) = 14 n + 175</strong>" },
      { label: "B", text: "<strong>f (n) = 14 n + 525</strong>" },
      { label: "C", text: "<strong>f (n) = 35 n − 350</strong>" },
      { label: "D", text: "<strong>f (n) = 14 n + 21</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. A tour group with <strong>n</strong> people, where <strong>n > or = 25</strong>, can be split into two subgroups: the first <strong>25</strong> people and the additional <strong>n − 25</strong> people. Since the museum charges <strong>dollar sign 21</strong> per person for the first <strong>25</strong> people and <strong>dollar sign 14</strong> for each additional person, the charge for the first <strong>25</strong> people is <strong>dollar sign 21 (25)</strong> and the charge for the additional <strong>n − 25</strong> people is <strong>dollar sign 14 (n − 25)</strong>. Therefore, the total charge, in dollars, is given by the function <strong>f (n) = 21 (25) + 14 (n − 25)</strong>, or <strong>f (n) = 14 n + 175</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: "be9cb6a2",
    type: "mcq",
    questionHtml:
      "The cost of renting a backhoe for up to <strong>10</strong> days is <strong>dollar sign 270</strong> for the first day and <strong>dollar sign 135</strong> for each additional day. Which of the following equations gives the cost <strong>y</strong>, in dollars, of renting the backhoe for <strong>x</strong> days, where <strong>x</strong> is a positive integer and <strong>x < or = 10</strong>?",
    choices: [
      { label: "A", text: "<strong>y = 270 x − 135</strong>" },
      { label: "B", text: "<strong>y = 270 x + 135</strong>" },
      { label: "C", text: "<strong>y = 135 x + 270</strong>" },
      { label: "D", text: "<strong>y = 135 x + 135</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that the cost of renting a backhoe for up to <strong>10</strong> days is <strong>dollar sign 270</strong> for the first day and <strong>dollar sign 135</strong> for each additional day. Therefore, the cost <strong>y</strong>, in dollars, for <strong>x</strong> days, where <strong>x < or = 10</strong>, is the sum of the cost for the first day, <strong>dollar sign 270</strong>, and the cost for the additional <strong>x − 1</strong> days, <strong>dollar sign 135 (x − 1)</strong>. It follows that <strong>y = 270 + 135 (x − 1)</strong>, which is equivalent to <strong>y = 270 + 135 x − 135</strong>, or <strong>y = 135 x + 135</strong>.<br>Choice A is incorrect. This equation represents a situation where the cost of renting a backhoe is <strong>dollar sign 135</strong> for the first day and <strong>dollar sign 270</strong> for each additional day.<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: "daad7c32",
    type: "mcq",
    questionHtml:
      "An object hangs from a spring. The formula <strong>l = 30 + 2 w</strong> relates the length <strong>l</strong>, in centimeters, of the spring to the weight w, in newtons, of the object. Which of the following describes the meaning of the 2 in this context?",
    choices: [
      {
        label: "A",
        text: "The length, in centimeters, of the spring with no weight attached",
      },
      {
        label: "B",
        text: "The weight, in newtons, of an object that will stretch the spring 30 centimeters",
      },
      {
        label: "C",
        text: "The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring",
      },
      {
        label: "D",
        text: "The increase in the length, in centimeters, of the spring for each one-newton increase in the weight of the object",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The value 2 is multiplied by w, the weight of the object. When the weight is 0, the length is 30 + 2(0) = 30 centimeters. If the weight increases by w newtons, the length increases by 2w centimeters, or 2 centimeters for each one-newton increase in weight.<br> <br><br>Choice A is incorrect because this describes the value 30. Choice B is incorrect because 30 represents the length of the spring before it has been stretched. Choice C is incorrect because this describes the value w.",
    hasFigure: false,
  },
];
