Linear Functions practice questions

Linear Functions questions

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1. 1Linear functionsFoundation

   For the line y = 2x - 3, what is the value of y when x = 4?For the line y = 2x - 3, what is the value of y when x = 4?

   • A88
   • B22
   • C1111
   • D55
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   Answer: 55

   Substitute 4 for x: y = 2(4) - 3 = 8 - 3 = 5. Evaluating a line at a point is only substitution — there is nothing to solve.Substitute 4 for x: y = 2(4) - 3 = 8 - 3 = 5. Evaluating a line at a point is only substitution — there is nothing to solve.

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2. 2Linear functionsFoundation

   A line passes through (0, 7) and has slope -2. What is the value of y when x = 4?A line passes through (0, 7) and has slope -2. What is the value of y when x = 4?

   • A11
   • B1515
   • C-1-1
   • D-15-15
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   Answer: -1-1

   The line is y = -2x + 7. Substituting x = 4 gives y = -8 + 7 = -1.The line is y = -2x + 7. Substituting x = 4 gives y = -8 + 7 = -1.

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3. 3Linear functionsFoundation

   The cost C, in dollars, of renting a bicycle for h hours is C = 4 + 3h. Enter the number of hours that gives a cost of 25 dollars.The cost C, in dollars, of renting a bicycle for h hours is C = 4 + 3h. Enter the number of hours that gives a cost of 25 dollars.

   Student-produced response: write your own answer rather than choosing one.

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   Answer: 77

   Set the rule equal to the total and solve: 4 + 3h = 25, so 3h = 21 and h = 7 hours. The 4 is a fixed charge, not part of the hourly rate, so it comes off before dividing.Set the rule equal to the total and solve: 4 + 3h = 25, so 3h = 21 and h = 7 hours. The 4 is a fixed charge, not part of the hourly rate, so it comes off before dividing.

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4. 4Linear functionsStandard

   A line has slope 4 and passes through (2, 3). Which equation describes the line?A line has slope 4 and passes through (2, 3). Which equation describes the line?

   • Ay = 4x + 3y = 4x + 3
   • By - 3 = 4(x - 2)y - 3 = 4(x - 2)
   • Cy + 3 = 4(x + 2)y + 3 = 4(x + 2)
   • Dy - 2 = 4(x - 3)y - 2 = 4(x - 3)
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   Answer: y - 3 = 4(x - 2)y - 3 = 4(x - 2)

   Point-slope form is y - y₁ = m(x - x₁). Substituting m = 4 and the point (2, 3) gives y - 3 = 4(x - 2). Note that the point's coordinates are subtracted, so both signs are negative here.Point-slope form is y - y1 = m(x - x1). Substituting m = 4 and the point (2, 3) gives y - 3 = 4(x - 2). Note that the point's coordinates are subtracted, so both signs are negative here.

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5. 5Linear functionsStandard

   A line has x-intercept 4 and y-intercept -6. What is its slope?A line has x-intercept 4 and y-intercept -6. What is its slope?

   • A232/3
   • B-23-2/3
   • C-1.5-1.5
   • D1.51.5
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   Answer: 1.51.5

   The line passes through (4, 0) and (0, -6). Slope is 0 - (-6)4 - 0 = 64 = 1.5.The line passes through (4, 0) and (0, -6). Slope is (0 - (-6))/(4 - 0) = 6/4 = 1.5.

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6. 6Linear functionsStandard

   A line passes through (0, -3) and (4, 5). What is its equation?A line passes through (0, -3) and (4, 5). What is its equation?

   • Ay = 2x + 3y = 2x + 3
   • By = -2x - 3y = -2x - 3
   • Cy = 2x - 3y = 2x - 3
   • Dy = (12)x - 3y = (1/2)x - 3
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   Answer: y = 2x - 3y = 2x - 3

   The slope is 5 - (-3)4 - 0 = 84 = 2, and the point (0, -3) is the y-intercept, so y = 2x - 3. Inverting the slope calculation gives the 12, and dropping the sign on the intercept gives y = 2x + 3.The slope is (5 - (-3))/(4 - 0) = 8/4 = 2, and the point (0, -3) is the y-intercept, so y = 2x - 3. Inverting the slope calculation gives the 1/2, and dropping the sign on the intercept gives y = 2x + 3.

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7. 7Linear functionsStandard

   A line has a y-intercept of 5 and a slope of -3. What is y when x = 4?A line has a y-intercept of 5 and a slope of -3. What is y when x = 4?

   • A-7-7
   • B1717
   • C22
   • D-12-12
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   Answer: -7-7

   Substituting into y = -3x + 5 gives -12 + 5 = -7. Dropping the sign on the slope gives 17, and forgetting the intercept gives -12.Substituting into y = -3x + 5 gives -12 + 5 = -7. Dropping the sign on the slope gives 17, and forgetting the intercept gives -12.

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8. 8Linear functionsChallenge

   A line is perpendicular to y = -2x + 5 and passes through the origin. What is its equation?A line is perpendicular to y = -2x + 5 and passes through the origin. What is its equation?

   • Ay = (12)xy = (1/2)x
   • By = 2xy = 2x
   • Cy = -2xy = -2x
   • Dy = -(12)xy = -(1/2)x
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   Answer: y = (12)xy = (1/2)x

   Perpendicular slopes are negative reciprocals: the negative reciprocal of -2 is 12. A line through the origin has no constant term, so y = (12)x.Perpendicular slopes are negative reciprocals: the negative reciprocal of -2 is 1/2. A line through the origin has no constant term, so y = (1/2)x.

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9. 9Linear functionsChallenge

   A line passes through (-2, 7) and (4, -5). At what value of x does it cross the x-axis?A line passes through (-2, 7) and (4, -5). At what value of x does it cross the x-axis?

   • A3.03.0
   • B1.51.5
   • C-1.5-1.5
   • D2.02.0
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   Answer: 1.51.5

   The slope is -5 - 74 - (-2) = -126 = -2. Using (4, -5): -5 = -2(4) + b gives b = 3, so y = -2x + 3. Setting y = 0 gives x = 1.5.The slope is (-5 - 7)/(4 - (-2)) = -12/6 = -2. Using (4, -5): -5 = -2(4) + b gives b = 3, so y = -2x + 3. Setting y = 0 gives x = 1.5.

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10. 10Linear functionsChallenge

    Two lines are perpendicular. One has equation y = (34)x - 2. What is the slope of the other?Two lines are perpendicular. One has equation y = (3/4)x - 2. What is the slope of the other?

    • A-34-3/4
    • B-43-4/3
    • C343/4
    • D434/3
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    Answer: -43-4/3

    Perpendicular slopes are negative reciprocals: flip 34 to 43 and change the sign, giving -43.Perpendicular slopes are negative reciprocals: flip 3/4 to 4/3 and change the sign, giving -4/3.

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11. 11Linear functionsChallenge

    A linear function f satisfies f(2) = 11 and f(7) = 26. What is f(0)?A linear function f satisfies f(2) = 11 and f(7) = 26. What is f(0)?

    • A88
    • B33
    • C1111
    • D55
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    Answer: 55

    The slope is 26 - 117 - 2 = 3. Working back from f(2) = 11 by two steps of 3 gives f(0) = 11 - 6 = 5. The choice 3 is the slope itself and 11 is the value at x = 2.The slope is (26 - 11)/(7 - 2) = 3. Working back from f(2) = 11 by two steps of 3 gives f(0) = 11 - 6 = 5. The choice 3 is the slope itself and 11 is the value at x = 2.

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