SAT Math Quiz: Linear Functions
20 questions · exam conditions
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Linear FunctionsQuestion 1 of 20

For the linear function g(x)=−4x+9g(x)=-4x+9, for what value of xx does g(x)=−7g(x)=-7?

−74-\frac{7}{4}
44
74\frac{7}{4}
−4-4
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SAT Math Quiz

SAT Math Quiz: Linear Functions

Practice Linear Functions in SAT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Linear Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

For the linear function g(x)=−4x+9g(x)=-4x+9, for what value of xx does g(x)=−7g(x)=-7?

  1. −74-\frac{7}{4}
  2. 44 (correct answer)
  3. 74\frac{7}{4}
  4. −4-4
Explanation: We need to find x when g(x) = -7, where g(x) = -4x + 9. Set up the equation: -4x + 9 = -7. Solve for x by first subtracting 9 from both sides: -4x = -7 - 9 = -16. Then divide both sides by -4: x = -16/(-4) = 4. We can verify: g(4) = -4(4) + 9 = -16 + 9 = -7 ✓. A common error is sign confusion when dividing by negative numbers. Remember that a negative divided by a negative gives a positive result.

Question 2

Let f(x)=2.5x−6f(x)=2.5x-6. What is the value of f(8)f(8)?

  1. 1414 (correct answer)
  2. 2626
  3. −26-26
  4. 88
Explanation: We need to evaluate f(8) where f(x) = 2.5x - 6. To find f(8), substitute 8 for x in the function: f(8) = 2.5(8) - 6. Calculate step by step: 2.5 × 8 = 20, then 20 - 6 = 14. Therefore, f(8) = 14. A common error is arithmetic mistakes when multiplying decimals; converting 2.5 to 5/2 can help avoid this. When evaluating functions, always substitute carefully and perform operations in the correct order.

Question 3

A line has slope −32-\frac{3}{2} and passes through the point (4,7)(4,7). Which equation represents the line in point-slope form?

  1. y+7=−32(x−4)y+7=-\frac{3}{2}(x-4)
  2. y−7=−32(x−4)y-7=-\frac{3}{2}(x-4) (correct answer)
  3. y−4=−32(x−7)y-4=-\frac{3}{2}(x-7)
  4. y−7=32(x−4)y-7=\frac{3}{2}(x-4)
Explanation: The question asks for point-slope form of a line with slope -3/2 passing through (4, 7). Point-slope form is y - y1 = m(x - x1), so substituting m = -3/2, x1 = 4, and y1 = 7 gives y - 7 = -3/2(x - 4), the equation that subtracts 7 on the left and 4 inside the parentheses while keeping the negative coefficient. The version written with y + 7 mishandles the sign, since subtracting a positive 7 yields y - 7; that form would correspond to the point (4, -7). The version with y - 4 and (x - 7) swaps the coordinates, dropping the x-value into the y-slot. And the version with a positive 3/2 out front loses the negative sign, describing a line that rises rather than falls.

Question 4

A line has equation y=13x+4y=\tfrac{1}{3}x+4. Which statement correctly interprets the slope in this context of yy changing with xx?

  1. When xx increases by 4, yy increases by 1
  2. When xx increases by 1, yy increases by 3
  3. When xx increases by 3, yy decreases by 1
  4. When xx increases by 3, yy increases by 1 (correct answer)
Explanation: In y = (1/3)x + 4 the slope is the coefficient 1/3, and slope means change in y divided by change in x. So a run of 3 produces a rise of 3 times 1/3, which equals 1. That is exactly the statement that when x increases by 3, y increases by 1. The statement that y increases by 3 when x increases by 1 flips the fraction over, treating the slope as 3 rather than 1/3. The statement that y increases by 1 when x increases by 4 pulls the 4 from the y-intercept; the intercept tells you where the line sits, not how steeply it climbs. The statement that y decreases by 1 when x increases by 3 gets the size right but the direction wrong, since a positive slope means y rises as x rises.

Question 5

A phone plan charges a base fee plus a constant amount per minute. A 60-minute bill is 17 dollars, and a 110-minute bill is 27 dollars. According to the linear model, what will a 200-minute bill cost?

  1. 40
  2. 45 (correct answer)
  3. 50
  4. 42
Explanation: The rate is (27-17)/(110-60) = 0.2 dollars/minute, so the base fee is 17 - 0.2(60) = 5. The cost for 200 minutes is 0.2(200) + 5 = 45.

Question 6

A line passes through (3,8)(3,8) and has slope 13\frac{1}{3}. Which equation is the line in point-slope form?

  1. y−8=13(x−3)y-8=\frac{1}{3}(x-3) (correct answer)
  2. y−3=13(x−8)y-3=\frac{1}{3}(x-8)
  3. y−8=3(x−3)y-8=3(x-3)
  4. y+8=13(x+3)y+8=\frac{1}{3}(x+3)
Explanation: The question asks for the point-slope form of a line through (3,8) with slope 1/3. The form is y - y1 = m(x - x1), so y - 8 = (1/3)(x - 3). This directly uses the given point and slope. It models the linear equation without expanding. A common error is switching x and y coordinates, like y-3=(1/3)(x-8). Another mistake might be using reciprocal slope like 3. For point-slope, plug in the known point and slope carefully.

Question 7

A line has yy-intercept 5 and passes through the point (2,1)(2,1). Which equation represents the line?

  1. y=−2x+5y=-2x+5 (correct answer)
  2. y=2x+5y=2x+5
  3. y=−12x+5y=-\frac{1}{2}x+5
  4. y=12x−5y=\frac{1}{2}x-5
Explanation: The question asks for the equation of a line with y-intercept 5 passing through (2,1). Find slope m=(1-5)/(2-0)=-4/2=-2. Thus, y=-2x+5. This models the line accurately. A key error is using (2,1) as intercept, miscounting slope as 1/2. Another mistake could be positive slope calculation. When given intercept and point, compute slope then write in slope-intercept form.

Question 8

A line is graphed on a coordinate plane. Moving from the point (1,3)(1,3) to (5,1)(5,1) on the line, what is the slope?

  1. −2-2
  2. −12-\frac{1}{2} (correct answer)
  3. 12\frac{1}{2}
  4. 22
Explanation: The question asks for the slope of a line from (1,3) to (5,1). Use m=(1-3)/(5-1)=-2/4=-1/2. This negative slope indicates decrease. The calculation shows a drop of 2 over run of 4. A common error is sign reversal, getting positive 1/2. Another mistake might be dividing incorrectly like -2/2=-1. For slope from points, always subtract y's over x's consistently to understand direction.

Question 9

Which function gives the total paid P (in dollars) after m months?

  1. P(m)=20m+45P(m)=20m+45
  2. P(m)=45m+20P(m)=45m+20 (correct answer)
  3. P(m)=45m+65P(m)=45m+65
  4. P(m)=225m+65P(m)=225m+65
Explanation: Slope = (290 - 65)/(6 - 1) = 45 and using P(1) = 65 gives 45(1) + b = 65, so b = 20; thus P(m) = 45m + 20. Other options swap slope/intercept or use the difference 225 as a slope.

Question 10

A company's profit PP (in dollars) is linear in the number of items sold xx. If P=200P=200 when x=50x=50 and P=440P=440 when x=110x=110, what is the rate of change of profit per item?

  1. 22
  2. 66
  3. 33
  4. 44 (correct answer)
Explanation: This problem asks for the rate of change of profit per item sold. Given P = 200 when x = 50 and P = 440 when x = 110, the rate of change is the slope: (440 - 200)/(110 - 50) = 240/60 = 4. This means profit increases by $4 for each additional item sold. In context, rate of change represents how much one quantity changes per unit change in another. Don't confuse the rate of change with the total change or the profit values themselves.

Question 11

A salesperson earns a weekly base pay plus a commission per item sold. In a week with 12 items sold, the pay is $460, and in a week with 20 items sold, the pay is $620. Assuming a linear relationship, what is the commission per item?

  1. $18
  2. $22
  3. $20 (correct answer)
  4. $16
Explanation: Let b be the base pay and c be the commission per item. We have two equations: b+12c=460b + 12c = 460 and b+20c=620b + 20c = 620. Subtracting the first from the second: 8c=1608c = 160, so c=20c = 20. The commission per item is 2020. To verify: b=460−12(20)=460−240=220b = 460 - 12(20) = 460 - 240 = 220, and checking: 220+20(20)=220+400=620220 + 20(20) = 220 + 400 = 620 ✓. A common error is setting up the equations incorrectly or making arithmetic mistakes when solving the system. When dealing with pay structures, clearly define your variables before writing equations.

Question 12

A streaming plan follows y=5x+12y = 5x + 12, where xx is the number of add-on playlists purchased in a month and yy is the total monthly cost in dollars. What is the total monthly cost if no add-on playlists are purchased?

  1. 12 (correct answer)
  2. 5
  3. 17
  4. 60
Explanation: When x=0x=0, the cost is the y-intercept, 12 dollars. The other values use the slope, add incorrectly, or multiply the slope and intercept.

Question 13

A car rental company charges a fixed daily fee plus a constant amount per mile. For a 1-day rental, 100 miles cost 95 dollars, and 220 miles cost 131 dollars. What is the per-mile charge (in dollars per mile)?

  1. 0.36
  2. 0.3 (correct answer)
  3. 36
  4. 0.12
Explanation: The slope is $(131-95)/(220-100) = 36/120 = 0.30 dollars per mile. The other choices come from miscomputing the slope or using incorrect units.

Question 14

A line has slope m=−34m=-\frac{3}{4} and passes through the point (4,2)(4,2). Which equation is the line in point-slope form?

  1. y−2=−43(x−4)y-2=-\frac{4}{3}(x-4)
  2. y+2=−34(x−4)y+2=-\frac{3}{4}(x-4)
  3. y−2=−34(x−4)y-2=-\frac{3}{4}(x-4) (correct answer)
  4. y−4=−34(x−2)y-4=-\frac{3}{4}(x-2)
Explanation: The question asks for the point-slope form of a line with slope m = -3/4 passing through the point (4, 2). The point-slope form is y - y1 = m(x - x1), so substituting gives y - 2 = -3/4 (x - 4), which is choice C. This form highlights the slope and a specific point on the line. A common computational error is inverting the fraction in the slope, like using -4/3 instead of -3/4, as in choice A. Another mistake is swapping the x and y coordinates in the formula, leading to something like y - 4 = -3/4 (x - 2) as in choice D. To verify, you can convert to slope-intercept form: y = -3/4 x + 3 + 2 = -3/4 x + 5, and check the point satisfies it. A strategy for these problems is to remember point-slope is useful when you know a point and slope directly.

Question 15

For the linear function f(x)=−4x+9f(x)=-4x+9, for what value of xx does f(x)=−7f(x)=-7?

  1. −12-\frac{1}{2}
  2. 44 (correct answer)
  3. −4-4
  4. −74-\frac{7}{4}
Explanation: We need to solve for xx when f(x)=−7f(x) = -7 given f(x)=−4x+9f(x) = -4x + 9. Setting up the equation: −7=−4x+9-7 = -4x + 9. Subtracting 9 from both sides: −16=−4x-16 = -4x. Dividing by -4: x=4x = 4. We can verify: f(4)=−4(4)+9=−16+9=−7f(4) = -4(4) + 9 = -16 + 9 = -7 ✓. Common errors include sign mistakes when solving or forgetting to divide by the coefficient of xx. When solving linear equations, always verify your answer by substituting back into the original function.

Question 16

A line has slope −32-\tfrac{3}{2} and passes through the point (4,1)(4,1). Which equation is the line in point-slope form?

  1. y−1=−32(x−4)y-1=-\frac{3}{2}(x-4) (correct answer)
  2. y−4=−32(x−1)y-4=-\frac{3}{2}(x-1)
  3. y+1=−32(x−4)y+1=-\frac{3}{2}(x-4)
  4. y−1=32(x−4)y-1=\frac{3}{2}(x-4)
Explanation: We need to write the equation of a line with slope −32-\frac{3}{2} passing through (4,1)(4,1) in point-slope form. Point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope. Substituting our values: y−1=−32(x−4)y - 1 = -\frac{3}{2}(x - 4). Common errors include using the wrong signs or confusing point-slope form with slope-intercept form. Remember that in point-slope form, we subtract the coordinates of the given point, not add them.

Question 17

The temperature TT (in degrees) changes linearly with time tt (in hours). At t=1t=1, T=68T=68, and at t=5t=5, T=60T=60. Which equation models TT as a function of tt?

  1. T=2t+66T=2t+66
  2. T=−2t+70T=-2t+70 (correct answer)
  3. T=−12t+68.5T=-\frac{1}{2}t+68.5
  4. T=12t+67.5T=\frac{1}{2}t+67.5
Explanation: Temperature T changes linearly with time t, with T = 68 when t = 1 and T = 60 when t = 5. The slope is m = (60 - 68)/(5 - 1) = -8/4 = -2 degrees per hour. Using point-slope form with (1, 68): T - 68 = -2(t - 1), which gives T = -2t + 2 + 68 = -2t + 70. The equation is T = -2t + 70. Common errors include mixing up which variable is dependent (T) versus independent (t), or sign errors in the slope calculation. In physics contexts, negative slope often indicates a decreasing quantity over time.

Question 18

A line passes through the points (−2,5)(-2,5) and (4,−1)(4,-1). What is the slope of the line?

  1. −1-1 (correct answer)
  2. 11
  3. −32-\frac{3}{2}
  4. −23-\frac{2}{3}
Explanation: To find the slope between two points, we use the formula m = (y₂ - y₁)/(x₂ - x₁). Using points (-2, 5) and (4, -1), we calculate: m = (-1 - 5)/(4 - (-2)) = -6/6 = -1. The key is to be consistent with which point is (x₁, y₁) and which is (x₂, y₂), and to be careful with negative signs when subtracting. A common error is forgetting that subtracting a negative becomes addition in the denominator. When you see two points, immediately think slope formula.

Question 19

For the linear function g(x)=5x−20g(x)=5x-20, for what value of xx does g(x)=0g(x)=0?

  1. 2020
  2. 00
  3. 44 (correct answer)
  4. −4-4
Explanation: We need to find the x-value where g(x) = 0 for the function g(x) = 5x - 20. Setting g(x) = 0: 5x - 20 = 0, so 5x = 20, which gives x = 4. This is the x-intercept of the function. A common error is to find the y-intercept instead (when x = 0, g(0) = -20) or to solve 5x - 20 = x, confusing the function value with the input. To find where a linear function equals zero, set the function equal to zero and solve for x.

Question 20

A linear function has slope −4-4 and yy-intercept 99. What is the value of the function when x=3x=3?

  1. −3-3 (correct answer)
  2. −21-21
  3. 2121
  4. −13-13
Explanation: We need to find the value of a linear function with slope -4 and y-intercept 9 when x = 3. The function is f(x) = -4x + 9. Evaluating at x = 3: f(3) = -4(3) + 9 = -12 + 9 = -3. A common error is to make arithmetic mistakes with negative numbers, such as computing -4(3) as -7 or forgetting the negative sign entirely. When evaluating linear functions, write out each step to avoid sign errors.