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

The graph of the linear function hh is shown in the xyxy-plane. If h(x)=mx+bh(x)=mx+b, what is the value of h(b)−h(m)h(b)-h(m)?

Question graphic
−14-14
1414
−10-10
−2-2
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PSAT Math Quiz

PSAT Math Quiz: Linear Functions

Practice Linear Functions in PSAT 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 PSAT 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

The graph of the linear function hh is shown in the xyxy-plane. If h(x)=mx+bh(x)=mx+b, what is the value of h(b)−h(m)h(b)-h(m)?

  1. −14-14 (correct answer)
  2. 1414
  3. −10-10
  4. −2-2

Explanation: The graph shows slope m=−2m=-2 and yy-intercept b=5b=5, so h(x)=−2x+5h(x)=-2x+5. Then h(b)=h(5)=−2(5)+5=−5h(b)=h(5)=-2(5)+5=-5 and h(m)=h(−2)=−2(−2)+5=9h(m)=h(-2)=-2(-2)+5=9. Therefore h(b)−h(m)=−5−9=−14h(b)-h(m)=-5-9=-14. Choice B is the magnitude with wrong sign. Choice C omits the constant in one evaluation. Choice D comes from computing h(b−m)h(b-m) instead.

Question 2

The graph shown represents a linear function ff on the interval −4≤x≤6-4\le x\le 6. Let g(x)=f(2x)+3g(x)=f(2x)+3. What is the maximum value of gg on the interval −2≤x≤3-2\le x\le 3?

  1. 1111 (correct answer)
  2. 88
  3. 1414
  4. 55

Explanation: From the graph, f(x)=−x+4f(x)=-x+4 (slope −1-1, yy-intercept 4). Then g(x)=−(2x)+4+3=−2x+7g(x)=-(2x)+4+3=-2x+7. Since gg is decreasing, its max on [−2,3][-2,3] occurs at x=−2x=-2: g(−2)=−2(−2)+7=11g(-2)=-2(-2)+7=11. Choice B omits the +3+3 shift. Choice C evaluates at wrong endpoint. Choice D evaluates at x=1x=1.

Question 3

The graph of a linear function gg is shown in the xyxy-plane. The function hh is defined by h(x)=g(x−3)+4h(x)=g(x-3)+4. What is the xx-intercept of the graph of hh?

  1. (5,0)(5,0) (correct answer)
  2. (−1,0)(-1,0)
  3. (7,0)(7,0)
  4. (1,0)(1,0)

Explanation: From the graph, line gg passes through (0,−8)(0,-8) and (4,0)(4,0), so it has slope 0−(−8)4−0=2\frac{0-(-8)}{4-0}=2 and equation g(x)=2x−8g(x)=2x-8. The transformed function is h(x)=g(x−3)+4=2(x−3)−8+4=2x−6−8+4=2x−10h(x)=g(x-3)+4=2(x-3)-8+4=2x-6-8+4=2x-10. To find the xx-intercept, set h(x)=0h(x)=0: 2x−10=02x-10=0, so x=5x=5. The xx-intercept is (5,0)(5,0).

Question 4

A line passes through the points (2,−1)(2, -1) and (−4,11)(-4, 11). What is the equation of the line in slope-intercept form y=mx+by = mx + b?

  1. y=−2x+3y = -2x + 3 (correct answer)
  2. y=2x−5y = 2x - 5
  3. y=−2x−5y = -2x - 5
  4. y=2x+3y = 2x + 3

Explanation: We need to find the equation of the line passing through (2, -1) and (-4, 11) in the form y = mx + b. First, calculate the slope: m = (11 - (-1))/(-4 - 2) = 12/(-6) = -2. Now use point-slope form with (2, -1): y - (-1) = -2(x - 2), which gives y + 1 = -2x + 4, so y = -2x + 3. The most common error is sign mistakes when subtracting negative numbers in the slope calculation. Always double-check by substituting both original points back into your final equation to verify they satisfy it.

Question 5

The function ff is linear and satisfies f(1)=9f(1)=9 and f(5)=1f(5)=1. What is the value of f(3)f(3)? Incorrect approaches often compute the slope correctly but apply it from the wrong starting point.

  1. 33
  2. 55 (correct answer)
  3. 77
  4. 99

Explanation: This question asks for f(3) given that f is linear with f(1) = 9 and f(5) = 1. First, find the slope: m = (1 - 9)/(5 - 1) = -8/4 = -2. Since f is linear, it decreases by 2 for each unit increase in x. From x = 1 to x = 3 is an increase of 2 units, so f(3) = f(1) + 2(-2) = 9 - 4 = 5. A common error is computing the slope correctly but then applying it from the wrong starting point, such as calculating from x = 5 instead of x = 1. When finding intermediate values of linear functions, work from the nearest known point to minimize calculation steps.

Question 6

A taxi ride costs a base fare of $4.50 plus $2.00 per mile. Let $mbethenumberofmilesandbe the number of miles andCbethetotalcostindollars.Whatdoesthebe the total cost in dollars. What does they−interceptrepresentinthiscontext?Choosetheinterpretationthatmatchesthemodel-intercept represent in this context? Choose the interpretation that matches the model C=2.00m+4.50$.

  1. The cost per mile
  2. The base fare when m=0m=0 (correct answer)
  3. The number of miles for a $4.50 ride
  4. The change in miles per dollar

Explanation: This question asks for the interpretation of the y-intercept in the context of a taxi fare model C = 2.00m + 4.50. In this equation, C is the total cost, m is the number of miles, 2.00 is the cost per mile (slope), and 4.50 is the y-intercept. The y-intercept represents the value of C when m = 0, which is the base fare charged before any miles are driven. This is the fixed cost that every passenger pays regardless of distance. Students often confuse the y-intercept with other quantities like the rate of change or incorrectly interpret it as a distance value. In real-world linear models, the y-intercept always represents the initial or fixed value when the independent variable equals zero.

Question 7

On a coordinate plane, a line passes through the points (−2,3)(-2,3) and (4,−9)(4,-9). Which equation represents the line in slope-intercept form y=mx+by=mx+b? Several options have the correct slope but an incorrect intercept.

  1. y=−2x−1y=-2x-1 (correct answer)
  2. y=2x−1y=2x-1
  3. y=−2x+1y=-2x+1
  4. y=12x−1y=\tfrac{1}{2}x-1

Explanation: This question asks for the slope-intercept form of a line passing through (-2,3) and (4,-9). First, calculate the slope: m = (-9 - 3)/(4 - (-2)) = -12/6 = -2. Now use point-slope form with either point to find the equation; using (-2,3): y - 3 = -2(x - (-2)), which simplifies to y - 3 = -2(x + 2) = -2x - 4, so y = -2x - 1. Common errors include getting the correct slope but making arithmetic mistakes when finding the y-intercept, resulting in equations like y = -2x + 1. Always verify your final equation by substituting both original points.

Question 8

A line crosses the yy-axis at −3-3 and also passes through (4,1)(4, 1). Which equation represents the line?

  1. y=x−3y=x-3 (correct answer)
  2. y=14x−3y=\tfrac{1}{4}x-3
  3. y=−x−3y=-x-3
  4. y=14x+3y=\tfrac{1}{4}x+3

Explanation: This problem provides the y-intercept and one point to find the equation of a line. We know the line crosses the y-axis at -3 and passes through (4, 1). The slope is m = (1 - (-3))/(4 - 0) = 4/4 = 1. Using slope-intercept form with slope 1 and y-intercept -3: y = 1·x + (-3) = x - 3. We can verify with the given point (4, 1): 1 = 4 - 3 = 1 ✓. A common mistake is confusing which point represents the y-intercept; remember that the y-intercept occurs when x = 0. Always double-check your equation by substituting the given point.

Question 9

A line has equation y=−4x+9y=-4x+9. Which of the following points lies on the line?

  1. (2,1)(2,1) (correct answer)
  2. (−1,5)(-1,5)
  3. (1,13)(1,13)
  4. (−2,−1)(-2,-1)

Explanation: The question asks which of the given points lies on the line with equation y = -4x + 9. To determine this, substitute the x-value of each point into the equation and check if it yields the corresponding y-value. For (2, 1): y = -4(2) + 9 = -8 + 9 = 1, which matches. The other points do not satisfy the equation; for example, (-1, 5) gives y=13, not 5. Thus, the point is (2, 1), choice A. A common error is a calculation mistake, such as multiplying -4 by x incorrectly or forgetting the sign. As a test-taking strategy, when given points and a line equation, systematically substitute each option to verify which one fits the linear relationship.

Question 10

A movie theater sells tickets for a fixed price plus a one-time online processing fee of $2. If 3 tickets cost $35 total and each additional ticket increases the total by $11, which equation gives the total cost yy for xx tickets?

  1. y=11x+2y=11x+2 (correct answer)
  2. y=2x+11y=2x+11
  3. y=11x−2y=11x-2
  4. y=11x+35y=11x+35

Explanation: This problem describes a movie theater pricing structure and asks for the cost equation. We're told that each additional ticket increases the total by $11, so the price per ticket is $11 (slope = 11). For 3 tickets at $11 each plus a $2 processing fee: 3 × $11 + $2 = $33 + $2 = $35 ✓, which matches the given information. Therefore, the equation is y = 11x + 2, where x is the number of tickets and y is the total cost. The processing fee is a one-time charge regardless of the number of tickets, making it the y-intercept. Always verify your equation using the given example in word problems.

Question 11

A music streaming service charges a one-time sign-up fee of $8 and then $3 per month. Let $mbethenumberofmonthsandletbe the number of months and letCbethetotalcostindollars.Whichequationrepresentsthisrelationshipbetweenbe the total cost in dollars. Which equation represents this relationship betweenCandandm$?

  1. C=8m+3C = 8m + 3
  2. C=3m+8C = 3m + 8 (correct answer)
  3. C=8m−3C = 8m - 3
  4. C=3m−8C = 3m - 8

Explanation: The question asks for the equation representing the total cost C in terms of the number of months m for a music streaming service with a one-time sign-up fee of $8 and a monthly charge of $3. To model this linear relationship, recognize that the fixed sign-up fee is the y-intercept and the monthly charge is the slope in the equation C = 3m + 8. This equation shows that for each additional month, the cost increases by $3, starting from the initial $8 fee. You can verify by plugging in m = 0, which gives C = 8, matching the sign-up fee, or m = 1, giving C = 11, which is the fee plus one month's charge. A common computational error is reversing the slope and intercept, such as writing C = 8m + 3, which would incorrectly imply $8 per month and a $3 fee. Another error might involve subtracting instead of adding the fee, but the context indicates addition for total cost. As a test-taking strategy, always identify the fixed and variable components in word problems to correctly assign the slope and y-intercept in linear equations.

Question 12

A music streaming service charges a one-time sign-up fee of $6 and then $3 per month. Let $mbethenumberofmonthsaftersign−upandbe the number of months after sign-up andCbethetotalcostindollars.Whichequationrepresentstherelationshipbetweenbe the total cost in dollars. Which equation represents the relationship betweenCandandm$? Be careful to distinguish the initial fee from the monthly rate.

  1. C=6m+3C=6m+3
  2. C=3m+6C=3m+6 (correct answer)
  3. C=6m−3C=6m-3
  4. C=3m−6C=3m-6

Explanation: This question asks for an equation representing the total cost C in terms of months m for a music streaming service. The service has a one-time sign-up fee of $6 (this is the starting cost when m = 0) and charges $3 per month thereafter. The total cost equals the initial fee plus the monthly charge times the number of months: C = 6 + 3m, which can be written as C = 3m + 6. A common error is confusing which value is the coefficient (monthly rate) and which is the constant (initial fee), leading to equations like C = 6m + 3. When dealing with linear cost models, identify the fixed cost (y-intercept) and variable cost (slope) before writing the equation.

Question 13

The graph shown models the total cost CC, in dollars, for a catering company to serve nn guests. Based on the graph, which statement best interprets the slope?

  1. Each additional guest increases the total cost by $18. (correct answer)
  2. The fixed setup fee for the event is $18.
  3. Each additional guest increases the total cost by $25.
  4. The company charges $150 per guest served.

Explanation: The line passes through (0,150)(0,150) and (50,1050)(50,1050), giving slope 1050−15050−0=18\frac{1050-150}{50-0}=18 dollars per guest. Choice B confuses slope with yy-intercept. Choice C uses an incorrect rise over wrong run. Choice D misidentifies the yy-intercept as per-guest cost.

Question 14

In the xyxy-plane, the graph of line ℓ\ell is shown. Line kk (not shown) is parallel to line ℓ\ell and passes through the point (4,−1)(4,-1). What is the yy-intercept of line kk?

  1. (0,−9)(0,-9) (correct answer)
  2. (0,−5)(0,-5)
  3. (0,7)(0,7)
  4. (0,3)(0,3)

Explanation: From the graph, line ℓ\ell passes through (0,−4)(0,-4) and (2,0)(2,0), giving slope 0−(−4)2−0=2\frac{0-(-4)}{2-0}=2. Since line kk is parallel to ℓ\ell, it also has slope 22. Using point-slope form with (4,−1)(4,-1): y−(−1)=2(x−4)y-(-1)=2(x-4), which simplifies to y=2x−9y=2x-9. The yy-intercept is (0,−9)(0,-9).

Question 15

A student earns $50 for signing up to tutor and then $18 per hour of tutoring. Let yy be total earnings after xx hours. How much will the student earn after 6 hours?

  1. $108
  2. $158 (correct answer)
  3. $168
  4. $300

Explanation: This problem involves calculating total earnings from a tutoring job with an initial sign-up bonus plus hourly pay. The student earns $50 for signing up plus $18 per hour, giving the equation y = 50 + 18x. For x = 6 hours: y = 50 + 18(6) = 50 + 108 = 158. Therefore, the student will earn $158 after 6 hours. This type of problem combines a fixed amount (y-intercept) with a variable rate (slope). Always double-check your arithmetic: 18 × 6 = 108, and 50 + 108 = 158. In linear cost/earning problems, identify the fixed component and the per-unit rate separately.

Question 16

A concert venue charges $15 to enter plus $5 per drink. If someone spent $55 total, how many drinks did they buy?

  1. 66
  2. 77
  3. 88 (correct answer)
  4. 1111

Explanation: This problem involves a linear cost function where we work backwards from total cost to find the number of drinks purchased. The venue charges $15 to enter plus $5 per drink, so the equation is y = 15 + 5x. Given that someone spent $55 total, we solve: 55 = 15 + 5x. Subtracting 15: 40 = 5x. Dividing by 5: x = 8 drinks. We can verify: 15 + 5(8) = 15 + 40 = 55 ✓. This type of problem requires identifying the fixed cost (entry fee) and variable cost (per drink), then solving algebraically for the unknown quantity. Always check your answer by substituting back into the original equation.

Question 17

A line is shown on the coordinate plane passing through (0,0)(0, 0) and (2,−6)(2, -6). What is the slope of the line?

  1. −3-3 (correct answer)
  2. 33
  3. −13-\tfrac{1}{3}
  4. 13\tfrac{1}{3}

Explanation: This problem asks for the slope of a line passing through two given points. Using the slope formula m = (y₂ - y₁)/(x₂ - x₁) with points (0, 0) and (2, -6): m = (-6 - 0)/(2 - 0) = -6/2 = -3. The negative slope indicates that the line decreases as we move from left to right. For every 1 unit we move to the right, the line drops 3 units. Since one point is the origin (0, 0), this line passes through the origin and has a simple relationship y = -3x. Always simplify your slope fraction to lowest terms for the clearest representation.

Question 18

A line passes through the points (−2,7)(-2, 7) and (4,−5)(4, -5). What is the slope of the line?

  1. −2-2 (correct answer)
  2. 22
  3. −12-\tfrac{1}{2}
  4. 12\tfrac{1}{2}

Explanation: This problem requires finding the slope of a line given two points using the slope formula. The slope formula is m = (y₂ - y₁)/(x₂ - x₁). Using points (-2, 7) and (4, -5), we get m = (-5 - 7)/(4 - (-2)) = -12/6 = -2. The numerator is the change in y-values: -5 - 7 = -12, and the denominator is the change in x-values: 4 - (-2) = 6. A common computational error is making sign mistakes when subtracting negative numbers. Always double-check your arithmetic with positive and negative values when calculating slope.

Question 19

A line is graphed. It passes through the points (−2,−3)(-2, -3) and (1,6)(1, 6). Which equation represents the line?

  1. y=3x+3y=3x+3 (correct answer)
  2. y=3x−3y=3x-3
  3. y=−3x+3y=-3x+3
  4. y=−3x−3y=-3x-3

Explanation: This problem asks for the equation of a line passing through two given points. First, we calculate the slope: m = (6 - (-3))/(1 - (-2)) = 9/3 = 3. Using point-slope form with point (-2, -3): y - (-3) = 3(x - (-2)), which gives y + 3 = 3(x + 2) = 3x + 6. Therefore, y = 3x + 6 - 3 = 3x + 3. We can verify with the other point (1, 6): 6 = 3(1) + 3 = 6 ✓. The equation has slope 3 and y-intercept 3. Always verify your final equation by substituting both given points to ensure they satisfy the equation.

Question 20

A phone plan charges $25 each month plus $0.10 per text message. Let yy be the monthly cost (in dollars) for sending xx texts. What is the cost for 180 texts?

  1. $43 (correct answer)
  2. $25.18
  3. $205
  4. $18

Explanation: This problem involves a linear cost function where we need to calculate the total cost for a specific number of texts. The phone plan has a fixed monthly charge of $25 plus $0.10 per text, so the equation is y = 25 + 0.10x. For x = 180 texts: y = 25 + 0.10(180) = 25 + 18 = 43. Therefore, the cost is 43.Thekeyisidentifyingthefixedcost(43. The key is identifying the fixed cost (25) and the variable rate ($0.10 per text), then substituting the given value. Double-check your decimal multiplication: 0.10 × 180 = 18.