ACT Math Quiz: Linear Functions

20 questions · exam conditions

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Linear FunctionsQuestion 1 of 20

In the standard $(x, y)$ coordinate plane, point $A$ has coordinates $(4, -3)$ and point $B$ has coordinates $(-2, 7)$ . What are the coordinates of the midpoint of line segment $\overline{AB}$ ?

(1, 2)

(2, 4)

(3, 5)

(6, -10)

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ACT Math Quiz

ACT Math Quiz: Linear Functions

Practice Linear Functions in ACT 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 ACT 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

In the standard $(x, y)$ coordinate plane, point $A$ has coordinates $(4, -3)$ and point $B$ has coordinates $(-2, 7)$ . What are the coordinates of the midpoint of line segment $\overline{AB}$ ?

$(1, 2)$ (correct answer)
$(2, 4)$
$(3, 5)$
$(6, -10)$

Explanation: The correct answer is A (1, 2). The midpoint formula averages the x-coordinates and the y-coordinates separately: x = (4 + (−2)) ÷ 2 = 2 ÷ 2 = 1; y = (−3 + 7) ÷ 2 = 4 ÷ 2 = 2. Midpoint = (1, 2). B (2, 4) uses the sums without dividing by 2: (4 + (−2)) = 2 and (−3 + 7) = 4. C (3, 5) drops the negative signs: (4 + 2) ÷ 2 = 3 and (3 + 7) ÷ 2 = 5. D (6, −10) likely comes from subtracting or doubling instead of averaging. Always divide both sums by 2 — the midpoint is literally the average of the two endpoints.

Question 2

A hiker's elevation changes at a constant rate along a trail. The trail's elevation line has slope $-2$ and passes through the point $(3,7)$ . Which equation represents this line?

$y=\dfrac{1}{2}x+\dfrac{11}{2}$
$y=2x+1$
$y=-2x+13$ (correct answer)
$y=-2x+1$

Explanation: When a problem hands you a slope and one point, point-slope form

y-y_1=m(x-x_1)

is the fastest route to the equation. Substituting

m=-2

and

(3,7)

gives

y-7=-2(x-3)

, and distributing the

-2

yields

y-7=-2x+6

, so adding 7 to both sides produces

y=-2x+13

; the hiker descends 2 units of elevation per unit along the trail. The equation

y=-2x+1

has the right slope but the wrong intercept, which comes from computing

7-6

instead of

7+6

, that is, from failing to negate the

-2\cdot(-3)

correctly. The equation

y=2x+1

additionally reverses the sign of the slope, turning a descent into a climb, and

y=\dfrac{1}{2}x+\dfrac{11}{2}

uses the negative reciprocal of the slope, which describes a perpendicular line rather than this one. Distribute the slope through the parentheses carefully, watching that a negative slope times a negative

x_1

becomes positive, then verify by plugging the given point back into your final equation.

Question 3

What is the y-intercept of the line $y = -2x + 7$ ?

0
2
-2
7 (correct answer)

Explanation: The

y

-intercept is the output when the input is zero, and in slope-intercept form

y=mx+b

it is simply the constant

b

. For

y=-2x+7

, substituting

x=0

gives

y=-2(0)+7=7

, so the line crosses the vertical axis at

(0,7)

and the

y

-intercept is 7. The values

-2

and

2

come from reading the coefficient of

x

, which is the slope and describes steepness rather than a crossing point, with the positive version also dropping the sign. The value 0 confuses the

y

-intercept with the

x

-coordinate of the point where the line crosses, since

x=0

there but

y

does not. When you need an intercept, substitute zero for the other variable, so set

x=0

for a

y

-intercept and

y=0

for an

x

-intercept.

Question 4

A company's profit is modeled by the linear equation $y=5x-20$ , where $x$ is the number of items sold and $y$ is profit in dollars. What is the y-intercept of this line?

$5$
$-20$ (correct answer)
$20$
$-5$

Explanation: We need to find the y-intercept of the line y = 5x - 20. In slope-intercept form y = mx + b, the y-intercept is b, which is the constant term. From y = 5x - 20, we see that b = -20. The y-intercept is -20, representing the profit when no items are sold (a $20 loss). Choice A (5) incorrectly identifies the slope as the y-intercept, while choice C (20) has the wrong sign.

Question 5

A linear function models the balance in a gift card after buying snacks. The function is $f(x)=-4x+20$ , where $x$ is the number of snacks purchased. What is $f(0)$ ?

$-4$
$0$
$16$
$20$ (correct answer)

Explanation: This question asks for f(0) in the linear function f(x) = -4x + 20, which models the initial gift card balance before purchases. f(0) is the y-intercept, found by substituting x = 0 into the equation: f(0) = -4(0) + 20 = 20. This value represents the starting balance, with the slope -4 indicating the cost per snack. The structure y = mx + b directly shows b as the intercept. The correct answer is 20, which is choice D. A key distractor is choice A, -4, confusing the slope with the intercept. Another is choice C, 16, possibly from miscalculating -4(1) + 20 or another small error.

Question 6

A delivery drone's height changes at a constant rate as it flies. The drone is at height 2 meters when it is 1 second into the flight and at height 14 meters when it is 5 seconds into the flight. What is the slope of the line through points $(1,2)$ and $(5,14)$ ?

$\dfrac{1}{3}$
$3$ (correct answer)
$-3$
$\dfrac{1}{4}$

Explanation: This question asks for the slope of the line passing through the points (1,2) and (5,14), which represents the constant rate of change in the drone's height over time. To find the slope, use the formula m = (y₂ - y₁)/(x₂ - x₁), substituting the given points. Here, m = (14 - 2)/(5 - 1) = 12/4 = 3, emphasizing how the change in y over the change in x gives the rate. This calculation shows the slope is positive 3, meaning the height increases by 3 meters per second. The correct answer is 3, which is choice B. A key distractor is choice A, which is 1/3, likely from reversing the numerator and denominator in the slope formula. Another common error is choice C, -3, which might result from confusing the order of subtraction and adding a negative sign incorrectly.

Question 7

A straight road's elevation above sea level changes linearly with distance. The elevation line is $y=5x-10$ . What is the y-intercept of this line?

$5$
$-10$ (correct answer)
$10$
$-5$

Explanation: This question seeks the y-intercept of the line y = 5x - 10, which models the road's elevation starting point. In the slope-intercept form y = mx + b, the y-intercept b is directly -10. This represents the elevation at distance x=0, with slope 5 as the rate of change. Verifying by setting x=0 gives y=-10, confirming the structure. The correct answer is -10, which is choice B. A key distractor is choice D, -5, possibly halving the slope or misreading. Another is choice A, 5, confusing the slope with the intercept and ignoring the sign.

Question 8

Which equation represents a line with a slope of 5 and a y-intercept of -1?

$y = 5x - 1$ (correct answer)
$y = -5x + 1$
$y = 1x + 5$
$y = 5x + 1$

Explanation: We need to identify the equation with slope 5 and y-intercept -1. Using the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept, we get y = 5x + (-1), which simplifies to y = 5x - 1. Looking at the choices, option A correctly shows y = 5x - 1. Choice D incorrectly has y-intercept +1 instead of -1, while choice B has the wrong slope sign (negative instead of positive).

Question 9

A straight road's elevation above sea level changes linearly with distance. The elevation line is $y=5x-10$ . What is the y-intercept of this line?

$5$
$-10$ (correct answer)
$10$
$-5$

Question 10

Which equation has slope 4 and y-intercept -2?

$y = 4x + 2$
$y = -2x + 4$
$y = 4x - 2$ (correct answer)
$y = -4x + 2$

Explanation: We need to identify which equation has slope 4 and y-intercept -2. In the slope-intercept form y = mx + b, m represents the slope and b represents the y-intercept. We need m = 4 and b = -2, so the equation is y = 4x + (-2) = y = 4x - 2. This matches choice C exactly. Choice A has the wrong y-intercept (2 instead of -2), and choices B and D have the wrong slopes.

Question 11

$\dfrac{1}{3}$
$3$ (correct answer)
$-3$
$\dfrac{1}{4}$

Question 12

A line models the temperature $y$ (in °C) as a function of time $x$ (in hours) during a cooling process. The line passes through $(0,8)$ and $(6,-4)$ . Which equation gives the line in slope-intercept form?

$y=-2x-8$
$y=2x+8$
$y=-\dfrac{1}{2}x+8$
$y=-2x+8$ (correct answer)

Explanation: The question seeks the slope-intercept form of the line passing through (0,8) and (6,-4), modeling temperature over time. First, calculate the slope m = (y₂ - y₁) / (x₂ - x₁) = (-4 - 8) / (6 - 0) = (-12) / 6 = -2, representing the cooling rate. The y-intercept is 8 from the point (0,8), so the equation is y = -2x + 8 using y = mx + b form. This structure emphasizes how linear equations describe constant change processes. The correct equation is y=-2x+8. Choice C might result from halving the slope incorrectly, leading to -1/2, or confusing points.

Question 13

In the standard $(x, y)$ coordinate plane, what is the slope of the line given by the equation $5x + 2y = 10$ ?

$-5$
$-\frac{5}{2}$ (correct answer)
$-\frac{2}{5}$
$\frac{5}{2}$

Explanation: This is a slope-intercept form question testing equation rewriting. Choice B (−5/2) is correct — isolate y: 2y = −5x + 10, so y = (−5/2)x + 5. The slope is the coefficient of x: −5/2. Choice A (−5) reads the coefficient of x from the original equation before dividing by 2 — correctly identifying the −5 but forgetting that both sides must be divided by 2 when solving for y. Choice C (−2/5) inverts the slope fraction, giving the negative reciprocal of the correct answer (which would actually be the slope of a perpendicular line). Choice D (5/2) inverts the fraction AND drops the negative sign — two errors compounded. Pro tip: Never read the slope from an equation unless y is completely isolated. Rewrite 5x + 2y = 10 as y = (−5/2)x + 5 first, then the slope is right there as the coefficient of x.

Question 14

In the standard $(x, y)$ coordinate plane, point $A$ has coordinates $(3, -4)$ and point $B$ has coordinates $(-5, 8)$ . What are the coordinates of the midpoint of line segment $\overline{AB}$ ?

$(-1, 2)$ (correct answer)
$(-1, 4)$
$(-2, 2)$
$(8, -12)$

Explanation: This is a midpoint formula question. Choice A (−1, 2) is correct — midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((3 + (−5))/2, (−4 + 8)/2) = (−2/2, 4/2) = (−1, 2). Choice B (−1, 4) correctly finds the x-coordinate but forgets to divide the y-sum by 2: −4 + 8 = 4, but the midpoint y-coordinate is 4/2 = 2, not 4. Choice C (−2, 2) correctly finds the y-coordinate but forgets to divide the x-sum by 2: 3 + (−5) = −2, but the midpoint x-coordinate is −2/2 = −1, not −2. Choice D (8, −12) doubles the coordinates rather than halving them — applying the wrong operation entirely. Pro tip: The midpoint is just the average of the x-coordinates and the average of the y-coordinates. If your answer has a value larger in magnitude than both of the input coordinates for that axis, you've multiplied instead of divided — that's a sign you need to go back.

Question 15

A student saves $40 in month 1, and $10 more each subsequent month. What is the total amount saved after 6 months?

$90
$240
$390 (correct answer)
$450

Explanation: This is an arithmetic series question testing careful addition of a finite sequence. Choice C ($390) is correct — list the monthly savings: Month 1: $40, Month 2: $50, Month 3: $60, Month 4: $70, Month 5: $80, Month 6: $90. Sum = 40 + 50 + 60 + 70 + 80 + 90 = $390. Using the series formula: Sum = (n/2)(first + last) = (6/2)(40 + 90) = 3 × 130 =

390. Choice A (

90) reports only the 6th month's savings — the last term rather than the cumulative total. Choice B ($240) computes 6 × $40 = $240 — treating all months as equal to the first month, ignoring the

10 monthly increase. Choice D (

450) uses an incorrect series: perhaps 6 × $75 = $450, where $75 is incorrectly computed as the average. Pro tip: For arithmetic sequences, the sum equals (number of terms)/2 × (first term + last term). Here: 6/2 × (40 + 90) = 3 × 130 = 390. Always find the last term first: a₆ = 40 + 5(10) = 90, then apply the formula.

Question 16

The first 4 terms of an arithmetic sequence are 12, 19, 26, and 33. What is the value of $t_{20}$ ?

138
145 (correct answer)
152
159

Explanation: This is an arithmetic sequences question testing the nth-term formula. Choice B (145) is correct — the common difference d = 19 − 12 = 7. Apply the formula: tₙ = a₁ + (n − 1)d → t₂₀ = 12 + (20 − 1)(7) = 12 + 19 × 7 = 12 + 133 = 145. Choice A (138) uses n − 2 = 18 instead of n − 1 = 19: 12 + 18 × 7 = 12 + 126 = 138 — an off-by-one error in the formula. Choice C (152) uses n = 20 instead of n − 1 = 19: 12 + 20 × 7 = 12 + 140 = 152 — adding one too many common differences. Choice D (159) uses n + 1 = 21: 12 + 21 × 7 = 12 + 147 = 159. Pro tip: The nth-term formula is tₙ = a₁ + (n − 1)d, not tₙ = a₁ + nd. The (n − 1) reflects that the first term has already been "given" — you only add d a total of n − 1 times to get from the 1st term to the nth. For t₂₀, you add d exactly 19 times to 12.

Question 17

What is the slope of the line through points (-3, -2) and (1, 6)?

2 (correct answer)
4
1
3

Explanation: We need to find the slope of the line through points (-3, -2) and (1, 6). Using the slope formula m = (y₂ - y₁)/(x₂ - x₁), we substitute: m = (6 - (-2))/(1 - (-3)) = (6 + 2)/(4) = 8/4 = 2. The slope is 2, which matches choice A. Choice B would result from incorrectly calculating the denominator as just the difference in x-coordinates without considering the signs properly.

Question 18

A linear function models the balance in a gift card after buying snacks. The function is $f(x)=-4x+20$ , where $x$ is the number of snacks purchased. What is $f(0)$ ?

$-4$
$0$
$16$
$20$ (correct answer)

Question 19

What is the y-intercept of the line $y = -3x + 4$ ?

4 (correct answer)
3
0
-4

Explanation: We need to find the y-intercept of the line y = -3x + 4. In slope-intercept form y = mx + b, the y-intercept is the constant term b. Looking at y = -3x + 4, we can see that b = 4. The y-intercept represents where the line crosses the y-axis at the point (0, 4). Choice A correctly identifies the y-intercept as 4. Choice B confuses the coefficient of x (-3) with the y-intercept.

Question 20

A linear function models the number of pages read: $y=7x-1$ , where $x$ is hours and $y$ is pages. What is the y-intercept of the line $y=7x-1$ ?

$7$
$-1$ (correct answer)
$1$
$0$

Explanation: We need to identify the y-intercept of the line y = 7x - 1. In the slope-intercept form y = mx + b, the y-intercept is the constant term b, which is -1. This represents the initial number of pages read (or a starting adjustment) when x = 0 hours. Choice A incorrectly identifies the slope (7) as the y-intercept, while choices C and D show incorrect values for the y-intercept.