ACT Math Quiz: Linear Functions

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.

Question 1

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

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

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

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

1. 0
2. 2
3. -2
4. 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

1. $-4$
2. $0$
3. $16$
4. $20$ (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 5

1. $\dfrac{1}{3}$
2. $3$ (correct answer)
3. $-3$
4. $\dfrac{1}{4}$

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 6

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

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 7

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

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 8

1. $\dfrac{1}{3}$
2. $3$ (correct answer)
3. $-3$
4. $\dfrac{1}{4}$

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 9

1. $y=-2x-8$
2. $y=2x+8$
3. $y=-\dfrac{1}{2}x+8$
4. $y=-2x+8$ (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 10

1. $-5$
2. $-\frac{5}{2}$ (correct answer)
3. $-\frac{2}{5}$
4. $\frac{5}{2}$

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 11

1. $(-1, 2)$ (correct answer)
2. $(-1, 4)$
3. $(-2, 2)$
4. $(8, -12)$

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 12

1. $90
2. $240
3. $390 (correct answer)
4. $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 13

1. 138
2. 145 (correct answer)
3. 152
4. 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 14

1. 2 (correct answer)
2. 4
3. 1
4. 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 15

1. $-4$
2. $0$
3. $16$
4. $20$ (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 16

1. 4 (correct answer)
2. 3
3. 0
4. -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 17

1. $7$
2. $-1$ (correct answer)
3. $1$
4. $0$

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.

Question 18

1. -6
2. 1 (correct answer)
3. 6
4. 0

Explanation: We need to find f(0) for the function f(x) = -6x + 1. To evaluate f(0), we substitute x = 0 into the function. f(0) = -6(0) + 1 = 0 + 1 = 1. The answer is 1, which corresponds to choice B. This represents the y-intercept of the linear function. Choice A incorrectly uses the coefficient of x instead of evaluating the function at x = 0.

Question 19

1. Graph with slope -3 and y-intercept 2 (correct answer)
2. Graph with slope 3 and y-intercept 2
3. Graph with slope -2 and y-intercept 3
4. Graph with slope 3 and y-intercept -2

Explanation: The equation y = -3x + 2 is in slope-intercept form y = mx + b, where the slope is -3 and the y-intercept is 2. This means the graph should show a line that falls 3 units vertically for every 1 unit horizontally, starting from the point (0, 2). Choice A correctly describes this with slope -3 and y-intercept 2. Choice B shows the wrong slope sign (positive instead of negative), while choice D has the wrong y-intercept sign.

Question 20

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

Explanation: To build a line in the form $y=mx+b$ you place the given slope in the $m$ position and then solve for $b$ using a point the line passes through. With $m=2$ and the point $(3,11)$, substitute to get $11=2(3)+b$, so $11=6+b$ and $b=5$, which makes the equation $y=2x+5$; here the $5$ is the flat fare charged before any miles and the $2$ is the cost per mile. The equation $y=5x+2$ swaps the slope and the intercept, treating the number you solved for as the rate. The equation $y=2x-5$ keeps the right slope but subtracts the intercept, which comes from writing $11=6-b$ or moving the $6$ the wrong way, and $y=-2x+5$ flips the sign of a rate the problem stated as positive. Substitute the point into $y=mx+b$ and isolate $b$, then check by plugging the point back in, since the rate multiplies the variable while the one-time starting amount stands alone as the constant.