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ACT Math Help: Linear Functions

Review real example questions for Linear Functions in ACT Math.

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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 $\overline{AB}$ ?

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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.