Linear Functions - ACT Math

A line on a coordinate plane passes through the points $(0,-2)$ and $(3,4)$ . What is the slope of the line shown by these two points?
Answer: $2$ The question asks for the slope of the line passing through (0,-2) and (3,4). Use the slope formula m = (y₂ - y₁) / (x₂ - x₁) to find the rate of change. Substituting, m = (4 - (-2)) / (3 - 0) = 6 / 3 = 2. This calculation underscores the linear nature where slope is constant between points. The slope is 2. Choice D reverses the sign to -1/2, perhaps from subtracting in the wrong order like ( -2 - 4 ) / (0 - 3) without simplifying correctly.
What is the slope of the line through points (4, -1) and (6, 5)?
Answer: 3 We need to find the slope of the line through points (4, -1) and (6, 5). Using the slope formula m = (y₂ - y₁)/(x₂ - x₁), we substitute: m = (5 - (-1))/(6 - 4) = (5 + 1)/(2) = 6/2 = 3. The slope is 3, which matches choice A. This represents a fairly steep positive slope where the line rises 3 units for every 1 unit moved to the right. Choice B would result from reversing the calculation to (x₂ - x₁)/(y₂ - y₁).
A student graphs a linear function representing temperature change over time and writes the equation $y=-2x+7$ . In this model, what is the y-intercept of the line?
Answer: $7$ The question asks for the y-intercept of the line given by y = -2x + 7. In slope-intercept form y = mx + b, the y-intercept b is the value of y when x = 0, representing the starting point on the y-axis. Here, the equation is already in this form, with m = -2 as the slope and b = 7 as the y-intercept. This structure emphasizes how the intercept is independent of the slope's rate of change. The y-intercept is 7, which is choice C. A key distractor is choice A, -2, which confuses the slope with the y-intercept by switching their roles in the equation.
What is $f(0)$ for the function $f(x) = 9x - 7$ ?
Answer: -7 To find f(0) for the function f(x) = 9x - 7, we substitute x = 0 into the function. f(0) = 9(0) - 7 = 0 - 7 = -7. This means when the input is 0, the output is -7, which is also the y-intercept of this linear function. The answer is -7. Choice B (9) incorrectly uses the coefficient of x instead of evaluating the function at x = 0.
What is $f(0)$ for the function $f(x) = 3x - 2$ ?
Answer: -2 We need to find f(0) for the function f(x) = 3x - 2. To evaluate f(0), we substitute x = 0 into the function. f(0) = 3(0) - 2 = 0 - 2 = -2. The answer is -2, which corresponds to choice C. This represents the y-intercept of the linear function where the graph crosses the y-axis below the origin. Choice B incorrectly uses the coefficient of x instead of the constant term.
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?
Answer: $y=-2x+13$ 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.
Which equation represents a line with a slope of 0 and a y-intercept of -3?
Answer: $y = -3$ We need to identify the equation with slope 0 and y-intercept -3. A line with slope 0 is horizontal, meaning y remains constant regardless of x value. Using the slope-intercept form y = mx + b, where m = 0 and b = -3, we get y = 0x + (-3), which simplifies to y = -3. This represents a horizontal line at y = -3. Choice A correctly shows y = -3, while choice B unnecessarily includes the 0x term.
A car rental price is modeled by a line with slope $-2$ that passes through the point $(3,5)$ . Which equation represents this line?
Answer: $y=-2x+11$ We need the equation of a line with slope -2 passing through point (3,5). Using point-slope form: y - 5 = -2(x - 3). Expanding: y - 5 = -2x + 6, so y = -2x + 11. The answer is A: y = -2x + 11. Choice C (y = -2x - 1) has the correct slope but calculates the y-intercept incorrectly.
Which equation has slope 4 and y-intercept -2?
Answer: $y = 4x - 2$ 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.
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 $(5,14)$ ?
Answer: $3$ 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.
What is the slope of the line through points (3, 5) and (-1, -1)?
Answer: 3/2 We need to find the slope of the line through points (3, 5) and (-1, -1). Using the slope formula m = (y₂ - y₁)/(x₂ - x₁), we substitute: m = (-1 - 5)/(-1 - 3) = (-6)/(-4) = 6/4 = 3/2. The slope is 3/2, which matches choice C. This means the line rises 3 units for every 2 units moved to the right. Choice A would result from incorrectly calculating the reciprocal of the correct slope.
A linear temperature model has slope $m=4$ and passes through the point $(2,-1)$ , where $x$ is hours and $y$ is degrees. What is the equation of the line in the form $y=mx+b$ ?
Answer: $y=4x-9$ We need to find the equation of a line with slope m=4 that passes through point (2,-1). Using the point-slope form and substituting into y = mx + b: -1 = 4(2) + b, so -1 = 8 + b, which gives b = -9. Therefore, the equation is y = 4x - 9. Choice A uses an incorrect y-intercept of 1, while choices C and D have incorrect slopes or arrangements of the coefficients.
A taxi charges a fixed pickup fee plus a constant rate per mile. The cost is modeled by a linear function with slope $m=4$ and $y$ -intercept $b=7$ . Which equation has slope $4$ and $y$ -intercept $7$ ?
Answer: $y=4x+7$ We need to write the equation of a line with slope m = 4 and y-intercept b = 7. Using the slope-intercept form y = mx + b, we substitute the given values: y = 4x + 7. The answer is C: y = 4x + 7. Choice B (y = 4x - 7) has the correct slope but the wrong y-intercept, using -7 instead of +7.
A delivery driver tracks their route on a map and notes that their position changed from $(2,5)$ to $(8,-1)$ on a coordinate grid. What is the slope of the line through points $(2,5)$ and $(8,-1)$ ?
Answer: $-\dfrac{1}{1}$ We need to find the slope of the line through points (2,5) and (8,-1). Using the slope formula m = (y₂-y₁)/(x₂-x₁), we substitute: m = (-1-5)/(8-2) = -6/6 = -1. We can write this as -1/1 to match the answer format. The answer is B: -1/1. Choice A shows -6/3 = -2, which incorrectly simplifies the fraction before reducing.
A line represents a constant-rate change in elevation. The line passes through points $(-2,3)$ and $(4,0)$ . What is the slope of the line through points $(-2,3)$ and $(4,0)$ ?
Answer: $-\dfrac{1}{2}$ We need to find the slope of the line through points (-2,3) and (4,0). Using the slope formula m = (y₂-y₁)/(x₂-x₁), we calculate m = (0-3)/(4-(-2)) = -3/6 = -1/2. The slope represents the rate of elevation change, which is -1/2 units per unit of horizontal distance. Choice A shows the positive version of the correct answer, while choices C and D show integer values that would result from calculation errors.
A runner's position is linear in time. The line passes through $(1,4)$ and $(5,-8)$ , where $x$ is seconds and $y$ is meters relative to a marker. What is the slope of the line through points $(1,4)$ and $(5,-8)$ ?
Answer: $-3$ Slope is the ratio of vertical change to horizontal change, computed as $m=\dfrac{y_2-y_1}{x_2-x_1}$ , and the two coordinate pairs must be subtracted in the same order. Using $(1,4)$ and $(5,-8)$ gives $m=\dfrac{-8-4}{5-1}=\dfrac{-12}{4}=-3$ , so the runner loses 3 meters of position per second and the slope is $-3$ . The value $3$ comes from subtracting the $y$ -values in one direction and the $x$ -values in the other, or from ignoring that the position drops from 4 to $-8$ . The values $\dfrac{1}{3}$ and $-\dfrac{1}{3}$ come from inverting the fraction and dividing the run by the rise instead of the rise by the run. Fix the order of your two points before you compute anything, keep rise on top and run on the bottom, and sanity-check the sign against whether $y$ increases or decreases as $x$ increases.
A taxi company charges a fixed fee plus a constant rate per mile. The total cost $y$ (in dollars) is a linear function of miles $x$ . The line has slope $m=3$ and passes through the point $(2,11)$ . Which equation represents this situation in slope-intercept form?
Answer: $y=3x+5$ The question requires finding the slope-intercept form of the line with slope m=3 passing through (2,11), modeling the taxi cost. Start with the slope-intercept form y = mx + b, where m is the rate and b is the fixed fee. Substitute the point: 11 = 3(2) + b, so 11 = 6 + b, and b = 5. Thus, the equation is y = 3x + 5, highlighting the linear structure with constant rate per mile. The equation representing this situation is y=3x+5. Choice B switches the slope and intercept, using y=5x+3, which confuses the rate and fixed fee roles.
A line on a coordinate grid passes through the points $(0,-2)$ and $(4,6)$ . What is the slope of the line through points $(0,-2)$ and $(4,6)$ ?
Answer: $2$ We need to find the slope of the line through points (0,-2) and (4,6). Using the slope formula m = (y₂-y₁)/(x₂-x₁): m = (6-(-2))/(4-0) = 8/4 = 2. The answer is A: 2. Choice B (-2) has the wrong sign, possibly from reversing the subtraction order in the numerator.
A taxi charges a fixed start fee plus a constant cost per mile. The cost is $11 after 2 miles and $23 after 6 miles. Which equation gives the total cost $y$ (in dollars) as a function of miles $x$ ?
Answer: $y=3x+5$ This question requires finding the equation of the line that models the taxi cost y in dollars as a function of miles x, using the points (2,11) and (6,23). First, calculate the slope m = (23 - 11)/(6 - 2) = 12/4 = 3, which represents the cost per mile. Then, use the point-slope form or substitute into y = mx + b; using (2,11), 11 = 3(2) + b, so b = 11 - 6 = 5, giving y = 3x + 5. This equation structure highlights the fixed fee as the y-intercept and the per-mile rate as the slope. The correct answer is y = 3x + 5, which is choice D. A key distractor is choice B, y = 5x + 3, which switches the slope and intercept values. Another is choice A, y = 3x - 5, possibly from subtracting instead of adding the intercept.
A line represents a constant change in a bank balance. The line has slope $m=-1$ and passes through the point $(6,2)$ . What is the equation of the line in the form $y=mx+b$ ?
Answer: $y=-x+8$ We need to find the equation of a line with slope m=-1 that passes through point (6,2). Using the point-slope form and substituting into y = mx + b: 2 = -1(6) + b, so 2 = -6 + b, which gives b = 8. Therefore, the equation is y = -x + 8. Choice B has the wrong sign for the slope, choice C has an incorrect y-intercept, and choice A has completely wrong slope and y-intercept values.

Start this quiz