Linear functions questions on Digital SAT Math go well beyond y = mx + b in isolation — they test function notation like f(x), finding slope from two points or a table, matching a function to its correct graph, choosing the best linear model for scattered real-world data, and interpreting what a function's output means in context. Because function notation shows up across nearly every other algebra topic on the exam, getting comfortable with it here pays off throughout the whole Math section.
The 29 problems below cover every version of this skill that shows up on real, recently administered Digital SAT exams, including several graph-matching questions, a scatterplot best-fit question, and perpendicular-slope problems. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. Pay close attention to the graph-identification problems in particular — the fastest way through them is usually to check the y-intercept first (it eliminates half the choices immediately) and only then compare slopes. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
Line k passes through the points (1, 1) and (2, 6) in the xy-plane. If the equation for line k is written in the form y = mx + b, where m and b are constants, what is the value of m?
Find the slope using the two given points: m = (6 − 1) / (2 − 1) = 5/1.
So m = 5.
In the linear function g, g(−2) = 3/4 and g(3) = 9/2. Which equation defines g?
Find the slope using the two given values: m = (9/2 − 3/4) / (3 − (−2)) = (15/4) / 5 = 3/4.
Use g(x) = (3/4)x + b with the point (−2, 3/4): 3/4 = (3/4)(−2) + b, so 3/4 = −3/2 + b, giving b = 3/4 + 3/2 = 9/4. That gives g(x) = (3/4)x + 9/4. The correct answer is A.
Line k in the xy-plane has slope −2p/5 and y-intercept (0, p), where p is a positive constant. What is the x-coordinate of the x-intercept of line k?
Write the equation of the line: y = (−2p/5)x + p. Set y = 0 to find the x-intercept: 0 = (−2p/5)x + p.
Solve for x: (2p/5)x = p, so x = p × (5/(2p)) = 5/2 (the p cancels since p ≠ 0). So the x-coordinate is 5/2.
The function f is defined by f(x) = (1/5)x + 9/10. For what value of x does f(x) = 1?
Set the function equal to 1: (1/5)x + 9/10 = 1.
Subtract 9/10 from both sides: (1/5)x = 1/10. Multiply both sides by 5: x = 1/2.
The function m is defined by m(x) = 30x + 120. What is the slope of the graph of y = m(x) in the xy-plane?
A linear function written as m(x) = ax + b has slope equal to the coefficient a.
Here, that coefficient is 30.
Lucia and John will work together to make 60 paper flowers for a school party. The graph shown represents the possible combinations of time, in hours, spent by Lucia and John to fulfill this task. According to the graph, on average, how many paper flowers will Lucia make per hour?
The graph's y-intercept shows that if John contributes 0 hours, Lucia alone needs 12 hours to finish all 60 flowers.
Lucia's rate is therefore 60 flowers ÷ 12 hours = 5 flowers per hour. The correct answer is B.
A business consultant charges $408 for the first hour and $204 for each additional hour of work. Which function gives the charge C(h), in dollars, for h hours of work, where h is a positive integer?
The first hour costs a flat $408. Each additional hour beyond the first costs $204, and there are (h − 1) additional hours: total = 408 + 204(h − 1).
Distribute: 408 + 204h − 204 = 204h + 204. The correct answer is A.
To make a bookcase, a worker charged a onetime fee plus $17 per hour worked. The equation 17h + 45 = 164 represents this situation, where h is the number of hours worked. Which of the following is the best interpretation of 164 in this context?
In the equation, 45 is the flat onetime fee and 17h is the hourly charge for h hours. Together, 17h + 45 represents the sum of both parts.
Since that sum is set equal to 164, this number represents the total amount charged. The correct answer is D.
There is a linear relationship between x and y. The table shows three values of x and their corresponding values of y in terms of a constant n: (0, n), (4, n + 19), (8, n + 38). What is the slope of the line that represents this relationship in the xy-plane?
| x | y |
|---|---|
| 0 | n |
| 4 | n + 19 |
| 8 | n + 38 |
Find the slope using the first two rows: slope = [(n + 19) − n] / (4 − 0) = 19/4.
Check with the third row: from x = 0 to x = 8 is a change of 8, and (19/4)(8) = 38, matching (n + 38) − n = 38. So the slope is 19/4.
What is the y-intercept of the graph of 3x + 2y = 96 in the xy-plane?
The y-intercept occurs where x = 0. Substitute x = 0: 3(0) + 2y = 96, so 2y = 96.
Divide by 2: y = 48. The y-intercept is (0, 48). The correct answer is D.
For the linear function f, the graph of y = f(x) in the xy-plane passes through the points (0, 2) and (3, 3). What is the slope of y = f(x)?
Find the slope using the two given points: slope = (3 − 2) / (3 − 0) = 1/3.
So the slope is 1/3.
The function f is defined by f(x) = (x + 11)/3, and f(a) = 16, where a is a constant. What is the value of a?
Set the function equal to 16: (a + 11)/3 = 16.
Multiply both sides by 3: a + 11 = 48. Subtract 11: a = 37. The correct answer is D.
Line n is shown in the xy-plane. Line k (not shown) is perpendicular to line n. What is the slope of line k?
Reading the graph, line n has a slope of 6 (it rises very steeply — about 6 units up for every 1 unit right).
Perpendicular lines have slopes that are negative reciprocals of each other: the negative reciprocal of 6 is −1/6. The correct answer is B.
A line in the xy-plane passes through the points (0, 12) and (1, 14). The equation of the line is y = mx + b, where m and b are constants. What is the value of b?
Since (0, 12) is the y-intercept, b = 12 directly — the y-value when x = 0 is always the constant b.
So b = 12.
p = 873 − 88t. The given equation models the approximate pressure p, in pounds per square inch, in an astronaut's oxygen tank during a spacewalk that lasts for t hours, where t ≤ 9. What would be the tank pressure, in pounds per square inch, at the end of a spacewalk that lasted 1 hour?
Substitute t = 1 into the equation: p = 873 − 88(1).
p = 873 − 88 = 785. The correct answer is B.
The function p is defined by p(x) = (−5/8)x + 21/2. What is the graph of y = p(x)?
The function has a y-intercept of 21/2 = 10.5 and a slope of −5/8 = −0.625, so the graph should start near y = 10.5 at x = 0 and decrease to about 10.5 − 0.625(10) = 4.25 by x = 10.
Only one graph starts near 10.5 and drops to approximately 4 by x = 10 — the others are either too steep, too shallow, or end at the wrong height. The correct answer is B.
The average price, in dollars, when adjusted for inflation, of a US domestic airline ticket each quarter of a year can be modeled by the equation y = −5x + 500, where x is the number of quarters after the last quarter of 1999, and x ≤ 20. Which graph represents this model?
The equation has a y-intercept of 500 and a gentle negative slope of −5. Over the full domain (x up to 20), the price only drops by 5(20) = 100 dollars, ending around y = 400.
Only one graph shows a gentle, gradual decline from 500 down to about 400 — the other graphs are either too steep, flat, or increasing. The correct answer is A.
The function w models the weight, in pounds, of a certain pig t months after it was born, where 1 ≤ t ≤ 36. Which statement is the best interpretation of w(7) = 20 in this context?
In function notation w(t) = weight, the input t = 7 represents time in months, and the output 20 represents the resulting weight in pounds.
So w(7) = 20 means that at 7 months of age, the pig weighed 20 pounds. The correct answer is A.
The total amount of food y, in pounds, that a zookeeper feeds a giraffe over x days is represented by the equation y = 30x. What is the total amount of food, in pounds, that the zookeeper feeds the giraffe over 3 days?
Substitute x = 3 into the equation: y = 30(3).
y = 90. The correct answer is C.
For the linear function f, the graph of y = f(x) in the xy-plane passes through the points (0, 5) and (7, 7). What is the slope of y = f(x)?
Find the slope using the two given points: slope = (7 − 5) / (7 − 0) = 2/7.
So the slope is 2/7.
The function f is defined by the equation f(x) = 100x + 2. What is the value of f(x) when x = 8?
Substitute x = 8 into the function: f(8) = 100(8) + 2.
f(8) = 800 + 2 = 802. The correct answer is D.
The graph of the linear equation 15x + By = 60 is shown, where B is a constant. What is the value of B?
Reading the graph, the line crosses the y-axis at approximately (0, 3) and the x-axis at (4, 0). Substitute the y-intercept into the equation: 15(0) + B(3) = 60, so 3B = 60.
Divide by 3: B = 20. (Check with the x-intercept: 15(4) + B(0) = 60 confirms x = 4 works for any B, so the y-intercept is the more useful point here.) The correct answer is D.
Which of the following equations is the most appropriate linear model for the data shown in the scatterplot?
The scatterplot shows points that generally increase together — as x increases, y increases at roughly the same rate, with points clustering near a line of slope close to 1 passing near the origin.
A negative slope (options A and B) would not match data that trends upward, and a slope of 2 (option D) would rise much faster than the data shows. The correct answer is C.
An equation of the line in the xy-plane that contains the points (0, 2) and (8, 0) is ax + by = 1, where a and b are constants. What is the value of a?
Using intercept form, a line with x-intercept 8 and y-intercept 2 can be written as x/8 + y/2 = 1.
Comparing this to ax + by = 1, the coefficient of x is a = 1/8.
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