These Digital SAT Algebra questions dress a slope-intercept line up in function notation — f(x) = mx + b — and ask you to evaluate it, build it from outputs, a table or a graph, or explain what a number in it means.
Linear functions is one of five skills in College Board's Algebra domain, along with linear equations in one variable, linear equations in two variables, systems of two linear equations and linear inequalities. Algebra is about 35% of SAT Math, or 13 to 15 questions, shared among those five.
What sets this skill apart is the input-output idea. In a linear function, College Board's framework says, “one of the variables is designated as the input and determines a unique value of the other variable, which is the output.” Its list for the skill covers evaluating a function, finding “the input value for a corresponding output,” writing “the rule for a linear function given two input/output pairs or one input/output pair and the rate of change,” interpreting each part “based on the context,” and connecting tables, equations and graphs. Its two sample questions test evaluation, with two functions combined in one expression, and reading a graph in context.
Math is two adaptive 35-minute modules of 22 questions. About 25% of the questions are student-produced response, where you type the answer, and about 30% are set in context. College Board's Skills Insight ties skill statements to Math score bands. These are the Algebra statements that apply to linear functions, quoted without a note that limits part of the top one to particular tests.
Each band adds a move: plug in, read a graph, interpret a number, build the function, then handle unknown constants. Sections 03 to 06 work through them.
Function notation intimidates students who could solve the exact same equation written as y = mx + b without blinking. f(x) is just a label meaning "the output of the line, in terms of x." Once you translate the notation, everything you already know about slope and intercept applies directly.
Three moves handle almost every linear function question. First, read the notation: f(a) means "plug a in for x and compute the output" — it is a value, not a multiplication of f and a. Second, identify the slope and intercept exactly as before: in f(x) = mx + b, m is the rate of change and b is f(0), the output when the input is zero. Third, evaluate by substituting the given input for every x and simplifying with order of operations.
Walk through a context example — a ride-share fare that depends on distance.
If f(x) = −3x + 11, what is f(4)?
Evaluating is the easy half. The harder skill — building a function when you're only given outputs — comes next.
Some questions never show you m or b directly — instead you get two input-output pairs, like f(2) = 7 and f(5) = 16, and you have to reconstruct the function. This is exactly the two-points-to-a-line problem from linear equations, just written with function notation.
Find the slope the same way you would from two points: divide the change in output by the change in input, m = (f(b) − f(a)) / (b − a). Then plug either known pair back into f(x) = mx + b and solve for the one remaining unknown, b.
A tree's height h(t), in feet, is a linear function of t, the number of years since it was planted. If h(2) = 7 and h(5) = 16, which equation gives h(t)?
A table works the same way. Any two rows give the slope, and a third row is a free check that the function really is linear.
Given one pair and the rate, skip straight to b: a plant 14 cm tall after 4 weeks, growing 2.5 cm per week, gives 14 = 2.5(4) + b, so h(w) = 2.5w + 4.
If the point (a, b) is on the graph of y = f(x), then f(a) = b. Read three things from a graph. The y-intercept is f(0). The slope is rise over run between two points where the line crosses grid corners exactly. The x-intercept is the input that makes the output zero, the solution to f(x) = 0.
The graph of the linear function f is shown above. Which equation defines f?
In context, the x-intercept has a meaning of its own, such as the moment a tank runs empty. Keep the axes straight, too: in College Board's sample question about a graph in context, one wrong answer is the input value read back as the output.
Interpretation questions describe a real situation with a linear function and ask what a specific number in it means — no computation required, just connecting the number to its role in f(x) = mx + b.
b, which equals f(0), is always the output before the input has changed at all — an initial amount, a starting balance, a base price. m is always a rate: how much the output changes for each one-unit increase in the input, and its sign tells you the direction (positive means increasing, negative means decreasing). Neither number is a total, and neither is a specific point in time unless the question says the input equals that value.
A company's profit, in thousands of dollars, after t months in business is given by P(t) = 12t − 30. What does the number 12 represent?
Harder versions use what Skills Insight calls a “complex linear function” (610–670), often built around a shifted input like C(n) = 40 + 0.15(n − 500). Don't expand it. Ask what each piece does: n − 500 counts the minutes over 500, so 500 is the allowance and 40 is the cost at exactly 500 minutes.
For a phone plan, the monthly cost, in dollars, for n minutes of calls, where n ≥ 500, is C(n) = 40 + 0.15(n − 500). What is the best interpretation of 500 in this context?
Combined functions. In 3f(4) − g(4), evaluate each function first, then do the arithmetic. The 3 multiplies the output f(4), not the input. College Board's sample question for evaluating linear functions is built this way, with wrong answers that drop the multiplier or apply it to the input.
The functions f and g are defined by f(x) = 2x − 5 and g(x) = −x + 8. What is the value of 3f(4) − g(4)?
Translations. The 550–600 statement includes “details about a translation of a given function.” f(x) + k moves the line up k units; f(x − h) moves it right h units. For f(x) = 2x + 1, down 4 gives 2x − 3 and right 4 gives 2(x − 4) + 1 = 2x − 7. The slope never changes, so for this line (slope 2), right 4 is the same as down 8.
The function f is defined by f(x) = 3x − 2. The graph of y = g(x) is the graph of y = f(x) shifted 5 units to the right. Which equation defines g?
The built-in Desmos calculator is available on every Math question. On this skill it handles the arithmetic and the graph reading.
f(x) = 2.25x + 3.5, then f(8) on a new line. Desmos shows 21.5.y = f(x) and y = 29 and click the crossing point; its x-coordinate is the input.y₁ ~ mx₁ + b. Desmos reports m and b; for exactly linear data they're the slope and intercept.f(x) = ax + 6 with a slider for a and drag until the line crosses the x-axis at 3. Confirm: 3a + 6 = 0, so a = −2.Most points lost on this skill come from misreading notation under time pressure, not from the algebra itself. Three patterns cause almost all of the damage.
First, f(x + 2) is not the same as f(x) + 2. The first shifts the input before the function acts on it — substitute x + 2 for every x in the rule. The second adds 2 to the output after the function has already run. For f(x) = 4x − 5: f(x + 2) = 4(x + 2) − 5 = 4x + 3, while f(x) + 2 = 4x − 5 + 2 = 4x − 3 — different expressions.
Second, f(a + b) almost never equals f(a) + f(b). Evaluate the sum of the inputs first, then apply the function once — don't apply the function twice and add. Third, when a function shows up with a variable other than x — f(d), C(t), h(n) — that letter is just the input's name; the rule works exactly the same way.
Three more slips show up in the answer choices:
f(0) versus f(x) = 0. The first is an output, the y-intercept. The second asks for an input, the x-intercept.P(t) = 12t − 30 is in thousands of dollars, the rate is $12,000 per month, not $12.What role does each number play?
Speed on this skill comes from no longer pausing to "translate" function notation — your eyes should go straight from f(a) to a substitution. Mix evaluating, reading intercepts, and the addition traps below.
Try to answer before revealing each one.
If g(x) = 4x − 9, what is g(2)?
If h(x) = −2x + 5, find h(0).
If p(t) = 6t + 3, what is p(4) − p(1)?
If f(x) = 3x − 2, is f(5) equal to f(2) + f(3)?
A linear function satisfies f(0) = 9 and has slope −3. Write f(x).
If f(x) = 2x + 7, what is f(x + 1) − f(x)?
If f(x) = −4x + 10, for what x is f(x) = −6?
A table shows x = 0, 2, 4 and f(x) = 7, 3, −1. Write f(x).
If f(x) = 5x − 3, write f(x − 2).
Where does the graph of f(x) = 2x + 6 cross the x-axis?
If f(x) = 3x + 1 and g(x) = x − 4, what is 2f(3) + g(3)?
Once function notation feels automatic, the next place these skills combine is systems of linear equations, where you'll set two expressions — sometimes written as functions — equal to each other. Quadratic and exponential functions follow the same input-output rules; see our nonlinear functions guide.
Here are three Linear Functions questions from our problem sets, from easier to harder, in the same format as the Digital SAT. Answer each one to see the full explanation.
Shorter questions on the same skills. Questions 6 and 7 make you build the function before you use it.
If f(x) = 5x − 8, what is f(−2)?
A linear function f satisfies f(1) = 4 and f(3) = 14. What is f(0)?
If f(x) = 4x − 5, what is f(x + 2) − f(x)?
A company's profit is modeled by P(t) = 12t − 30, in thousands of dollars, where t is months in business. What is P(5)?
A rideshare fare is modeled by f(d) = 2.5d + 4, where d is the distance in miles and f(d) is the fare in dollars. If a ride costs $29, how many miles was the ride?
For the linear function f, f(3) = 10, and the graph of y = f(x) has a slope of −2. What is the x-intercept of the graph?
The function f is defined by f(x) = ax + b, where a and b are constants. If f(2) = 11 and f(−1) = 2, what is the value of f(a)?
Related: Linear equations in one variable · Linear equations in two variables · Systems of linear equations · Linear inequalities · Nonlinear functions · SAT Math overview