Math · Algebra

Linear Functions on the Digital SAT

Linear functions are where the SAT stops asking you to just solve and starts asking you to interpret. You'll see f(x) = mx + b in equations, tables, and word problems about gym memberships, water tanks, and delivery fees, and the test wants to know whether you understand what the slope and the starting value actually mean.

This skill sits inside Algebra, the biggest Math domain on the Digital SAT, and it rewards one specific habit: reading m and b as real things, not just letters. The slope is a rate, something per something, like dollars per hour or gallons per minute. The y-intercept is the starting amount before anything happens. Students who miss these questions almost always have the two mixed up.

The other half of the skill is building a linear function from limited information, usually two points or two rows of a table. Find the slope from the two points, use one point to pin down the intercept, and always verify with the point you didn't use.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Evaluating a function like f(x) = 3x - 4 at a given input, and finding the input that produces a given output.
Interpreting the slope and y-intercept of a linear model in a real-world context.
Writing the equation of a linear function from two points, a table of values, or a verbal description.
Identifying slope and y-intercept from an equation, including forms that need rearranging first.
Reading slope, direction, and x- and y-intercepts from a graph, and matching an equation to its graph.
Finding an unknown constant in a function from a single condition, such as using g(4) = 31 to solve for a missing coefficient.
Comparing rates of change between two linear functions or models.
Evaluating composed functions, such as f(f(x)) or g(x) = f(2x - 1), by substituting the inner function first.
Shifting a function with h(x) = f(x - a) + b and working with inverse linear functions to find an input from a given output.
Understanding function notation, such as what f(5) = 11 says about the function.

Strategies that work

Attach units to the slope

Whenever a linear model appears in a word problem, immediately write the slope with its units, like 45 dollars per hour. Interpretation questions become almost automatic once the units are explicit, because only one answer choice will describe that exact rate.

Use slope first, then one point

Given two points or two table rows, compute the slope as the change in output divided by the change in input. Then substitute one point into y = mx + b to find b. Finish by plugging in the second point as a check: if it doesn't satisfy your equation, you've caught your own error before the test did.

Translate function notation into a point

Every statement like f(2) = 7 is just the point (2, 7) in disguise. When a question hands you function values, rewrite them as points right away and the problem turns into a familiar two-points question instead of an abstract notation puzzle.

Test answer choices against both given points

On which-equation-defines-f questions, wrong choices are often built to pass through exactly one of the given points. Checking a single point can leave you confident in a wrong answer. A quick substitution of both points eliminates every trap.

Mistakes to avoid

Mixing up slope and y-intercept, such as reading C(h) = 45h + 60 as a 60 dollar hourly rate with a 45 dollar fee.
Computing slope as change in x over change in y instead of change in y over change in x.
Using a given y-value directly as the y-intercept without solving for b.
Forgetting the constant when evaluating, for example finding f(5) for f(x) = 3x - 4 and answering 15.
Verifying a candidate equation against only one of the two given points.

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Linear Functions questions in a few clicks. Here is the move that applies, step by step.

Read a Line's Slope and Intercepts from Its GraphSlope, x-intercept, y-intercept: three answers, one graph.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

The function f is defined by f(x) = 3x - 4.

What is the value of f(5)?

Pick an answer to get instant coaching from Nova.

Question 2Medium

A plumber charges C(h) = 45h + 60 dollars for a job that takes h hours to complete.

What is the best interpretation of the number 60 in this context?

Pick an answer to get instant coaching from Nova.

Question 3Hard

The table gives two values of the linear function f.

Selected Values of f
x	f(x)
2	7
5	16

Which equation defines f?

Pick an answer to get instant coaching from Nova.

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The rest of Algebra

Linear Functions is one of 4 skills in Algebra. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Algebra fits together.

Other Math skills to explore

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