Algebra · SAT Math

Linear Functions on the Digital SAT

Linear functions appear in roughly 1 in 5 Digital SAT math questions — the most-tested concept in the algebra domain. This lesson covers the three question types you'll see on test day (evaluating, writing, and interpreting), the most common traps, and 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~10 min read

The formula: f(x) = mx + b

A linear function is a rule that takes an input x and produces an output of the form f(x) = mx + b. The graph is always a straight line. Two numbers control everything:

m is the slope — the constant rate of change. Every time x increases by 1, the output changes by exactly m.

b is the y-intercept — the output when x = 0. In real-world problems, it's usually a starting amount, flat fee, or initial position.

Examples of linear functions: f(x) = 3x + 7, g(x) = -2x + 100, h(x) = (1/4)x - 9. Each one graphs as a straight line on the coordinate plane.

What you'll see on the test

The Digital SAT tests linear functions in three recurring formats. Master all three and you'll handle the vast majority of linear-function questions in both math modules.

  1. Evaluating a function — given a rule, find the output for a specific input (or the input that produces a given output).
  2. Writing a linear equation from a word problem — given a real-world scenario, build the equation that models it.
  3. Interpreting slope or y-intercept in context — given an equation tied to a real scenario, translate the numbers back into English.

Type 1: Evaluating a linear function

Evaluating means substituting a number for x and simplifying. The SAT tests this two ways:

  • Forward: given x, find f(x). Replace every x in the rule with the input, then simplify.
  • Backward: given f(x), find x. Set the rule equal to the given output and solve for x.

Worked example. If f(x) = 2x - 1, what is f(5)?

Substitute 5 for every x: f(5) = 2(5) - 1 = 10 - 1 = 9. The answer is 9.

The number one trap on these questions is treating f(5) as f times 5. f(5) is not multiplication — it means "plug 5 into the rule."

Type 2: Writing a linear function from a word problem

The SAT loves to dress linear functions up as real-world stories. The recipe is always the same:

  1. Name the variables. What does x measure? What does f(x) measure?
  2. Find the rate of change (m). Look for phrases like "per day," "for each additional," "every hour." The number attached to that unit rate is m.
  3. Find the starting value (b). What is the value when x = 0? An initial fee, weight at birth, base price — that's b.
  4. Assemble: f(x) = mx + b. If the quantity is decreasing, m is negative.

Common twist. The starting value isn't always the first number in the problem. Read carefully — separate "what happens at the start" from "what happens per unit."

Type 3: Interpreting slope and y-intercept

Once you have an equation, the SAT loves to ask "what does m mean?" or "what does b represent?" These questions aren't asking you to calculate — they're asking you to translate math into English.

Interpretation template

Slope: "For each additional [x-unit], [y-quantity] [increases/decreases] by [|m|] [y-units]."

Y-intercept: "When [x] is 0, [y-quantity] is [b] [y-units]."

Example. A plumber charges C(h) = 12h + 50 dollars to work for h hours. The slope 12 means "the plumber charges $12 per hour." The y-intercept 50 means "there's a $50 flat fee before any hours are worked."

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Solving with Desmos

The Digital SAT has the Desmos graphing calculator built into Bluebook. For linear-function questions, Desmos can turn 90-second algebra into 15-second graph reads. Three techniques cover most of what you'll need:

1. Forward evaluation — graph it, hover the input

Type f(x) = 2x - 1 into Desmos, then click the line near x = 5. Desmos shows the exact (5, 9) coordinate. Faster and less sign-error-prone than mental math.

2. Backward evaluation — graph two lines, find the intersection

To find x when f(x) = 11 in f(x) = 3x - 7: enter both y = 3x - 7 and y = 11. Desmos automatically marks the intersection — read the x-coordinate.

3. Word problems — verify with a table

Once you've written the equation, create a Desmos table with input/output pairs from the problem. If your equation is right, the table values will match the scenario's data.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Treating f(5) as f × 5

f(5) means substitute 5 for x in the function rule, not multiply f by 5. The most-missed easy question on the SAT is the one that tests this.

Sign errors with negative slopes

When m is negative, f(x) = -3x + 10 evaluated at x = 4 gives -12 + 10 = -2, not 22. Watch the sign on every step.

Mixing up slope and y-intercept in interpretation

In C(h) = 12h + 50, the slope is 12 (the per-hour rate) and the y-intercept is 50 (the flat fee). Swap them and you'll pick the wrong answer.

Forgetting that the 'first unit' may cost more

If a service charges $270 for day 1 and $135 per additional day, the rate for x days isn't 270x + 135. The total is 135x + 135 — base + per-additional structure.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

If f(x) = 2x - 1, what is the value of f(5)?

  1. 7
  2. 9
  3. 10
  4. 11

Click to reveal solution →

Answer: (B) 9

Substitute 5 for every x: f(5) = 2(5) - 1 = 10 - 1 = 9. Multiply first, then subtract.

2

The function h is defined by h(x) = 3x - 7. What is h(-2)?

  1. -13
  2. -10
  3. 10
  4. 13

Click to reveal solution →

Answer: (A) -13

h(-2) = 3(-2) - 7 = -6 - 7 = -13. The sign trap: 3 × (-2) is -6, not 6.

3

If f(x) = (2x - 1) / 3, what is f(5)?

  1. 4/3
  2. 7/3
  3. 3
  4. 9

Click to reveal solution →

Answer: (C) 3

Substitute 5 for x: f(5) = (2(5) - 1) / 3 = (10 - 1) / 3.

Simplify: 9 / 3 = 3.

4

The functions f and g are defined as f(x) = (1/4)x - 9 and g(x) = (3/4)x + 21. If h(x) = f(x) + g(x), what is the x-coordinate of the x-intercept of y = h(x)?

Click to reveal solution →

Answer: -12

Combine: h(x) = (1/4)x - 9 + (3/4)x + 21 = x + 12.

Set output to 0: 0 = x + 12, so x = -12.

5

The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Which equation gives the cost y, in dollars, of renting for x days, where x ≤ 10?

  1. y = 270x − 135
  2. y = 270x + 135
  3. y = 135x + 270
  4. y = 135x + 135

Click to reveal solution →

Answer: (D) y = 135x + 135

Day 1 costs $270. Each extra day costs $135, and there are (x - 1) extra days.

y = 270 + 135(x - 1) = 270 + 135x - 135 = 135x + 135.

6

The boiling point of water at sea level is 212°F. For every 550 feet above sea level, the boiling point drops by about 1°F. Which equation finds the boiling point B at x feet above sea level?

  1. B = 550 + x/212
  2. B = 550 − x/212
  3. B = 212 + x/550
  4. B = 212 − x/550

Click to reveal solution →

Answer: (D) B = 212 − x/550

Starting value: at sea level B = 212.

Rate: every 550 ft lowers B by 1, so per foot the change is -1/550.

Equation: B = 212 - x/550.

7

Sean rents a tent at $11 per day plus a one-time insurance fee of $10. Which equation represents the total cost c to rent the tent with insurance for d days?

  1. c = 11(d + 10)
  2. c = 10(d + 11)
  3. c = 11d + 10
  4. c = 10d + 11

Click to reveal solution →

Answer: (C) c = 11d + 10

Slope = $11/day; y-intercept = $10. So c = 11d + 10. Choice A wrongly charges the insurance every day.

8

The function f(x) = 33 - 0.18x gives the estimated height of a liquid in a container x days into an evaporation experiment. Which is the best interpretation of 33?

  1. Height at the start of the experiment.
  2. Height at the end of the experiment.
  3. Change in height each day.
  4. Number of days until full evaporation.

Click to reveal solution →

Answer: (A) Height at the start of the experiment.

The y-intercept is the output when x = 0. At x = 0 days, f(0) = 33 cm — the starting height before any time passes.

Frequently asked questions

What is a linear function on the SAT?

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A linear function is a rule of the form f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept (the output when x = 0). The graph is always a straight line. On the Digital SAT, linear functions appear in roughly 1 in 5 math questions.

What does f(x) mean?

+
f(x) is read as 'f of x' and means 'plug the input x into the function rule.' It is not multiplication. For example, if f(x) = 2x + 3, then f(5) = 2(5) + 3 = 13.

How do you find the slope of a linear function?

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If the function is given in the form f(x) = mx + b, the slope is m — the number multiplied by x. If you have two points (x1, y1) and (x2, y2) on the line, the slope is (y2 − y1) divided by (x2 − x1).

What is the difference between slope and y-intercept?

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The slope tells you how much the output changes for each 1-unit increase in the input. The y-intercept tells you the output when the input is zero — often a starting amount, flat fee, or initial value.

How many SAT questions involve linear functions?

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Roughly 1 in 5 Digital SAT math questions involve linear functions or linear equations. They appear in both math modules and at all difficulty levels — easy questions test evaluation, medium questions test writing equations from word problems, and hard questions test interpretation in context.

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