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Linear functions: SAT practice

A linear function changes by the same amount every time x increases by 1. That amount is the slope. If you know two points, you know the whole function.

By Eren ŞimşirUpdated Questions checked How we write questions

Key numbers

Section
SAT Math
Domain
Algebra
Questions here
10 per set, unlimited sets
Explanations
Every choice

The rules that solve most questions

  • Slope m = (y₂ − y₁) / (x₂ − x₁). Subtract in the same order on top and bottom.
  • Find the intercept b by substituting one point into y = mx + b.
  • f(a) means "replace x with a."
  • In a word problem, the slope is the rate (per hour, per item) and the intercept is the starting value.

Traps the wrong answers are built on

  • Subtracting the points in opposite orders, which flips the slope's sign.
  • Forgetting the intercept and computing only m · x.
  • Mixing up which value is the input and which is the output.

Worked example

A linear function f satisfies f(−2) = 2 and f(3) = 12. What is the value of f(6)?

  1. A. 12
  2. B. −6
  3. C. 18
  4. D. 6

Answer: C. Slope: m = (12 − 2) / (3 − (−2)) = 10 / 5 = 2. Intercept: 2 = 2(−2) + b, so b = 6. So f(x) = 2x + 6, and f(6) = 2(6) + 6 = 18.

Practice set

Answer each question to see whether you're right and why. When you finish, draw a new set: the numbers change every time.

0 of 10 answered · 0 correct · set 1

  1. Question 1Medium

    A linear function f satisfies f(−2) = 2 and f(3) = 12. What is the value of f(6)?

    Show answer and explanation

    Answer: C

    Slope: m = (12 − 2) / (3 − (−2)) = 10 / 5 = 2.

    Intercept: 2 = 2(−2) + b, so b = 6.

    So f(x) = 2x + 6, and f(6) = 2(6) + 6 = 18.

  2. Question 2Medium

    A linear function f satisfies f(−2) = 1 and f(3) = 26. What is the value of f(6)?

    Show answer and explanation

    Answer: D

    Slope: m = (26 − 1) / (3 − (−2)) = 25 / 5 = 5.

    Intercept: 1 = 5(−2) + b, so b = 11.

    So f(x) = 5x + 11, and f(6) = 5(6) + 11 = 41.

  3. Question 3Medium

    A linear function f satisfies f(−1) = −15 and f(3) = 9. What is the value of f(12)?

    Show answer and explanation

    Answer: B

    Slope: m = (9 − (−15)) / (3 − (−1)) = 24 / 4 = 6.

    Intercept: −15 = 6(−1) + b, so b = −9.

    So f(x) = 6x − 9, and f(12) = 6(12) − 9 = 63.

  4. Question 4Medium

    A linear function f satisfies f(3) = −10 and f(9) = −40. What is the value of f(12)?

    Show answer and explanation

    Answer: B

    Slope: m = (−40 − (−10)) / (9 − 3) = −30 / 6 = −5.

    Intercept: −10 = −5(3) + b, so b = 5.

    So f(x) = −5x + 5, and f(12) = −5(12) + 5 = −55.

  5. Question 5Medium

    A linear function f satisfies f(−3) = 16 and f(−1) = 12. What is the value of f(7)?

    Show answer and explanation

    Answer: D

    Slope: m = (12 − 16) / (−1 − (−3)) = −4 / 2 = −2.

    Intercept: 16 = −2(−3) + b, so b = 10.

    So f(x) = −2x + 10, and f(7) = −2(7) + 10 = −4.

  6. Question 6Medium

    A linear function f satisfies f(2) = −6 and f(8) = −24. What is the value of f(5)?

    Show answer and explanation

    Answer: C

    Slope: m = (−24 − (−6)) / (8 − 2) = −18 / 6 = −3.

    Intercept: −6 = −3(2) + b, so b = 0.

    So f(x) = −3x, and f(5) = −3(5) + 0 = −15.

  7. Question 7Medium

    A linear function f satisfies f(2) = 0 and f(5) = 6. What is the value of f(10)?

    Show answer and explanation

    Answer: D

    Slope: m = (6 − 0) / (5 − 2) = 6 / 3 = 2.

    Intercept: 0 = 2(2) + b, so b = −4.

    So f(x) = 2x − 4, and f(10) = 2(10) − 4 = 16.

  8. Question 8Medium

    A linear function f satisfies f(−3) = −9 and f(−1) = −5. What is the value of f(11)?

    Show answer and explanation

    Answer: C

    Slope: m = (−5 − (−9)) / (−1 − (−3)) = 4 / 2 = 2.

    Intercept: −9 = 2(−3) + b, so b = −3.

    So f(x) = 2x − 3, and f(11) = 2(11) − 3 = 19.

  9. Question 9Medium

    A linear function f satisfies f(2) = 16 and f(3) = 19. What is the value of f(11)?

    Show answer and explanation

    Answer: D

    Slope: m = (19 − 16) / (3 − 2) = 3 / 1 = 3.

    Intercept: 16 = 3(2) + b, so b = 10.

    So f(x) = 3x + 10, and f(11) = 3(11) + 10 = 43.

  10. Question 10Medium

    A linear function f satisfies f(3) = 6 and f(7) = 10. What is the value of f(4)?

    Show answer and explanation

    Answer: D

    Slope: m = (10 − 6) / (7 − 3) = 4 / 4 = 1.

    Intercept: 6 = 1(3) + b, so b = 3.

    So f(x) = x + 3, and f(4) = 1(4) + 3 = 7.

More Algebra practice

Questions people ask

Is this linear functions practice like the real SAT?

The questions follow the format of the digital SAT's Algebra domain: four answer choices built from common mistakes. They are written for this site; they are not official College Board questions.

How many practice questions are there?

As many as you want. Each math question is generated from a template with new numbers, and its answer is checked by code. Use "New set" for another ten.

Where is the domain on the test?

Algebra is one of the four content domains of the SAT Math section, alongside Advanced Math, Problem-Solving and Data Analysis, Geometry and Trigonometry.