SAT · Math · Algebra

Linear functions

Linear functions is part of Algebra in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include meaning of slope and intercept, equation from two points, rate of change from a table and parallel and perpendicular lines.

Common traps: swapping slope and intercept; missing the units of the rate; treating the intercept as a rate.

Sample questions

Question 1
For the linear function f, f(0) = 1 and f(1) = 6. What is the value of f(2)?
- A.
  10
- B.
  11
- C.
  12
- D.
  16
Answer: B. 11
The slope is (6 - (1))/(1 - 0) = 5, so f(x) = 5x + 1. Then f(2) = 5(2) + (1) = 11.
Question 2
At a charity auction, a fixed amount from entry fees is counted before bidding opens, and the total raised is then updated as bids are won.
Hours since bidding opened Total raised ($)
0 400
2 1,000
5 2,200
What was the average rate of change of the total raised, in dollars per hour, from hour 2 to hour 5?
- A.
  400
- B.
  440
- C.
  500
- D.
  1200
Answer: A. 400
Average rate of change = (2,200 − 1,000)/(5 − 2) = 1,200/3 = 400 dollars per hour. B divides the hour-5 total by 5, using an end value only; C divides the hour-2 total by 2, using the wrong interval; D is the change in dollars and was never divided by the 3 hours.
Question 3
Gym A charges a $25 joining fee plus $8 per visit. Gym B charges a $10 joining fee plus $11 per visit. For how many visits is the total cost the same at both gyms?
- A.
  3
- B.
  5
- C.
  11.7
- D.
  15
Answer: B. 5
B is correct: 25 + 8v = 10 + 11v gives 15 = 3v, so v = 5 visits (both cost $65). A is the difference in the per-visit prices. C divides the sum of the fees by the price difference. D is the difference in the joining fees.
Question 4
A cooking class charges a total fee F, in dollars, that is a linear function of the number of attendees n. For 2 attendees the fee is $190, and for 6 attendees it is $390. For how many attendees is the fee $840?
Enter your own answer.
Question 5
The line ℓ has the equation y = −(2/3)x + 5. The line m is perpendicular to ℓ and passes through the point (6, 1). What is the y-intercept of line m?
- A.
  -8
- B.
  -3
- C.
  5
- D.
  10
Answer: A. -8
A perpendicular slope is the negative reciprocal, 3/2. Then 1 = (3/2)(6) + b gives b = −8 (A). Choice B uses 2/3, changing only the sign; C uses the parallel slope −2/3 (b = 5); D uses −3/2, taking only the reciprocal.

At a charity auction, a fixed amount from entry fees is counted before bidding opens, and the total raised is then updated as bids are won.
Hours since bidding opened	Total raised ($)
0	400
2	1,000
5	2,200

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