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
In the xy-plane, a line passes through the points (8, 2) and (4, -7). What is the slope of the line?
- A.
  -9/4
- B.
  -4/9
- C.
  4/9
- D.
  9/4
Answer: D. 9/4
Slope = (y₂ - y₁)/(x₂ - x₁) = (-7 - (2))/(4 - (8)) = -9/-4 = 9/4.
Question 2
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 3
For the linear function f, f(0) = -5 and f(4) = -21. What is the value of f(-3)?
- A.
  3
- B.
  7
- C.
  12
- D.
  63
Answer: B. 7
The slope is (-21 - (-5))/(4 - 0) = -4, so f(x) = -4x - 5. Then f(-3) = -4(-3) + (-5) = 7.
Question 4
A dog run is built along a fence. The table shows the length y, in meters, of wire still left after x posts are set.
x y
2 31
6 19
If y is a linear function of x, which equation relates y and x?
- A.
  y = −3x + 31
- B.
  y = −3x + 37
- C.
  y = 3x + 25
- D.
  y = −12x + 55
Answer: B. y = −3x + 37
Slope = (19 − 31)/(6 − 2) = −3, and 31 = −3(2) + b gives b = 37, so y = −3x + 37 (B). A uses the first y-value 31 as the intercept; C drops the negative sign of the slope (and then fits the first point); D does not divide the change −12 by 4.
Question 5
A bicycle courier compares two pay plans. Plan X pays $8 per delivery plus a $60 daily bonus. Plan Y pays $11 per delivery plus a $24 daily bonus. What is the least number of deliveries in a day for which Plan Y pays strictly more than Plan X?
- A.
  3
- B.
  12
- C.
  13
- D.
  36
Answer: C. 13
Plan X pays 8d + 60 and Plan Y pays 11d + 24. They are equal when 3d = 36, so d = 12. Y pays more only when d > 12, so the least whole number is 13 (C). A is the $3 rate difference, B is the tie point (not strictly more), and D is the $36 bonus gap.

A dog run is built along a fence. The table shows the length y, in meters, of wire still left after x posts are set.
x	y
2	31
6	19

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