ACT · Math · Preparing for Higher Math: Functions

Linear functions and slope

Linear functions and slope is part of Preparing for Higher Math: Functions in the ACT 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 slope from two points, equation from a slope and a point, parallel or perpendicular line and interpret slope and intercept in context.

Common traps: subtracting x over y; subtracting in inconsistent orders; a sign error with a negative coordinate.

Sample questions

Question 1
Line ℓ is defined by y = -(1/5)x - 5. Line k is perpendicular to line ℓ in the xy-plane. What is the slope of line k?
- A.
  -5
- B.
  -1/5
- C.
  1/5
- D.
  5
Answer: D. 5
Line ℓ has slope -1/5. Perpendicular lines have slopes that are negative reciprocals, so the slope is 5.
Question 2
Line ℓ is defined by y = (1/7)x - 9. Line k is parallel to line ℓ in the xy-plane. What is the slope of line k?
- A.
  -7
- B.
  -1/7
- C.
  1/7
- D.
  7
Answer: C. 1/7
Line ℓ has slope 1/7. Parallel lines have equal slopes, so the slope is 1/7.
Question 3
In the xy-plane, a line passes through the points (8, -6) and (-6, 7). What is the slope of the line?
- A.
  -14/13
- B.
  -13/14
- C.
  13/14
- D.
  14/13
Answer: B. -13/14
Slope = (y₂ - y₁)/(x₂ - x₁) = (7 - (-6))/(-6 - (8)) = 13/-14 = -13/14.
Question 4
In the xy-plane, a line passes through the points (-2, -4) and (9, 8). What is the slope of the line?
- A.
  -12/11
- B.
  -11/12
- C.
  11/12
- D.
  12/11
Answer: D. 12/11
Slope = (y₂ - y₁)/(x₂ - x₁) = (8 - (-4))/(9 - (-2)) = 12/11 = 12/11.
Question 5
In the standard (x, y) coordinate plane, a line passes through the points (2, 7) and (−4, −2). At what point does the line cross the y-axis?
- A.
  (0, 3)
- B.
  (0, 4)
- C.
  (0, 10)
- D.
  (4, 0)
Answer: B. (0, 4)
The slope is (7 − (−2))/(2 − (−4)) = 9/6 = 3/2. Substituting (2, 7) into y = (3/2)x + b gives 7 = 3 + b, so b = 4 and the line crosses the y-axis at (0, 4). Choice C adds 3 to 7 instead of subtracting; choice A stops at (3/2)(2) = 3, the amount subtracted; choice D puts the intercept on the x-axis.

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