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
A row of raised beds in a school garden follows the line 4x − 3y = 12. A path is to be laid perpendicular to this row. What is the slope of the path?
- A.
  −4/3
- B.
  −3/4
- C.
  3/4
- D.
  4/3
Answer: B. −3/4
The row has slope 4/3 (y = (4/3)x − 4). A perpendicular slope is the negative reciprocal, −3/4. A only negates; C only flips; D is the slope of the row itself, which is parallel.
Question 3
A quilt uses square patches. The table shows the number of patches y in a quilt with x rows.
Rows (x) Patches (y)
2 12
4 22
6 32
Which equation gives y in terms of x?
- A.
  y = 6x
- B.
  y = 2x + 5
- C.
  y = 5x + 2
- D.
  y = 10x + 2
Answer: C. y = 5x + 2
From x = 2 to x = 4, y rises 10 while x rises 2, so the slope is 5. Then 12 = 5(2) + b gives b = 2, so y = 5x + 2. Choice A (y = 6x) uses the first row's y ÷ x as the slope. Choice B (y = 2x + 5) swaps the slope and the intercept. Choice D (y = 10x + 2) uses the change in y (10) without dividing by the change in x.
Question 4
What is the slope of the line through (−2, 5) and (4, −1)?
- A.
  −1
- B.
  1
- C.
  −2
- D.
  2
Answer: A. −1
Slope = (−1−5)/(4−(−2)) = −6/6 = −1. Choice B ignores the negative; C doubles the correct answer; D reverses sign and doubles.
Question 5
A bakery in Cairo records the cost y, in Egyptian pounds, of baking x loaves in a batch.
Loaves x Cost y
20 95
50 170
80 245
The relationship is linear. What is the cost of baking 100 loaves?
- A.
  250
- B.
  295
- C.
  345
- D.
  475
Answer: B. 295
The slope is (170 − 95) ÷ (50 − 20) = 2.5 pounds per loaf, so y = 2.5x + 45. For x = 100, y = 250 + 45 = 295 pounds. Choice A (250) multiplies the 2.5 pounds per loaf by 100 and omits the fixed cost. Choice C (345) uses the first cost, 95, as the fixed cost. Choice D (475) treats the cost as proportional to the number of loaves.

Rows (x)	Patches (y)
2	12
4	22
6	32

Loaves x	Cost y
20	95
50	170
80	245

Find your real gaps

A few questions show the style. The free scan shows which ACT skills cost you the most points, then builds a plan around them.

Take the free scan