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Average Rate of Change

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

The average rate of change of f(x) on the interval [a, b] is (f(b) - f(a)) / (b - a) — the slope of the line through the two endpoints. For linear functions, this slope is the same on every interval. For nonlinear functions, it varies.

What you'll learn

Why it matters: Average speed (distance per time) over a trip, average growth rate of a population between two years, and average exam-score change across a semester are all average rates of change.

Worked example

Problem. For f(x) = x^2, find the average rate of change from x = 1 to x = 3.

  1. f(3) - f(1) = 9 - 1 = 8.
  2. 3 - 1 = 2.
  3. ARC = 8 / 2 = 4.

Answer: 4

Practice problems

1. For f(x) = 2x + 5, find the ARC from x = 1 to x = 4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = 2x + 5, find the ARC from x = 1 to x = 4.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(4) - f(1) = 13 - 7 = 6.
  4. 4 - 1 = 3. ARC = 6/3 = 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. For f(x) = x^2, find the ARC from x = 2 to x = 4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = x^2, find the ARC from x = 2 to x = 4.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(4) - f(2) = 16 - 4 = 12.
  4. 4 - 2 = 2. ARC = 12/2 = 6.
  5. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

3. For f(x) = 3x, find the ARC from x = 0 to x = 5.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For f(x) = 3x, find the ARC from x = 0 to x = 5.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(5) - f(0) = 15.
  4. 5 - 0 = 5. ARC = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. For f(x) = x^3, find the ARC from x = 1 to x = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = x^3, find the ARC from x = 1 to x = 2.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(2) - f(1) = 8 - 1 = 7.
  4. 2 - 1 = 1. ARC = 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

5. For f(x) = x^2 + 1, find the ARC from x = 0 to x = 3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For f(x) = x^2 + 1, find the ARC from x = 0 to x = 3.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(3) - f(0) = 10 - 1 = 9.
  4. 3 - 0 = 3. ARC = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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