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Average Rate of Change

A free Precalculus lesson from the “Function Foundations and Behavior” unit, with a worked example and practice problems including step-by-step solutions.

Average rate of change is slope between two function values: change in output divided by change in input. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Function behavior is the language behind Calculus, physics graphs, economic models, and computer-generated curves.

Worked example

Problem. Find the average rate of change of f(x) = 2x + 2 from x = 1 to x = 4.

  1. Worked Example: First identify exactly what the question is asking: Find the average rate of change of f(x) = 2x + 2 from x = 1 to x = 4.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. A linear function's average rate of change equals its slope.
  4. The slope of 2x + 2 is 2.
  5. So the average rate is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

Practice problems

1. Find the average rate of change of f(x) = 2x + 2 from x = 1 to x = 4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the average rate of change of f(x) = 2x + 2 from x = 1 to x = 4.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. A linear function's average rate of change equals its slope.
  4. The slope of 2x + 2 is 2.
  5. So the average rate is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. Find the average rate of change of f(x) = x^2 from x = 2 to x = 5.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the average rate of change of f(x) = x^2 from x = 2 to x = 5.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. f(5) - f(2) = 25 - 4 = 21.
  4. Divide by 5 - 2 = 3.
  5. 21/3 = 7.
  6. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

3. Find the average rate of change of f(x) = x^2 + 4 from x = 0 to x = 3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the average rate of change of f(x) = x^2 + 4 from x = 0 to x = 3.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. f(3) - f(0) = (9 + 4) - (0 + 4) = 9.
  4. The +4 cancels; divide by 3.
  5. So the average rate is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. Find the average rate of change of f(x) = 1x^2 from x = 1 to x = 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the average rate of change of f(x) = 1x^2 from x = 1 to x = 4.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. f(4) - f(1) = 1(16) - 1(1) = 15.
  4. Divide by 3.
  5. So the average rate is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Find the average rate of change of f(x) = x^3 from x = 2 to x = 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the average rate of change of f(x) = x^3 from x = 2 to x = 5.
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. f(5) - f(2) = 125 - 8 = 117.
  4. Divide by 5 - 2 = 3.
  5. So the average rate is 39.
  6. Check the result by substituting or estimating: the response should match 39 and make sense in the original problem.

Answer: 39

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