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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. In Functions, Linear Relationships, and Rate of Change, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.

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.
  4. Connect the result back to Average Rate of Change so the method and meaning are both clear.

Answer: 4

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

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

Show solution
  1. f(4) - f(1) = 13 - 7 = 6.
  2. 4 - 1 = 3. ARC = 6/3 = 2.

Answer: 2

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

Show solution
  1. f(4) - f(2) = 16 - 4 = 12.
  2. 4 - 2 = 2. ARC = 12/2 = 6.

Answer: 6

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

Show solution
  1. f(5) - f(0) = 15.
  2. 5 - 0 = 5. ARC = 3.

Answer: 3

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

Show solution
  1. f(2) - f(1) = 8 - 1 = 7.
  2. 2 - 1 = 1. ARC = 7.

Answer: 7

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

Show solution
  1. f(3) - f(0) = 10 - 1 = 9.
  2. 3 - 0 = 3. ARC = 3.

Answer: 3

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