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Slope and 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.

Slope measures how steep a line is by comparing vertical change to horizontal change. It is often described as rise over run: change in y divided by change in x. Positive slopes rise from left to right, negative slopes fall, zero slopes are horizontal, and vertical lines have undefined slope because the run is zero. Slope matters because it represents a rate of change, such as dollars per hour or miles per minute. In practice, carefully subtract y-values and x-values in the same order. The most common mistake is reversing the points or putting run over rise.

What you'll learn

Why it matters: Slope describes rates and steepness: pay per hour, miles per minute, ramp design, roof pitch, and trends on a graph.

Worked example

Problem. Find the slope through (1, 3) and (4, 9).

  1. Find the change in y: 9 - 3 = 6.
  2. Find the change in x: 4 - 1 = 3.
  3. Slope = 6/3 = 2.
  4. Connect the result back to Slope and Rate of Change so the method and meaning are both clear.

Answer: 2

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. Find the slope through (0, 0) and (2, 6).

Show solution
  1. Change in y is 6.
  2. Change in x is 2.
  3. 6/2 = 3.

Answer: 3

2. Find the slope through (1, 5) and (3, 5).

Show solution
  1. The y-values are the same.
  2. A horizontal line has slope 0.

Answer: 0

3. A line that rises left to right has what kind of slope?

Choices: Positive · Negative · Zero · Undefined

Show solution
  1. Rising left to right means y increases as x increases.

Answer: Positive

4. Find the slope through (2, 4) and (6, 12).

Show solution
  1. Change in y: 12 - 4 = 8.
  2. Change in x: 6 - 2 = 4.
  3. 8/4 = 2.

Answer: 2

5. Find the slope through (3, 10) and (5, 4).

Show solution
  1. Change in y: 4 - 10 = -6.
  2. Change in x: 5 - 3 = 2.
  3. -6/2 = -3.

Answer: -3

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