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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.

Answer: 2

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (0, 0) and (2, 6).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Change in y is 6.
  4. Change in x is 2.
  5. 6/2 = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (1, 5) and (3, 5).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. The y-values are the same.
  4. A horizontal line has slope 0.
  5. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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

Choices: Positive · Negative · Zero · Undefined

Show solution
  1. Warm-up: First identify exactly what the question is asking: A line that rises left to right has what kind of slope?
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Rising left to right means y increases as x increases.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Positive

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the slope through (2, 4) and (6, 12).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Change in y: 12 - 4 = 8.
  4. Change in x: 6 - 2 = 4.
  5. 8/4 = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the slope through (3, 10) and (5, 4).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Change in y: 4 - 10 = -6.
  4. Change in x: 5 - 3 = 2.
  5. -6/2 = -3.
  6. 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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