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Linear Relationships Checkpoint

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.

This checkpoint mixes function notation, slope, graphing, and line forms. Expect to move between tables, equations, points, intercepts, and parallel or perpendicular relationships.

What you'll learn

Why it matters: Linear-relationship fluency supports comparing plans, graphing data, checking map routes, and explaining why two quantities move together.

Worked example

Problem. Write the line through (2, 5) with slope 3 in slope-intercept form.

  1. Start with point-slope form: y - 5 = 3(x - 2).
  2. Distribute: y - 5 = 3x - 6.
  3. Add 5 to get y = 3x - 1.

Answer: y = 3x - 1

Practice problems

1. If f(x) = 2x - 5, find f(8).

Show solution
  1. Functions: First identify exactly what the question is asking: If f(x) = 2x - 5, find f(8).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Substitute 8 for x.
  4. 2(8) - 5 = 11.
  5. Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.

Answer: 11

2. Which relation is not a function?

Choices: (2, 1), (2, 5), (3, 6) · (1, 4), (2, 4), (3, 4) · (0, 0), (1, 2), (2, 4) · (-1, 3), (0, 3), (1, 3)

Show solution
  1. Functions: First identify exactly what the question is asking: Which relation is not a function?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The input 2 has two different outputs.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (2, 1), (2, 5), (3, 6)

3. Find the slope through (1, 4) and (5, 12).

Show solution
  1. Slope: First identify exactly what the question is asking: Find the slope through (1, 4) and (5, 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 is 8.
  4. Change in x is 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

4. A line has slope -3 and x changes from 2 to 6. What is the change in y?

Show solution
  1. Slope: First identify exactly what the question is asking: A line has slope -3 and x changes from 2 to 6. What is the change in y?
  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 change in x is 4.
  4. -3 x 4 = -12.
  5. Check the result by substituting or estimating: the response should match -12 and make sense in the original problem.

Answer: -12

5. In y = -4x + 9, what is the y-intercept?

Show solution
  1. Slope-Intercept: First identify exactly what the question is asking: In y = -4x + 9, what is the y-intercept?
  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-intercept is b.
  4. Here b = 9.
  5. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

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