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Linear Functions Introduction

A free Pre-Algebra lesson from the “Coordinate Plane, Functions, and Slope” unit, with a worked example and practice problems including step-by-step solutions.

A function pairs each input with exactly one output. A linear function has a constant rate of change — its graph is a straight line and its equation looks like y = mx + b (or f(x) = mx + b). Function notation f(x) lets you name the rule and ask for its value at any input. In Coordinate Plane, Functions, and Slope, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

What you'll learn

Why it matters: Cell-phone plans with a flat fee plus a per-minute charge, taxi fares (drop fee plus per-mile rate), and any constant-rate scenario produce linear functions.

Worked example

Problem. If f(x) = 3x + 2, find f(4).

  1. Substitute 4 for x: f(4) = 3(4) + 2.
  2. = 12 + 2 = 14.
  3. Connect the calculation back to Linear Functions Introduction so the method, not just the arithmetic, is clear.
  4. Check the result against the original question before writing the final answer.

Answer: 14

Practice problems

1. Is y = 2x + 5 a linear function?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = 2x + 5 a linear function?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Form y = mx + b with m=2 and b=5.
  4. That's a linear function.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

2. Is y = x^2 a linear function?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is y = x^2 a linear function?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. x^2 is degree 2 (a parabola).
  4. Linear functions are degree 1.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No

3. If f(x) = x + 7, find f(3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = x + 7, find f(3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(3) = 3 + 7 = 10.
  4. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.
  5. Write the final response in the form requested by the prompt.

Answer: 10

4. If f(x) = 2x, find f(-3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(x) = 2x, find f(-3).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(-3) = 2 * (-3) = -6.
  4. Check the result by substituting or estimating: the response should match -6 and make sense in the original problem.
  5. Write the final response in the form requested by the prompt.

Answer: -6

5. A table has (1,3), (2,5), (3,7), (4,9). Linear?

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: A table has (1,3), (2,5), (3,7), (4,9). Linear?
  2. Look for a constant rate of change and connect the equation, table, or graph back to that rate.
  3. Outputs increase by 2 each step.
  4. Constant difference means linear.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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