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Reflections and Stretches

A free Precalculus lesson from the “Transformations and Combinations of Functions” unit, with a worked example and practice problems including step-by-step solutions.

Multiplying outside changes y-values; multiplying inside changes x-values. A negative sign reflects across an axis. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Transformations let students predict how a model changes when a situation is shifted, scaled, reflected, or combined.

Worked example

Problem. Compared with f(x), y = -f(x) is reflected across the:

  1. Worked Example: First identify exactly what the question is asking: Compared with f(x), y = -f(x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative outside changes y-values.
  4. Output signs flip.
  5. That reflects across the x-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-axis

Practice problems

1. Compared with f(x), y = -f(x) is reflected across the:

Choices: x-axis · y-axis · line y = x · origin only

Show solution
  1. Warm-up: First identify exactly what the question is asking: Compared with f(x), y = -f(x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative outside changes y-values.
  4. Output signs flip.
  5. That reflects across the x-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-axis

2. Compared with f(x), y = -f(x) is reflected across the: (variation 2)

Choices: x-axis · y-axis · line y = x · origin only

Show solution
  1. Warm-up: First identify exactly what the question is asking: Compared with f(x), y = -f(x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative outside changes y-values.
  4. Output signs flip.
  5. That reflects across the x-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-axis

3. Compared with f(x), y = f(-x) is reflected across the:

Choices: y-axis · x-axis · line y = x · horizontal line y = 1

Show solution
  1. Core Practice: First identify exactly what the question is asking: Compared with f(x), y = f(-x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative inside changes input signs.
  4. Left and right positions swap.
  5. That reflects across the y-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y-axis

4. Compared with f(x), y = f(-x) is reflected across the: (variation 2)

Choices: y-axis · x-axis · line y = x · horizontal line y = 1

Show solution
  1. Core Practice: First identify exactly what the question is asking: Compared with f(x), y = f(-x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative inside changes input signs.
  4. Left and right positions swap.
  5. That reflects across the y-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y-axis

5. Compared with f(x), y = 3f(x) is a:

Choices: vertical stretch by 3 · horizontal stretch by 3 · shift up 3 · reflection only

Show solution
  1. Core Practice: First identify exactly what the question is asking: Compared with f(x), y = 3f(x) is a:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Multiplying outside changes y-values.
  4. Each output is multiplied by 3.
  5. That is a vertical stretch.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: vertical stretch by 3

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