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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 y = f(x), the graph of y = -f(x) is reflected across the:

  1. A negative sign OUTSIDE the function multiplies every output by -1.
  2. Each y-value flips to its opposite, while x-values stay put.
  3. Flipping y-values reflects the graph across the x-axis.

Answer: x-axis

Practice problems

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

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

Show solution
  1. A negative sign OUTSIDE the function multiplies every output by -1.
  2. Each y-value flips to its opposite, while x-values stay put.
  3. Flipping y-values reflects the graph across the x-axis.

Answer: x-axis

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

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

Show solution
  1. A negative sign INSIDE the function negates the input.
  2. Each point's x-value flips to its opposite, while y-values stay put.
  3. Flipping x-values reflects the graph across the y-axis.

Answer: y-axis

3. g(x) = 5f(x) is a vertical stretch of f. If f(5) = 5, find g(5).

Show solution
  1. Core Practice: First identify exactly what the question is asking: g(x) = 5f(x) is a vertical stretch of f. If f(5) = 5, find g(5).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Multiplying outside scales each output: g(x) = 5f(x).
  4. g(5) = 5 * f(5) = 5 * 5.
  5. So g(5) = 25.
  6. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.

Answer: 25

4. A reflection across the x-axis maps (a, b) to (a, -b). Where does the point (3, 6) land? Give the y-coordinate of the image.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A reflection across the x-axis maps (a, b) to (a, -b). Where does the point (3, 6) land? Give the y-coordinate of the image.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Reflecting across the x-axis keeps x and negates y.
  4. (3, 6) maps to (3, -6).
  5. So the image's y-coordinate is -6.
  6. Check the result by substituting or estimating: the response should match -6 and make sense in the original problem.

Answer: -6

5. A reflection across the y-axis maps (a, b) to (-a, b). Where does the point (4, 2) land? Give the x-coordinate of the image.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A reflection across the y-axis maps (a, b) to (-a, b). Where does the point (4, 2) land? Give the x-coordinate of the image.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Reflecting across the y-axis negates x and keeps y.
  4. (4, 2) maps to (-4, 2).
  5. So the image's x-coordinate is -4.
  6. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

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