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Combining Transformations

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

A transformed function is easier to read when shifts, stretches, and reflections are separated before graphing. 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. Read every transformation in y = 3f(x - 2) + 3.

  1. Worked Example: First identify exactly what the question is asking: Read every transformation in y = 3f(x - 2) + 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x - 2 inside shifts right 2.
  4. The 3 outside stretches vertically by 3.
  5. + 3 outside shifts up 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: right 2, stretch by 3, up 3

Practice problems

1. Read every transformation in y = 3f(x - 2) + 3.

Choices: right 2, stretch by 3, up 3 · left 2, stretch by 3, up 3 · right 2, shrink by 3, down 3 · right 2, stretch by 3, down 3

Show solution
  1. Warm-up: First identify exactly what the question is asking: Read every transformation in y = 3f(x - 2) + 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x - 2 inside shifts right 2.
  4. The 3 outside stretches vertically by 3.
  5. + 3 outside shifts up 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: right 2, stretch by 3, up 3

2. Read every transformation in y = -4f(x - 3) + 4.

Choices: right 3, stretch by 4, reflect over the x-axis, up 4 · right 3, stretch by 4, up 4 · left 3, reflect over the y-axis, up 4 · right 3, reflect over the x-axis, down 4

Show solution
  1. Warm-up: First identify exactly what the question is asking: Read every transformation in y = -4f(x - 3) + 4.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. x - 3 shifts right 3.
  4. The -4 outside both reflects over the x-axis and stretches by 4.
  5. + 4 shifts up 4.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: right 3, stretch by 4, reflect over the x-axis, up 4

3. When several transformations appear at once, a useful first move is to:

Choices: separate the inside changes from the outside changes · multiply every constant together first · ignore the horizontal change · graph only the final answer

Show solution
  1. Inside changes (with x) act horizontally.
  2. Outside changes (the a and k) act vertically.
  3. Separating them keeps horizontal and vertical moves from mixing.

Answer: separate the inside changes from the outside changes

4. The point (2, 2) lies on y = f(x). Find the x-coordinate of its image on y = 3f(x - 1) + 6.

Show solution
  1. Core Practice: First identify exactly what the question is asking: The point (2, 2) lies on y = f(x). Find the x-coordinate of its image on y = 3f(x - 1) + 6.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Only the inside x - 1 moves the input.
  4. A point at x = 2 shifts to x = 2 + 1.
  5. So the new x-coordinate is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. The point (3, 3) lies on y = f(x). Find the y-coordinate of its image on y = 4f(x - 2) + 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: The point (3, 3) lies on y = f(x). Find the y-coordinate of its image on y = 4f(x - 2) + 2.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The outside acts on the output: y becomes 4y + 2.
  4. 4(3) + 2 = 12 + 2.
  5. So the new y-coordinate is 14.
  6. Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.

Answer: 14

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