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
- Apply multiple transformations in a sensible order
- Use combining transformations in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Read the transformations in y = -2f(x - 2) + 3.
- Worked Example: First identify exactly what the question is asking: Read the transformations in y = -2f(x - 2) + 3.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- x - 2 shifts right 2.
- -2 outside reflects over the x-axis and stretches by 2.
- + 3 shifts up 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: right 2, vertical stretch by 2, reflect over x-axis, up 3
Practice problems
1. Read the transformations in y = -2f(x - 2) + 3.
Choices: right 2, vertical stretch by 2, reflect over x-axis, up 3 · left 2, vertical stretch by 2, up 3 · right 2, reflect over y-axis, down 3 · up 2, right 3, reflect over x-axis
Show solution
- Warm-up: First identify exactly what the question is asking: Read the transformations in y = -2f(x - 2) + 3.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- x - 2 shifts right 2.
- -2 outside reflects over the x-axis and stretches by 2.
- + 3 shifts up 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: right 2, vertical stretch by 2, reflect over x-axis, up 3
2. Read the transformations in y = -2f(x - 3) + 4.
Choices: right 3, vertical stretch by 2, reflect over x-axis, up 4 · left 3, vertical stretch by 2, up 4 · right 3, reflect over y-axis, down 4 · up 3, right 4, reflect over x-axis
Show solution
- Warm-up: First identify exactly what the question is asking: Read the transformations in y = -2f(x - 3) + 4.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- x - 3 shifts right 3.
- -2 outside reflects over the x-axis and stretches by 2.
- + 4 shifts up 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: right 3, vertical stretch by 2, reflect over x-axis, up 4
3. Read the transformations in y = -2f(x - 4) + 5.
Choices: right 4, vertical stretch by 2, reflect over x-axis, up 5 · left 4, vertical stretch by 2, up 5 · right 4, reflect over y-axis, down 5 · up 4, right 5, reflect over x-axis
Show solution
- Core Practice: First identify exactly what the question is asking: Read the transformations in y = -2f(x - 4) + 5.
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- x - 4 shifts right 4.
- -2 outside reflects over the x-axis and stretches by 2.
- + 5 shifts up 5.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: right 4, vertical stretch by 2, reflect over x-axis, up 5
4. When several transformations appear, a useful first move is to:
Choices: separate inside changes from outside changes · multiply all constants together immediately · ignore horizontal changes · graph only the final answer choice
Show solution
- Inside changes affect x-values.
- Outside changes affect y-values.
- Separating them prevents mixing horizontal and vertical moves.
Answer: separate inside changes from outside changes
5. When several transformations appear, a useful first move is to: (variation 2)
Choices: separate inside changes from outside changes · multiply all constants together immediately · ignore horizontal changes · graph only the final answer choice
Show solution
- Inside changes affect x-values.
- Outside changes affect y-values.
- Separating them prevents mixing horizontal and vertical moves.
Answer: separate inside changes from outside changes
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