Advanced Function Transformations
A free College Algebra lesson from the “Transformations of Functions” unit, with a worked example and practice problems including step-by-step solutions.
Advanced transformation problems combine shifts, stretches, and reflections. Work one feature at a time and keep track of whether changes affect inputs or outputs.
What you'll learn
- Use combined transformations
- Read transformed function notation
- Connect transformations to domain and range
Worked example
Problem. Describe y = -2f(x + 1) - 3.
- x + 1 shifts left 1.
- -2 outside reflects over the x-axis and stretches vertically by 2.
- -3 shifts down 3.
Answer: left 1, vertical stretch by 2, reflect over x-axis, down 3
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. y = 4f(x) creates a...
Choices: Vertical stretch by 4 · Right shift by 4 · Left shift by 4 · Horizontal reflection
Show solution
- Outside multiplier scales outputs.
Answer: Vertical stretch by 4
2. y = f(2x) changes the graph horizontally by a factor of...
Choices: 1/2 · 2 · 4 · -2
Show solution
- Inside multiplier compresses horizontally.
Answer: 1/2
3. y = -f(x - 5) + 1 includes...
Choices: Right 5, reflect over x-axis, up 1 · Left 5, reflect over y-axis, down 1 · Right 1, up 5 · Only a vertical stretch
Show solution
- Read each transformation separately.
Answer: Right 5, reflect over x-axis, up 1
4. g(x) = f(2x) creates a...
Choices: Horizontal compression by factor 1/2 · Horizontal stretch by factor 2 · Vertical stretch by factor 2 · Shift right 2
Show solution
- Multiplying the input compresses horizontally.
Answer: Horizontal compression by factor 1/2
5. g(x) = f(-x) reflects f across the...
Choices: y-axis · x-axis · line y = x · origin only
Show solution
- The negative inside changes input signs.
Answer: y-axis
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