Transformations of Functions
A free College Algebra lesson from the “Transformations of Functions” unit, with a worked example and practice problems including step-by-step solutions.
Function transformations change a graph predictably. Outside changes affect outputs, inside changes affect inputs, and negative factors create reflections.
What you'll learn
- Identify shifts
- Identify reflections and stretches
- Apply transformation notation
Worked example
Problem. Describe y = f(x - 3) + 4.
- x - 3 inside shifts the graph right 3.
- +4 outside shifts up 4.
- The graph keeps the same basic shape.
Answer: right 3 and up 4
Practice problems
1. y = f(x) - 6 shifts the graph...
Choices: Down 6 · Up 6 · Right 6 · Left 6
Show solution
- Warm-up: First identify exactly what the question is asking: y = f(x) - 6 shifts the graph...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Outside subtraction shifts down.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Down 6
2. y = f(x + 2) shifts the graph...
Choices: Left 2 · Right 2 · Up 2 · Down 2
Show solution
- Core Practice: First identify exactly what the question is asking: y = f(x + 2) shifts the graph...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Inside changes feel backward.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Left 2
3. y = -f(x) reflects across the...
Choices: x-axis · y-axis · line y = x · origin only
Show solution
- Challenge: First identify exactly what the question is asking: y = -f(x) reflects across the...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Outputs change sign.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x-axis
4. g(x) = f(x) + 5 shifts f...
Choices: Up 5 · Down 5 · Left 5 · Right 5
Show solution
- Vertical Shifts: First identify exactly what the question is asking: g(x) = f(x) + 5 shifts f...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Adding outside moves the graph vertically.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Up 5
5. g(x) = f(x - 3) shifts f...
Choices: Right 3 · Left 3 · Up 3 · Down 3
Show solution
- Horizontal Shifts: First identify exactly what the question is asking: g(x) = f(x - 3) shifts f...
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Inside -3 shifts right.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Right 3
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