Vertical and Horizontal Shifts
A free Precalculus lesson from the “Transformations and Combinations of Functions” unit, with a worked example and practice problems including step-by-step solutions.
Changes outside the function move outputs; changes inside the function move inputs in the opposite-looking direction. 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
- Predict how f(x) changes under vertical and horizontal shifts
- Use vertical and horizontal shifts in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Compared with y = f(x), the graph of y = f(x) + 3 shifts:
- Worked Example: First identify exactly what the question is asking: Compared with y = f(x), the graph of y = f(x) + 3 shifts:
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Adding outside the function changes every output.
- Each output increases by 3.
- So the graph moves up 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: up 3
Practice problems
1. Compared with y = f(x), the graph of y = f(x) + 3 shifts:
Choices: up 3 · down 3 · left 3 · right 3
Show solution
- Warm-up: First identify exactly what the question is asking: Compared with y = f(x), the graph of y = f(x) + 3 shifts:
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Adding outside the function changes every output.
- Each output increases by 3.
- So the graph moves up 3.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: up 3
2. Compared with y = f(x), the graph of y = f(x) - 4 shifts:
Choices: down 4 · up 4 · left 4 · right 4
Show solution
- Warm-up: First identify exactly what the question is asking: Compared with y = f(x), the graph of y = f(x) - 4 shifts:
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Subtracting outside the function changes every output.
- Each output decreases by 4.
- So the graph moves down 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: down 4
3. Compared with y = f(x), the graph of y = f(x - 5) shifts:
Choices: right 5 · left 5 · up 5 · down 5
Show solution
- A change inside f moves inputs horizontally.
- x - 5 looks like minus, but the graph moves right 5.
- Horizontal shifts go opposite to the sign inside.
Answer: right 5
4. Compared with y = f(x), the graph of y = f(x + 2) shifts:
Choices: left 2 · right 2 · up 2 · down 2
Show solution
- A change inside f moves inputs horizontally.
- x + 2 looks like plus, but the graph moves left 2.
- Horizontal shifts go opposite to the sign inside.
Answer: left 2
5. Describe the shift from y = f(x) to y = f(x - 3) + 2.
Choices: right 3 and up 2 · left 3 and up 2 · right 3 and down 2 · left 3 and down 2
Show solution
- Core Practice: First identify exactly what the question is asking: Describe the shift from y = f(x) to y = f(x - 3) + 2.
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Inside: x - 3 shifts right 3.
- Outside: + 2 shifts up 2.
- Combine: right 3 and up 2.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: right 3 and up 2
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