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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

Why it matters: Transformations let students predict how a model changes when a situation is shifted, scaled, reflected, or combined.

Worked example

Problem. Compared with y = f(x), y = f(x) + 3 shifts:

  1. Worked Example: First identify exactly what the question is asking: Compared with y = f(x), y = f(x) + 3 shifts:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Adding outside changes every output.
  4. Outputs increase by 3.
  5. That is a vertical shift up.
  6. 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), y = f(x) + 3 shifts:

Choices: up 3 · down 3 · left 3 · right 3

Show solution
  1. Warm-up: First identify exactly what the question is asking: Compared with y = f(x), y = f(x) + 3 shifts:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Adding outside changes every output.
  4. Outputs increase by 3.
  5. That is a vertical shift up.
  6. 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), y = f(x) + 4 shifts:

Choices: up 4 · down 4 · left 4 · right 4

Show solution
  1. Warm-up: First identify exactly what the question is asking: Compared with y = f(x), y = f(x) + 4 shifts:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Adding outside changes every output.
  4. Outputs increase by 4.
  5. That is a vertical shift up.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: up 4

3. Compared with y = f(x), y = f(x - 5) shifts:

Choices: right 5 · left 5 · up 5 · down 5

Show solution
  1. Changing inside the input moves horizontally.
  2. x - 5 means the graph moves right 5.
  3. Horizontal shifts look opposite inside the parentheses.

Answer: right 5

4. Compared with y = f(x), y = f(x - 6) shifts:

Choices: right 6 · left 6 · up 6 · down 6

Show solution
  1. Changing inside the input moves horizontally.
  2. x - 6 means the graph moves right 6.
  3. Horizontal shifts look opposite inside the parentheses.

Answer: right 6

5. The graph of y = f(x + 2) - 1 moves:

Choices: left 2 and down 1 · right 2 and down 1 · left 2 and up 1 · right 2 and up 1

Show solution
  1. Core Practice: First identify exactly what the question is asking: The graph of y = f(x + 2) - 1 moves:
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x + 2 shifts left 2.
  4. -1 outside shifts down 1.
  5. Combine the horizontal and vertical moves.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: left 2 and down 1

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