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Function Transformations

A free Algebra II lesson from the “Transformations of Functions” unit, with a worked example and practice problems including step-by-step solutions.

Transformations change a parent function. Adding outside shifts up or down, changing inside parentheses shifts left or right, and multiplying by a negative reflects a graph.

What you'll learn

Why it matters: Audio waveforms shifted in time, signal amplifiers, image filters, and graph design all stretch, shrink, shift, and reflect a parent function. Each parameter in y = a · f(b(x - h)) + k has one specific job, and learning them separately is what makes the combinations readable.

Worked example

Problem. Describe y = (x - 3)^2 + 4 compared with y = x^2.

  1. x - 3 inside the square shifts the graph right 3.
  2. +4 outside shifts the graph up 4.
  3. The shape is still a parabola.

Answer: right 3 and up 4

Practice problems

1. y = f(x) + 6 shifts the graph...

Choices: Up 6 · Down 6 · Right 6 · Left 6

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

Answer: Up 6

2. y = f(x - 4) shifts the graph...

Choices: Right 4 · Left 4 · Up 4 · Down 4

Show solution
  1. Warm-up: First identify exactly what the question is asking: y = f(x - 4) shifts the graph...
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Inside changes feel backward.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Right 4

3. y = -f(x) reflects the graph across the...

Choices: x-axis · y-axis · line y = x · origin only

Show solution
  1. Core Practice: First identify exactly what the question is asking: y = -f(x) reflects the graph across the...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The outputs change sign.
  4. 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. The vertex of y = (x + 2)^2 - 5 is...

Choices: (-2, -5) · (2, -5) · (-2, 5) · (2, 5)

Show solution
  1. Core Practice: First identify exactly what the question is asking: The vertex of y = (x + 2)^2 - 5 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Vertex form gives (h, k).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-2, -5)

5. y = 3f(x) creates a...

Choices: Vertical stretch by 3 · Horizontal shift right 3 · Reflection · Vertical shift up 3

Show solution
  1. Challenge: First identify exactly what the question is asking: y = 3f(x) creates a...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Multiplying outside scales outputs.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Vertical stretch by 3

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