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Unit 2 Review and Quiz

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

This checkpoint makes sure students can move and combine functions before studying inverses. 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. Unit review 1 (Vertical and Horizontal Shifts): Compared with y = f(x), y = f(x) + 3 shifts:

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

Show solution
  1. Unit Review: 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. Unit review 2 (Reflections and Stretches): Compared with f(x), y = -f(x) is reflected across the:

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

Show solution
  1. Unit Review: First identify exactly what the question is asking: Compared with f(x), y = -f(x) is reflected across the:
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. A negative outside changes y-values.
  4. Output signs flip.
  5. That reflects across the x-axis.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-axis

3. Unit review 3 (Combining Transformations): Read the transformations in y = -2f(x - 4) + 5.

Choices: right 4, vertical stretch by 2, reflect over x-axis, up 5 · left 4, vertical stretch by 2, up 5 · right 4, reflect over y-axis, down 5 · up 4, right 5, reflect over x-axis

Show solution
  1. Unit Review: First identify exactly what the question is asking: Read the transformations in y = -2f(x - 4) + 5.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. x - 4 shifts right 4.
  4. -2 outside reflects over the x-axis and stretches by 2.
  5. + 5 shifts up 5.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: right 4, vertical stretch by 2, reflect over x-axis, up 5

4. Unit review 4 (Even and Odd Functions): Which function is odd?

Choices: f(x) = x^3 - x · f(x) = x^2 + 1 · f(x) = x^3 + 2 · f(x) = x^4 - x^2

Show solution
  1. Unit Review: First identify exactly what the question is asking: Which function is odd?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Odd functions satisfy f(-x) = -f(x).
  4. Both x^3 and x change sign.
  5. So x^3 - x is odd.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: f(x) = x^3 - x

5. Unit review 5 (Function Operations): For (f/g)(x), if g(x) = x - 1, what value is excluded?

Show solution
  1. Unit Review: First identify exactly what the question is asking: For (f/g)(x), if g(x) = x - 1, what value is excluded?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The denominator function cannot be zero.
  4. Set x - 1 = 0.
  5. The excluded value is 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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