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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), the graph of y = f(x) + 3 shifts:

  1. Worked Example: First identify exactly what the question is asking: Compared with y = f(x), the graph of 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 the function changes every output.
  4. Each output increases by 3.
  5. So the graph moves up 3.
  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), the graph of 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), the graph of 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 the function changes every output.
  4. Each output increases by 3.
  5. So the graph moves up 3.
  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 y = f(x), the graph of y = f(-x) is reflected across the:

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

Show solution
  1. A negative sign INSIDE the function negates the input.
  2. Each point's x-value flips to its opposite, while y-values stay put.
  3. Flipping x-values reflects the graph across the y-axis.

Answer: y-axis

3. Unit review 3 (Combining Transformations): When several transformations appear at once, a useful first move is to:

Choices: separate the inside changes from the outside changes · multiply every constant together first · ignore the horizontal change · graph only the final answer

Show solution
  1. Inside changes (with x) act horizontally.
  2. Outside changes (the a and k) act vertically.
  3. Separating them keeps horizontal and vertical moves from mixing.

Answer: separate the inside changes from the outside changes

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

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

Show solution
  1. An even function contains only even-power terms.
  2. x^4 - 3x^2 has all even powers, so f(-x) = f(x).
  3. The other choices contain an odd-power term.

Answer: f(x) = x^4 - 3x^2

5. Unit review 5 (Function Operations): If f(x) = 3x and g(x) = x, find (f/g)(4).

Show solution
  1. Unit Review: First identify exactly what the question is asking: If f(x) = 3x and g(x) = x, find (f/g)(4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Find f(4) = 3(4) = 12.
  4. Find g(4) = 4, which is not zero.
  5. Divide: 12 / 4 = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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