Algebra II: Translations on Parent Functions Review worksheet with graphs and function equations.

Problem Analysis:

The task involves translating verbal descriptions of functions into mathematical expressions and determining their domains. The goal is to match each description with the correct function and domain.

#### Given Descriptions:
1. Description 1: A function that takes a number, adds 2, then squares the result.
2. Description 2: A function that takes a number, squares it, then adds 2.
3. Description 3: A function that takes a number, subtracts 2, then squares the result.
4. Description 4: A function that takes a number, squares it, then subtracts 2.

#### Functions Provided:
- \( g(x) = (x + 2)^2 \)
- \( h(x) = x^2 - 2 \)
- \( k(x) = x^2 + 2 \)
- \( m(x) = (x - 2)^2 \)

#### Domains Provided:
- Domain 1: All real numbers (\( \mathbb{R} \))
- Domain 2: All real numbers (\( \mathbb{R} \))
- Domain 3: All real numbers (\( \mathbb{R} \))
- Domain 4: All real numbers (\( \mathbb{R} \))

Step-by-Step Solution:

#### Description 1: "A function that takes a number, adds 2, then squares the result."
- Translation: First, add 2 to the input \( x \), then square the result.
\[
f(x) = (x + 2)^2
\]
- Matching Function: \( g(x) = (x + 2)^2 \)
- Domain: Squaring any real number is always defined, so the domain is all real numbers (\( \mathbb{R} \)).
- Matched Domain: Domain 1

#### Description 2: "A function that takes a number, squares it, then adds 2."
- Translation: First, square the input \( x \), then add 2.
\[
f(x) = x^2 + 2
\]
- Matching Function: \( k(x) = x^2 + 2 \)
- Domain: Squaring any real number is always defined, so the domain is all real numbers (\( \mathbb{R} \)).
- Matched Domain: Domain 2

#### Description 3: "A function that takes a number, subtracts 2, then squares the result."
- Translation: First, subtract 2 from the input \( x \), then square the result.
\[
f(x) = (x - 2)^2
\]
- Matching Function: \( m(x) = (x - 2)^2 \)
- Domain: Squaring any real number is always defined, so the domain is all real numbers (\( \mathbb{R} \)).
- Matched Domain: Domain 3

#### Description 4: "A function that takes a number, squares it, then subtracts 2."
- Translation: First, square the input \( x \), then subtract 2.
\[
f(x) = x^2 - 2
\]
- Matching Function: \( h(x) = x^2 - 2 \)
- Domain: Squaring any real number is always defined, so the domain is all real numbers (\( \mathbb{R} \)).
- Matched Domain: Domain 4

Final Matches:

1. Description 1 → \( g(x) = (x + 2)^2 \) → Domain 1
2. Description 2 → \( k(x) = x^2 + 2 \) → Domain 2
3. Description 3 → \( m(x) = (x - 2)^2 \) → Domain 3
4. Description 4 → \( h(x) = x^2 - 2 \) → Domain 4

Boxed Answer:

\[
\boxed{
\begin{array}{lll}
\text{Description 1} & \rightarrow & g(x) = (x + 2)^2 \quad \text{(Domain 1)} \\
\text{Description 2} & \rightarrow & k(x) = x^2 + 2 \quad \text{(Domain 2)} \\
\text{Description 3} & \rightarrow & m(x) = (x - 2)^2 \quad \text{(Domain 3)} \\
\text{Description 4} & \rightarrow & h(x) = x^2 - 2 \quad \text{(Domain 4)}
\end{array}
}
\]