Question 390·Medium·Math

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The function $f$ is defined by $f(x)=\sqrt{x+4}-3$ . The graph of $y=g(x)$ is obtained from the graph of $y=f(x)$ by first reflecting it across the $x$ -axis and then translating it 2 units to the right. Which equation defines $g$ ?

A function transformation changes a graph by changing its inputs or outputs. Reflecting across the $x$ -axis negates the entire output; shifting right changes the input inside the function. Apply the moves in order, then substitute the given rule and simplify. Don’t leave the outside constant unchanged when you reflect: its sign changes too.

Hints

Hint 1
An $x$ -axis reflection sends every output to its opposite. Put a negative sign around the whole rule for $f$ , including the $-3$ . What happens when you distribute that sign?
Hint 2
A horizontal shift changes the input. To move a graph $2$ units right, use $x-2$ as the input: the old input $0$ now occurs at $x=2$ .
Hint 3
Put $x-2$ wherever $x$ appears in the reflected rule. Keep the number outside the square root where it is, then combine the numbers inside the root.

Step-by-step

Reflect, then shift the input
Step 1
Reflect the whole function
Type $f(x)=\sqrt{x+4}-3$ , then $h(x)=-f(x)$ in Desmos. An $x$ -axis reflection changes the sign of every output. Click the start of $h$ : Desmos shows $(-4,3)$ , across the axis from the start of $f$ at $(-4,-3)$ .
Step 2
Write the reflected rule
Replace $f(x)$ in $h(x)=-f(x)$ with its given rule. Keep parentheses around the whole output:
$h(x)=-(\sqrt{x+4}-3)$
Step 3
Change both signs
Distribute the negative sign to both terms:
$h(x)=-\sqrt{x+4}+3$
The original $-3$ becomes $+3$ because the reflection negates it too.
Step 4
Shift the reflected graph right
For 2 units right, replace the input with $x-2$ . Type $g(x)=h(x-2)$ below the earlier Desmos lines. Click $g$ ’s start: Desmos shows $(-2,3)$ , two units right of $h$ ’s $(-4,3)$ . To shift right, subtract inside the function.
Step 5
Substitute the new input
Put $x-2$ in place of the $x$ inside the square root in $h(x)$ . The $+3$ stays outside:
$g(x)=-\sqrt{(x-2)+4}+3$
Step 6
Simplify the rule for $g$
Combine $-2+4$ under the root:
$g(x)=-\sqrt{x+2}+3$
This equation defines the reflected, right-shifted graph. Choice C.

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Question 390·Medium·Math

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