Question

find $g(x)$, where $g(x)$ is the reflection across the $x$-axis of $f(x) = x^2$.
write your answer in the form $a(x - h)^2 + k$, where $a$, $h$, and $k$ are integers.
$g(x) = \square$

Explanation:

Step1: Recall x-axis reflection rule

To reflect a function $f(x)$ across the x-axis, we take $-f(x)$.

Step2: Apply rule to given function

Substitute $f(x)=x^2$ into the reflection rule:
$g(x) = -f(x) = -x^2$

Step3: Rewrite in vertex form

Express $-x^2$ as $a(x-h)^2 + k$ where $a=-1$, $h=0$, $k=0$:
$g(x) = -1(x-0)^2 + 0$

Answer:

$g(x) = -1(x-0)^2 + 0$ (or simplified as $g(x) = -x^2$)