Reflecting functions across the x-axis

A reflection across the $x$-axis mirrors a graph from top to bottom.
You form it by negating the whole function, giving $-f(x)$.
Each point $(x, y)$ moves to $(x, -y)$, keeping the same horizontal position.

The figure shows a function and its reflection across the $x$-axis.
Each point keeps its $x$-value but flips to the opposite height.
The $x$-axis acts like a mirror between the two curves.

To reflect a function across the $x$-axis, put a negative sign in front of the whole thing.
For $f(x) = x^2 - 4$, the reflection is $-(x^2 - 4) = -x^2 + 4$.
Every output changes sign.

Compare this with a $y$-axis reflection.
An $x$-axis reflection changes the output, $-f(x)$, flipping the graph up-down.
A $y$-axis reflection changes the input, $f(-x)$, flipping the graph left-right.

Reflect $f(x) = x^2$ across the $x$-axis. What is the new function?
Negate the whole function: $-f(x) = -x^2$.
So the reflection is $-x^2$.

A point $(4, 3)$ lies on a graph. Where does it go when the graph is reflected across the $x$-axis?
An $x$-axis reflection sends $(x, y)$ to $(x, -y)$.
So $(4, 3)$ moves to $(4, -3)$.

Reflect $f(x) = x^2 - 9$ across the $x$-axis. What is the result?
Put a negative sign in front of the whole function: $-(x^2 - 9)$.
So the reflection is $-x^2 + 9$.