Reflections

What is a Reflection?

A reflection flips a graph across a line, like a mirror image.

Reflection Across the x-axis

$-f(x)$

Negate the output to flip vertically.

Every point $(x, y)$ becomes $(x, -y)$.

Example: $f(x) = x^2$ becomes $-f(x) = -x^2$

• The original opens upward
• The reflection opens downward

Points above the x-axis go below, and vice versa.

Reflection Across the y-axis

$f(-x)$

Negate the input to flip horizontally.

Every point $(x, y)$ becomes $(-x, y)$.

Example: $f(x) = \sqrt{x}$ becomes $f(-x) = \sqrt{-x}$

• The original extends to the right
• The reflection extends to the left

Points on the right side go to the left, and vice versa.

How to Remember

Outside = vertical, Inside = horizontal

This pattern applies to all transformations.

Both Reflections

$-f(-x)$

Negate both input and output to flip across both axes.

This is the same as rotating 180° around the origin.

Connection to Even and Odd Functions

Remember:

• Even functions: $f(-x) = f(x)$ → symmetric about y-axis
• Odd functions: $f(-x) = -f(x)$ → symmetric about origin

Even functions look the same when reflected across the y-axis.

Odd functions look the same when reflected across both axes.