Reflecting Functions | Moosa Academy

Reflecting Functions — Moosa Academy

A reflection flips a graph across an axis. Which axis you flip across depends on where the minus sign goes: outside the function, or inside next to the variable. The two produce completely different results.

Concept The two reflections

g(x) = -f(x) \qquad\qquad g(x) = f(-x)

−f(x) — multiply the whole function by −1. Everything above the x-axis goes below it, and the reverse.

f(−x) — replace

x

with

-x

. Everything on the right goes to the left, and the reverse.

Note that the minus in $-f(x)$ applies to the entire equation, not just to the term containing $x$ .

Example Reflecting a composite function

Reflect $f(x) = x^2 + 3x + 4$ in each axis.

In the x-axis, multiply everything by −1:

-f(x) = -x^2 - 3x - 4

In the y-axis, replace

x

with

-x

f(-x) = (-x)^2 + 3(-x) + 4 = x^2 - 3x + 4

⟹ −x² − 3x − 4 and x² − 3x + 4

In the second case the constant 4 is unchanged, because it contains no $x$ . The even power $x^2$ also survives, since $(-x)^2 = x^2$ . Only the odd term flipped sign.

Note A reflection can move the domain

Reflecting $f(x) = \sqrt{x}$ in the y-axis gives $f(-x) = \sqrt{-x}$ . The original was defined only for $x \geq 0$ ; the reflection is defined only for $x \leq 0$ . The curve moved to the other side of the axis, and its domain moved with it.

Note Even and odd functions under reflection

Symmetric functions behave in a special way, which follows directly from their definitions:

An even function satisfies

f(-x) = f(x)

, so reflecting it in the y-axis leaves it unchanged. Reflecting

x^2

horizontally produces

x^2

again.

An odd function satisfies

f(-x) = -f(x)

, so both reflections give the same result. For

x^3

, flipping in either axis produces

-x^3

Reflecting an even function in the y-axis is therefore not an error but simply a null operation — it is precisely what "even" means.

Note The rules in brief

Reflection	Rule	Effect
In the x-axis	g(x) = −f(x)	Above ↔ below
In the y-axis	g(x) = f(−x)	Right ↔ left
Even function, y-axis	f(−x) = f(x)	No change
Odd function	f(−x) = −f(x)	Both reflections agree

Summary

To reflect in the x-axis, multiply the entire function by −1.
To reflect in the y-axis, replace x with −x throughout.
Constants and even powers survive a y-axis reflection; odd powers change sign.
An even function is unchanged by a y-axis reflection, and for an odd function both reflections agree.