Reflecting Functions

A reflection flips a graph across an axis. −f(x) flips across the x-axis; f(−x) flips across the y-axis — and the second one reverses the sign in a way that surprises most students.

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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

−f(x) — flips vertically f(−x) — flips horizontally
−f(x) — multiply the whole function by −1. Everything above the x-axis goes below it, and the reverse.
f(−x) — replace with . Everything on the right goes to the left, and the reverse.

Note that the minus in applies to the entire equation, not just to the term containing .

Example Reflecting a composite function

Reflect in each axis.

In the x-axis, multiply everything by −1:
In the y-axis, replace with :
⟹ −x² − 3x − 4 and x² − 3x + 4

In the second case the constant 4 is unchanged, because it contains no . The even power also survives, since . Only the odd term flipped sign.

Note A reflection can move the domain

Reflecting in the y-axis gives . The original was defined only for ; the reflection is defined only for . 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 , so reflecting it in the y-axis leaves it unchanged. Reflecting horizontally produces again.
An odd function satisfies , so both reflections give the same result. For , flipping in either axis produces .

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