Reflecting Functions Formula

Reflecting a function mirrors its graph across the x-axis (-f(x)), y-axis (f(-x)), or the line y = x (the inverse function).

The Formula

−f(x) reflects over x-axis; f(−x) reflects over y-axis

When to use: −f(x) flips over x-axis (upside down). f(−x) flips over y-axis (mirror).

Quick Example

f(x)=x2→−f(x)=−x2 (opens down).
f(−x)=(−x)2=x2 (unchanged—symmetric).

Notation

Even function: f(−x)=f(x) (symmetric about y-axis). Odd function: f(−x)=−f(x) (symmetric about origin).

What This Formula Means

Reflecting a function mirrors its graph across the x-axis (−f(x)), y-axis (f(−x)), or the line y=x (the inverse function).

−f(x) flips over x-axis (upside down). f(−x) flips over y-axis (mirror).

Formal View

−f(x): (x,y)↦(x,−y) (reflect over x-axis). f(−x): (x,y)↦(−x,y) (reflect over y-axis). Even: f(−x)=f(x)  ∀x. Odd: f(−x)=−f(x)  ∀x

Worked Examples

Example 1

easy
Given f(x)=x3−2, write the equations for (a) reflection over the x-axis and (b) reflection over the y-axis. Evaluate each at x=2.

Answer

(a) g(x)=−x3+2, g(2)=−6; (b) h(x)=−x3−2, h(2)=−10

First step

1
(a) Reflection over x-axis: negate the output → g(x)=−f(x)=−(x3−2)=−x3+2. g(2)=−8+2=−6.

Full solution

  1. 2
    (b) Reflection over y-axis: negate the input → h(x)=f(−x)=(−x)3−2=−x3−2. h(2)=−8−2=−10.
  2. 3
    Note: f(2)=8−2=6; the x-axis reflection negates the output (−6); the y-axis reflection changes the sign of x (−10).
Two fundamental reflections: −f(x) flips the graph over the x-axis (negates all outputs); f(−x) flips over the y-axis (reverses all inputs). They are distinct transformations that generally produce different results.

Example 2

medium
Show that f(x)=x2 is unchanged by reflection over the y-axis (even function) but f(x)=x3 is negated by this reflection (odd function).

Example 3

medium
For f(x)=x, write f(−x) and give its domain.

Common Mistakes

  • Swapping which negative flips which axis - outside −f(x) is the x-axis flip; inside f(−x) is the y-axis flip.
  • Calling y=x2 asymmetric - it's even, equal to its own y-axis reflection.
  • Confusing a reflection with a rotation - a flip mirrors across a line; it does not rotate the graph.

Why This Formula Matters

Reflection completes the transformation set and underlies even/odd symmetry (f(−x)=f(x) vs. f(−x)=−f(x)) and the inverse-function flip over y=x. Knowing which negative causes which flip is essential for graphing and for spotting symmetry. Recognizing it by "Is the graph a mirror image of the parent caused by a negative sign (not a slide or a stretch)?" — rather than by familiar numbers — is what lets a student tell it apart from outside vs. inside negative and even and odd functions and inverse function (reflect over y=x) in a mixed problem set.

Frequently Asked Questions

What is the Reflecting Functions formula?

Reflecting a function mirrors its graph across the x-axis (−f(x)), y-axis (f(−x)), or the line y=x (the inverse function).

How do you use the Reflecting Functions formula?

−f(x) flips over x-axis (upside down). f(−x) flips over y-axis (mirror).

What do the symbols mean in the Reflecting Functions formula?

Even function: f(−x)=f(x) (symmetric about y-axis). Odd function: f(−x)=−f(x) (symmetric about origin).

Why is the Reflecting Functions formula important in Math?

Reflection completes the transformation set and underlies even/odd symmetry (f(−x)=f(x) vs. f(−x)=−f(x)) and the inverse-function flip over y=x. Knowing which negative causes which flip is essential for graphing and for spotting symmetry. Recognizing it by "Is the graph a mirror image of the parent caused by a negative sign (not a slide or a stretch)?" — rather than by familiar numbers — is what lets a student tell it apart from outside vs. inside negative and even and odd functions and inverse function (reflect over y=x) in a mixed problem set.

What do students get wrong about Reflecting Functions?

The procedure for reflecting functions is the easy part; the trap is swapping which negative flips which axis. Asking "Is the graph a mirror image of the parent caused by a negative sign (not a slide or a stretch)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Reflecting Functions formula?

Before studying the Reflecting Functions formula, you should understand: transformation.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus →
← Back to Reflecting Functions See All Examples Practice Problems