Reflecting Functions Formula

Reflecting functions are 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)-f(x) reflects over xx-axis; f(−x)f(-x) reflects over yy-axis

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

Quick Example

f(x)=x2→−f(x)=−x2f(x) = x^2 \to -f(x) = -x^2 (opens down).
f(−x)=(−x)2=x2f(-x) = (-x)^2 = x^2 (unchanged—symmetric).

Notation

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

What This Formula Means

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

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

Formal View

−f(x)-f(x): (x,y)↦(x,−y)(x, y) \mapsto (x, -y) (reflect over xx-axis). f(−x)f(-x): (x,y)↦(−x,y)(x, y) \mapsto (-x, y) (reflect over yy-axis). Even: f(−x)=f(x)  ∀xf(-x) = f(x)\;\forall x. Odd: f(−x)=−f(x)  ∀xf(-x) = -f(x)\;\forall x

Worked Examples

Example 1

easy
Given f(x)=x3−2f(x)=x^3-2, write the equations for (a) reflection over the xx-axis and (b) reflection over the yy-axis. Evaluate each at x=2x=2.

Answer

(a) g(x)=−x3+2g(x)=-x^3+2, g(2)=−6g(2)=-6; (b) h(x)=−x3−2h(x)=-x^3-2, h(2)=−10h(2)=-10

First step

1
(a) Reflection over xx-axis: negate the output → g(x)=−f(x)=−(x3−2)=−x3+2g(x)=-f(x)=-(x^3-2)=-x^3+2. g(2)=−8+2=−6g(2)=-8+2=-6.

Full solution

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

Example 2

medium
Show that f(x)=x2f(x)=x^2 is unchanged by reflection over the yy-axis (even function) but f(x)=x3f(x)=x^3 is negated by this reflection (odd function).

Example 3

medium
For f(x)=xf(x) = \sqrt{x}, write f(−x)f(-x) and give its domain.

Common Mistakes

  • Swapping which negative flips which axis - outside −f(x)-f(x) is the xx-axis flip; inside f(−x)f(-x) is the yy-axis flip.
  • Calling y=x2y=x^2 asymmetric - it's even, equal to its own yy-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)f(-x)=f(x) vs. f(−x)=−f(x)f(-x)=-f(x)) and the inverse-function flip over y=xy=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=xy=x) in a mixed problem set.

Frequently Asked Questions

What is the Reflecting Functions formula?

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

How do you use the Reflecting Functions formula?

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

What do the symbols mean in the Reflecting Functions formula?

Even function: f(−x)=f(x)f(-x) = f(x) (symmetric about yy-axis). Odd function: f(−x)=−f(x)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)f(-x)=f(x) vs. f(−x)=−f(x)f(-x)=-f(x)) and the inverse-function flip over y=xy=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=xy=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.

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Functions and Graphs: Complete Foundations for Algebra and Calculus →
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