Reflecting Functions Formula

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) = x^2 \to -f(x) = -x^2 (opens down).
f(-x) = (-x)^2 = x^2 (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) \mapsto (x, -y) (reflect over x-axis). f(-x): (x, y) \mapsto (-x, y) (reflect over y-axis). Even: f(-x) = f(x)\;\forall x. Odd: f(-x) = -f(x)\;\forall x

Worked Examples

Example 1

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

Solution

  1. 1
    (a) Reflection over x-axis: negate the output → g(x)=-f(x)=-(x^3-2)=-x^3+2. g(2)=-8+2=-6.
  2. 2
    (b) Reflection over y-axis: negate the input → h(x)=f(-x)=(-x)^3-2=-x^3-2. h(2)=-8-2=-10.
  3. 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).

Answer

(a) g(x)=-x^3+2, g(2)=-6; (b) h(x)=-x^3-2, h(2)=-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)=x^2 is unchanged by reflection over the y-axis (even function) but f(x)=x^3 is negated by this reflection (odd function).

Common Mistakes

  • Confusing -f(x) with f(-x) — -f(x) reflects over the x-axis (flips output sign); f(-x) reflects over the y-axis (flips input sign)
  • Thinking reflection changes the shape of the graph — reflection only flips the graph; it preserves the shape exactly
  • Forgetting to check for symmetry — if f(-x) = f(x) the function is even (symmetric about y-axis); if f(-x) = -f(x) it is odd (symmetric about origin)

Why This Formula Matters

Reflections reveal the relationship between a function and its inverse (reflection across y = x) and connect even/odd symmetry to the function's formula.

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?

Reflections reveal the relationship between a function and its inverse (reflection across y = x) and connect even/odd symmetry to the function's formula.

What do students get wrong about Reflecting Functions?

Even functions: f(-x) = f(x). Odd functions: f(-x) = -f(x).

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 →
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