Symmetry for Axes and Points

Types of Symmetry

Symmetry about the Y-axis (Even Functions)

A graph is symmetric about the y-axis if replacing $x$ with $-x$ yields the same equation.

$\text{Test: } f(-x) = f(x)$

The point $(x, y)$ has a mirror image at $(-x, y)$.

Examples of even functions:

• $f(x) = x^2$
• $f(x) = |x|$
• $f(x) = \cos(x)$
• $f(x) = x^4 + x^2$

Symmetry about the X-axis

A graph is symmetric about the x-axis if replacing $y$ with $-y$ yields the same equation.

$\text{Test: Replace } y \text{ with } -y \text{ and simplify}$

The point $(x, y)$ has a mirror image at $(x, -y)$.

Note: Functions (passing vertical line test) cannot have x-axis symmetry except for $y = 0$.

Examples (relations):

• $x = y^2$
• $x^2 + y^2 = r^2$ (circle)

Symmetry about the Origin (Odd Functions)

A graph is symmetric about the origin if replacing both $x$ with $-x$ and $y$ with $-y$ yields the same equation.

$\text{Test: } f(-x) = -f(x)$

The point $(x, y)$ has a mirror image at $(-x, -y)$.

Examples of odd functions:

• $f(x) = x^3$
• $f(x) = x$
• $f(x) = \sin(x)$
• $f(x) = \frac{1}{x}$

Symmetry about the Line $y = x$

A graph is symmetric about $y = x$ if swapping $x$ and $y$ yields the same equation.

$\text{Test: Interchange } x \text{ and } y$

The point $(x, y)$ has a mirror image at $(y, x)$.

Examples:

• $x^2 + y^2 = r^2$ (circle centered at origin)
• $xy = 1$ (rectangular hyperbola)

Quick Reference Table

Symmetry	Test	Point Transformation
Y-axis	$f(-x) = f(x)$	$(x, y) \to (-x, y)$
X-axis	Replace $y$ with $-y$	$(x, y) \to (x, -y)$
Origin	$f(-x) = -f(x)$	$(x, y) \to (-x, -y)$
Line $y = x$	Swap $x$ and $y$	$(x, y) \to (y, x)$

Reflections

Original Point	Reflection across	Reflected Point
$(a, b)$	x-axis	$(a, -b)$
$(a, b)$	y-axis	$(-a, b)$
$(a, b)$	origin	$(-a, -b)$
$(a, b)$	$y = x$	$(b, a)$
$(a, b)$	$y = -x$	$(-b, -a)$

Determining Symmetry of Polynomial Functions

• Even degree polynomials with only even powers of $x$ are symmetric about the y-axis
• Odd degree polynomials with only odd powers of $x$ are symmetric about the origin
• Polynomials with mixed even and odd powers have no symmetry