Symmetry and Functions

Symmetry and Functions

Types of Symmetry

Symmetry with Respect to the Y-Axis

• Folding the graph along the y-axis results in the graph matching itself.
• If a point $(x, y)$ is on the graph, then the point $(-x, y)$ is also on the graph. The y-coordinate remains the same, while the x-coordinate takes the opposite value.

Symmetry with Respect to the X-Axis

• Reflecting the graph across the x-axis. The x-coordinate stays the same, and the y-coordinate takes the opposite value.
• If a point $(x, y)$ is on the graph, then the point $(x, -y)$ is also on the graph.

Symmetry with Respect to the Origin

• Both x and y coordinates are reflected around the origin.
• If a point $(x, y)$ is on the graph, then the point $(-x, -y)$ is also on the graph.

Examples

• Parabola Shape: Symmetry with respect to the y-axis. Y-coordinates stay the same; x-coordinates take the opposite value.
• Symmetry with respect to the x-axis: x-coordinates stay the same; y-coordinates take the opposite value.
• Symmetry with respect to the origin: Both x and y coordinates take opposite values.

Even and Odd Functions

Even Function

• Definition: A function $f(x)$ is even if $f(-x) = f(x)$.
• Symmetry: Even functions have symmetry with respect to the y-axis. Inputting the opposite x-value results in the same y-value.

Odd Function

• Definition: A function $f(x)$ is odd if $f(-x) = -f(x)$.
• Symmetry: Odd functions have symmetry with respect to the origin. Inputting the opposite x-value results in the opposite y-value.

Determining Symmetry Algebraically

To test for symmetry, substitute $-x$ into the function and observe the result.

Example 1

Given $f(x) = x^4 - 2x^2 - 20$, find $f(-x)$.

$f(-x) = (-x)^4 - 2(-x)^2 - 20$
$f(-x) = x^4 - 2x^2 - 20$

Since $f(-x) = f(x)$, the function is even and has symmetry with respect to the y-axis.

Example 2

Given $g(x) = x^5 - 3x + 1$, find $g(-x)$.

$g(-x) = (-x)^5 - 3(-x) + 1$
$g(-x) = -x^5 + 3x + 1$

Since $g(-x)$ is not equal to $g(x)$ or $-g(x)$, the function is neither even nor odd.

Example 3

Given $h(x) = {1 \over x^3} - x$, find $h(-x)$.

$h(-x) = {1 \over (-x)^3} - (-x)$
$h(-x) = -{1 \over x^3} + x$

$h(-x) = -( {1 \over x^3} - x )$

Since $h(-x) = -h(x)$, the function is odd and has symmetry with respect to the origin.

Graphical Confirmation

• The graph of $f(x) = x^4 - 2x^2 - 20$ confirms even symmetry with respect to the y-axis.
• The graph of $g(x) = x^5 - 3x + 1$ confirms that it has neither even nor odd symmetry.
• The graph of $h(x) = {1 \over x^3} - x$ confirms odd symmetry with respect to the origin.
• For every point $(x, y)$, there exists a point $(-x, -y)$.