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).