Symmetry in Graphs and Functions

Symmetry of Graphs

A graph is symmetric with respect to the x-axis if it has mirror symmetry with the x-axis as the mirror line.
If a point (x, y) is on the graph, then its mirror image (x, -y) is also on the graph.
Example: If (5, 3) is on the graph, then (5, -3) is also on the graph.
A graph is symmetric with respect to the x-axis if whenever a point (x, y) is on the graph, the point (x, -y) is also on the graph.

A graph is symmetric with respect to the y-axis if it has mirror symmetry with the y-axis as the mirror line.
If a point (x, y) is on the graph, then its mirror image (-x, y) is also on the graph.
Example: If (2, 3) is on the graph, then (-2, 3) is also on the graph.
A graph is symmetric with respect to the y-axis if whenever a point (x, y) is on the graph, the point (-x, y) is also on the graph.

A graph is symmetric with respect to the origin if it has a 180-degree rotational symmetry around the origin.
Rotating a graph by 180 degrees is the same as turning it upside down.
If a point (x, y) is on the graph, then its rotated point (-x, -y) is also on the graph.
Example: If (1, -2) is on the graph, then (-1, 2) is also on the graph.
A graph is symmetric with respect to the origin if whenever a point (x, y) is on the graph, the point (-x, -y) is also on the graph.

Graph A: Symmetric with respect to the origin (180-degree rotational symmetry). Not symmetric with respect to the x-axis or y-axis.
Graph B: Symmetric with respect to the x-axis, y-axis, and origin.
Graph C: Symmetric with respect to the y-axis only.
Graph D: Symmetric with respect to the x-axis only.

A function is even if its graph is symmetric with respect to the y-axis.
If a point (x, f(x)) is on the graph, then (-x, f(-x)) is also on the graph, and f(x) = f(-x).
A function f(x) is even if f(-x) = f(x) for all x values in its domain.
Example: f(x) = x^2 + 3
- f(-x) = (-x)^2 + 3 = x^2 + 3 = f(x)

A function is odd if its graph is symmetric with respect to the origin.
If a point (x, f(x)) is on the graph, then (-x, f(-x)) is also on the graph, and f(-x) = -f(x).
A function f(x) is odd if f(-x) = -f(x) for all x in its domain.
Example: f(x) = 5x - \!frac{1}{x}
- f(-x) = 5(-x) - \frac{1}{(-x)} = -5x + \frac{1}{x} = -(5x - \frac{1}{x}) = -f(x)