Symmetry in Graphs and Functions

Symmetry of Graphs

Symmetry with Respect to the X-Axis

• 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.

Symmetry with Respect to the Y-Axis

• 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.

Symmetry with Respect to the Origin

• 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.

Examples of Graphs and Their Symmetries

• 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.

Even Functions

• 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: $f(x) = x^2 + 3$
  • $f(-x) = (-x)^2 + 3 = x^2 + 3 = f(x)$

Odd Functions

• 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: $f(x) = 5x - \!frac{1}{x}$
  • $f(-x) = 5(-x) - \frac{1}{(-x)} = -5x + \frac{1}{x} = -(5x - \frac{1}{x}) = -f(x)$