Symmetry of Graphs and Even/Odd Functions

Graphs of Equations

Symmetry of Graphs

Understanding the symmetry of a graph helps in analyzing and sketching equations efficiently. Symmetry can be tested algebraically by substituting variables and observing if the equation remains unchanged.

• Symmetry with respect to the x-axis: Replace y with −y in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the x-axis. "If (x, y) is on the graph, so is (x, −y)."

• Symmetry with respect to the y-axis: Replace x with −x in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the y-axis. "If (x, y) is on the graph, so is (−x, y)."

• Symmetry with respect to the origin: Replace both x with −x and y with −y. If the resulting equation is equivalent to the original, the graph is symmetric about the origin. "If (x, y) is on the graph, so is (−x, −y)."

Example 1: Testing Symmetry

Determine whether the graph of $y^2 = x^3$ is symmetric with respect to the x-axis, the y-axis, and the origin.

• x-axis: Replace y with −y: $(-y)^2 = x^3 \implies y^2 = x^3$. The equation is unchanged, so the graph is symmetric about the x-axis.

• y-axis: Replace x with −x: $y^2 = (−x)^3 \implies y^2 = -x^3$. The equation is not equivalent, so the graph is not symmetric about the y-axis.

• Origin: Replace x with −x and y with −y: $(-y)^2 = (−x)^3 \implies y^2 = -x^3$. The equation is not equivalent, so the graph is not symmetric about the origin.

Example 2: Symmetric Points

Find the point symmetric to (−10, −7) with respect to the x-axis, y-axis, and the origin.

• x-axis: Reflect over the x-axis: (−10, 7)

• y-axis: Reflect over the y-axis: (10, −7)

• Origin: Reflect over the origin: (10, 7)

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Function: A function f is even if its graph is symmetric with respect to the y-axis. Algebraically, $f(x) = f(−x)$ for all x in the domain of f.

• Odd Function: A function f is odd if its graph is symmetric with respect to the origin. Algebraically, $f(−x) = −f(x)$ for all x in the domain of f.

Example 3: Even, Odd, or Neither

Determine if $f(x) = 2x^4 + 7x^2 - 3$ is even, odd, or neither.

• Compute $f(−x)$:

  • $f(−x) = 2(−x)^4 + 7(−x)^2 − 3 = 2x^4 + 7x^2 − 3 = f(x)$

  Therefore, f(x) is even because $f(−x) = f(x)$ for all x.

Additional info: If $f(−x) = −f(x)$, the function is odd. If neither condition holds, the function is neither even nor odd.