Symmetry of Graphs and Functions: Section 2.4 Study Notes

Symmetry of Graphs and Functions

Objectives

This section explores how to determine whether a graph is symmetric with respect to the x-axis, y-axis, or the origin, and how to classify functions as even, odd, or neither. Understanding symmetry helps in graphing and analyzing functions efficiently.

• Symmetry with respect to axes and origin: Learn to identify symmetry visually and algebraically.

• Even and odd functions: Classify functions based on their symmetry properties.

Symmetry in Graphs

Symmetry in graphs refers to the property where certain transformations (reflection or rotation) leave the graph unchanged. There are three main types:

• x-axis symmetry: The graph is unchanged when reflected across the x-axis.

• y-axis symmetry: The graph is unchanged when reflected across the y-axis.

• Origin symmetry: The graph is unchanged when rotated 180° about the origin.

Symmetry with Respect to the x-axis

If for every point $(x, y)$ on the graph, the point $(x, -y)$ is also on the graph, then the graph is symmetric with respect to the x-axis.

• Visual Test: Folding the graph along the x-axis should make the upper and lower parts coincide.

• Example: The graph of $x = y^2$ contains points $(4, 2)$ and $(4, -2)$, which are reflections across the x-axis.

Symmetry with Respect to the y-axis

If for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph, then the graph is symmetric with respect to the y-axis.

• Visual Test: Folding the graph along the y-axis should make the left and right parts coincide.

• Example: The graph of $y = x^2 - 5$ contains points $(3, 4)$ and $(-3, 4)$, which are reflections across the y-axis.

Symmetry with Respect to the Origin

If for every point $(x, y)$ on the graph, the point $(-x, -y)$ is also on the graph, then the graph is symmetric with respect to the origin.

• Visual Test: Rotating the graph 180° about the origin should make the figure coincide with itself.

• Example: The graph of $x^2 = y^2 + 2$ contains points $(-3, \\sqrt{7})$ and $(3, -\\sqrt{7})$, which are reflections through the origin.

Algebraic Tests of Symmetry

Algebraic tests allow us to determine symmetry by substituting variables in the equation:

• x-axis: Replace $y$ with $-y$. If the equation remains unchanged, the graph is symmetric with respect to the x-axis.

• y-axis: Replace $x$ with $-x$. If the equation remains unchanged, the graph is symmetric with respect to the y-axis.

• Origin: Replace $x$ with $-x$ and $y$ with $-y$. If the equation remains unchanged, the graph is symmetric with respect to the origin.

Example: Algebraic Test for Symmetry

Test $y = x^2 + 2$ for symmetry:

• x-axis: Replace $y$ with $-y$:

$y = x^2 + 2$ $-y = x^2 + 2$ $y = -x^2 - 2$

• The resulting equation is not equivalent to the original, so the graph is not symmetric with respect to the x-axis.

Even and Odd Functions

Functions can be classified based on their symmetry:

• Even function: The graph is symmetric with respect to the y-axis. For all $x$ in the domain, $f(-x) = f(x)$.

• Odd function: The graph is symmetric with respect to the origin. For all $x$ in the domain, $f(-x) = -f(x)$.

• Neither: If neither condition is satisfied, the function is neither even nor odd.

Steps to Determine Even or Odd Functions

1. Find $f(-x)$ and simplify.

2. If $f(-x) = f(x)$, the function is even.

3. If $f(-x) = -f(x)$, the function is odd.

4. Except for $f(x) = 0$, a function cannot be both even and odd.

Example: Classifying a Function

Given $f(x) = 5x^3 - 6x^2 - 2x$:

• Compute $f(-x)$:

$f(-x) = 5(-x)^3 - 6(-x)^2 - 2(-x)$ $= -5x^3 - 6x^2 + 2x$

• Compare with $-f(x)$:

$-f(x) = -[5x^3 - 6x^2 - 2x] = -5x^3 + 6x^2 + 2x$

• Since $f(-x) = -f(x)$, the function is odd.

Summary Table: Symmetry Tests

Type of Symmetry	Algebraic Test	Graphical Interpretation
x-axis	Replace $y$ with $-y$	Reflection across x-axis
y-axis	Replace $x$ with $-x$	Reflection across y-axis
Origin	Replace $x$ with $-x$ and $y$ with $-y$	Rotation 180° about origin

Additional info: These symmetry concepts are foundational for analyzing and graphing polynomial, rational, and trigonometric functions in precalculus.