Symmetry in Graphs: Intercepts and Testing for Symmetry

Section 2.2: Symmetry

Finding Intercepts of a Graph

Intercepts are points where a graph crosses the axes. They are useful for sketching and analyzing graphs of equations.

• x-intercepts: To find the x-intercept(s), if any, of the graph of an equation, let $y = 0$ in the equation and solve for $x$.

• y-intercepts: To find the y-intercept(s), if any, of the graph of an equation, let $x = 0$ in the equation and solve for $y$.

Example: Find intercepts from an equation. Graph the equation $2x - 3y = 6$ by finding its intercepts.

• To find the x-intercept: Set $y = 0$: $2x - 3(0) = 6$ $2x = 6$ $x = 3$ So, the x-intercept is $(3, 0)$.

• To find the y-intercept: Set $x = 0$: $2(0) - 3y = 6$ $-3y = 6$ $y = -2$ So, the y-intercept is $(0, -2)$.

Testing an Equation for Symmetry

Symmetry in graphs helps to understand their shape and properties. There are three main types of symmetry:

• y-axis symmetry: The graph is symmetric with respect to the y-axis if, for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.

• x-axis symmetry: The graph is symmetric with respect to the x-axis if, for every point $(x, y)$ on the graph, the point $(x, -y)$ is also on the graph.

• Origin symmetry: The graph is symmetric with respect to the origin if, for every point $(x, y)$ on the graph, the point $(-x, -y)$ is also on the graph.

How to Test for Symmetry

• y-axis symmetry: Replace $x$ with $-x$ in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the y-axis.

• x-axis symmetry: Replace $y$ with $-y$ in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the x-axis.

• Origin symmetry: Replace $x$ with $-x$ and $y$ with $-y$ in the equation. If the resulting equation is equivalent to the original, the graph is symmetric about the origin.

Example: Test an Equation for Symmetry

For the equation $x^2 + y^2 = 1$, test for symmetry.

1. y-axis symmetry: Replace $x$ with $-x$: $(-x)^2 + y^2 = 1$ $x^2 + y^2 = 1$ (same as original, so symmetric about the y-axis)

2. x-axis symmetry: Replace $y$ with $-y$: $x^2 + (-y)^2 = 1$ $x^2 + y^2 = 1$ (same as original, so symmetric about the x-axis)

3. Origin symmetry: Replace $x$ with $-x$ and $y$ with $-y$: $(-x)^2 + (-y)^2 = 1$ $x^2 + y^2 = 1$ (same as original, so symmetric about the origin)

Summary Table: Types of Symmetry

Type of Symmetry	Test	Result
y-axis	Replace $x$ with $-x$	Equation unchanged: symmetric about y-axis
x-axis	Replace $y$ with $-y$	Equation unchanged: symmetric about x-axis
Origin	Replace $x$ with $-x$ and $y$ with $-y$	Equation unchanged: symmetric about origin

Additional info: Symmetry tests are especially useful for quickly sketching graphs and understanding their properties without plotting many points.