Symmetry about the y-axis

Definition

A function is symmetric about the y-axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph. This implies that $f(x) = f(-x)$ for all x in the domain of the function.

5 Must Know Facts For Your Next Test

1. Symmetry about the y-axis means that reflecting the graph over the y-axis does not change its appearance.
2. If a function $f$ is even, then it is symmetric about the y-axis.
3. To test for symmetry about the y-axis algebraically, replace $x$ with $-x$ in the function and check if you get back the original function.
4. Graphs of polynomial functions with only even powers of x (e.g., $f(x) = x^2$, $f(x) = x^4 - 2x^2 + 1$) are symmetric about the y-axis.
5. Y-axis symmetry can simplify integration and finding areas under curves since you can integrate from 0 to a positive value and double the result.

Review Questions

• How can you determine if a function is symmetric about the y-axis by looking at its equation?
• Why are even functions always symmetric about the y-axis?
• Give an example of a polynomial function that is symmetric about the y-axis.

Related terms

Even Function: A function $f(x)$ is called even if $f(x) = f(-x)$ for all x in its domain.

Odd Function: $f(x)$ is called odd if $f(-x) = -f(x)$ for all x in its domain; these functions are symmetric about the origin.

$y$-intercept: The point where a graph intersects or crosses the y-axis of a coordinate plane.