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Function symmetry introduction

Learn what even and odd functions are, and how to recognize them in graphs.

What you will learn in this lesson

A shape has reflective symmetry if it remains unchanged after a reflection across a line.
For example, the pentagon above has reflective symmetry.
Notice how line l is a line of symmetry, and that the shape is a mirror image of itself across this line.
This idea of reflective symmetry can be applied to the shapes of graphs. Let's take a look.

Even functions

A function is said to be an even function if its graph is symmetric with respect to the y-axis.
For example, the function f graphed below is an even function.
Verify this for yourself by dragging the point on the x-axis from right to left. Notice that the graph remains unchanged after a reflection across the y-axis!

Check your understanding

1) Which of the graphs represent even functions?
Choose all answers that apply:
Choose all answers that apply:

An algebraic definition

Algebraically, a function f is even if f, left parenthesis, minus, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis for all possible x values.
For example, for the even function below, notice how the y-axis symmetry ensures that f, left parenthesis, x, right parenthesis, equals, f, left parenthesis, minus, x, right parenthesis for all x.

Odd functions

A function is said to be an odd function if its graph is symmetric with respect to the origin.
Visually, this means that you can rotate the figure 180, degrees about the origin, and it remains unchanged.
Another way to visualize origin symmetry is to imagine a reflection about the x-axis, followed by a reflection across the y-axis. If this leaves the graph of the function unchanged, the graph is symmetric with respect to the origin.
For example, the function g graphed below is an odd function.
Verify this for yourself by dragging the point on the y-axis from top to bottom (to reflect the function over the x-axis), and the point on the x-axis from right to left (to reflect the function over the y-axis). Notice that this is the original function!

Check your understanding

Which of the graphs represent odd functions?
Choose all answers that apply:
Choose all answers that apply:

An algebraic definition

Algebraically, a function f is odd if f, left parenthesis, minus, x, right parenthesis, equals, minus, f, left parenthesis, x, right parenthesis for all possible x values.
For example, for the odd function below, notice how the function's symmetry ensures that f, left parenthesis, minus, x, right parenthesis is always the opposite of f, left parenthesis, x, right parenthesis.

Reflection question

Can a function be neither even nor odd?
Choose 1 answer:
Choose 1 answer:

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