Symmetry

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even and odd

One of the benefits of graphs is that you can see a lot of information simultaneously. The visual patterns often suggest properties of the functions. Many times you can see how to move the graph so that the graph lands back on itself. Any time you can do something and it appears that nothing has been done, that is called symmetry.

1 Even Symmetry

Even Functions

A function is an even function if $f(x) = f(-x)$ for all $x$ in the domain.

If you flip the graph of an even function along the vertical axis, the graph lands on itself - the graph is a mirror image across the vertical axis.

You flip and it appears that nothing has happened - symmetry.

Even Functions

Let $g(k)$ be a function. The graph of $y= g(k)$ is displayed below.

Cosine

Cosine is an even function. The graph of $y= cos(\theta )$ is displayed below.

2 Odd Symmetry

Odd Functions

A function is an odd function if $-f(x) = f(-x)$ for all $x$ in the domain.

This means when the graph is rotated $180^\circ $ about the origin it lands back on itself.

Suppose $f(x)$ is an odd function.
Suppose the point $(a, b)$ is on the graph of $y = f(x)$. That means $b = f(a)$.

Now, rotate the graph $180^{\circ }$ about the origin.

The point plotted at $(a,b)$ rotates over to the position $(-a,-b)$. Is $(-a,-b)$ also on the graph? Does $-b = f(-a)$?

We know that $b = f(a)$, because the point $(a,b)$ is on the graph.
$f$ is an odd function. That tells us that $f(-a) = -f(a)$.
Puting these together gives us $f(-a) = -f(a) = -b$.
$f(-a) = -b$ means $(-a, -b)$ is on the graph.

If you rotate the graph of an odd function $180^{\circ }$ about the origin, then the graph lands back on itself. The points have moved from their original positions, but the their new positions are again positions on the graph.

Polynomials

Polynomials with only odd exponents are odd functions. The graph of $y= 0.1x^3-1.6x$ is displayed below.

Sine

Sine is an odd function. The graph of $y= sin(\theta )$ is displayed below.

3 Shift Symmetry

Periodic Functions

A periodic function, is one where the function values repeat at a constant distance in the domain.

\[ f(\theta + \omega ) = f(\theta ) \]

$\omega $ is called the period, also know as the wave length.

If you shift the graph of a periodic function horizontally by its wave length (or period), then you get the original graph back. It appears that nothing has happened - symmetry.

Sine and Cosine have shift symmetry. If you shift them by $2\pi $ radians, then the graph lands back on itself. You cannot tell that the graph has been shifted.

Cosine

The graph of $y= cos(\theta + 2\pi )$ is displayed below.

Sine

The graph of $y= sin(\theta + 2\pi )$ is displayed below.

Even

Half of the graph of $y = K(t)$ is displayed below. If $K(t)$ is an even function, then think of what the other half of the graph would look like.

Visualize what the full graph looks like, then click the arrow.

Odd

Half of the graph of $y = M(t)$ is displayed below. If $M(t)$ is an odd function, then think of what the other half of the graph would look like.

Visualize what the full graph looks like, then click the arrow.

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more examples can be found by following this link
More Examples of Graphical Analysis

2025-05-17 22:10:10