even and odd
Even Symmetry
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.
Odd Symmetry
This means when the graph is rotated about the origin it lands back on itself.
Suppose is an odd function.
Suppose the point is on the graph of . That means .
Now, rotate the graph about the origin.
The point plotted at rotates over to the position . Is also on the graph? Does ?
- We know that , because the point is on the graph.
- is an odd function. That tells us that .
- Puting these together gives us .
- means is on the graph.
If you rotate the graph of an odd function 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.
Shift Symmetry
A periodic function, is one where the function values repeat at a constant distance in the domain.
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 radians, then the graph lands back on itself. You cannot tell that the graph has been shifted.
Half of the graph of is displayed below. If 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.
Half of the graph of is displayed below. If 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