16 ## SYMMETRYTest for symmetry: Even and odd functions LET THIS BE THE RIGHT-HAND SIDE of the graph of a function: We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the In this case,
The height of the curve at − Again, let this be the right-hand side: We will now draw the left-hand side -- so that the graph will be symmetrical with respect to the origin: Every point on the right-hand side is reflected through the origin. In this case,
The value of (A reflection through the origin is equivalent to a reflection about the Test for symmetry: Even and odd functions Symmetry, then, depends on the behavior of Here is the test: If then the graph of To see the answer, pass your mouse over the colored area. If then the graph of A function symmetrical with respect to the A function that is symmetrical with respect to the origin is called an odd function. Example 1. Test this function for symmetry:
Since Example 2. Test this function for symmetry:
Since
Problem. Test each of the following for symmetry. Is a)
b)
c)
A polynomial will be an odd function when all the exponents are odd. But there are even and odd functions that are not polynomials. In trigonometry, Therefore, the issue is the test of Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: [email protected] |