Graphs and symmetric functions

Definition:  A function that is symmetric about the y-axis is called an even function

Conjecture: Polynomials without any odd exponents are even functions.

Proof. We’ll assume we have already shown that any function f(x) is symmetric about the y-axis if and only if f(x) = f(-x).

Software.  Using an automatic function graphing tool is a great way to explore functions.  One nice tool is Geogebra.  Graphing functions with GeoGebra is easy.  You just type in the function at the command line, on the bottom left of the Geogebra screen, then press return.  A graph of the function will appear on the right and the formula will appear at the left.  Below is an example (after return was pressed):

Graph of an even function

Symmetry in crafts. Can you see the symmetry in the rug below?

How many axes of symmetry does it have?

Oriental Rug

19th Century Carpet from Turkmenistan

Symmetry exploration:
When one part of an object looks like a reflection of another part of the same object we call that relation symmetry.  We’ve already discussed some forms of symmetry that occur in mathematical graphs.  More generally, there are many kinds of symmetry which occur in many different fields.  The following assignment gives you a chance to explore some different types of symmetry.
Part 1. The first part of your exploration is to use Geogebra to experiment with graphing even functions.  Save a screenshot of a particularly interesting graph that you create.
Part 2. Find examples of Symmetry.
Find examples of symmetry in shapes:
What are the axis of symmetry in a Rectangle? Square? Star (as a starfish).  Think of three others.
Find 4 examples of symmetry in architecture.  Describe the axes of symmetry.
Find 2 examples of symmetry in classical music and poetry composition forms.
Find 4 examples in the natural world.  Describe the axes of symmetry in each case.
Find 2 arithmetic operations which are symmetric (with respect to inputs).  Find 3 that aren’t, and
give examples demonstrating that the operations are not symmetric.

Exercise
Which of the following are even functions?  Graph them and give arguments to support your answers.

Problem
Part 1.   Is the sum of two even functions an even function?  How about the difference? the product? the quotient?  Give arguments to support your answers.
Part 2 (extra credit).   Definition:  f(x) is an odd function if f(-x) = -f(x).  For example, f(x) = x is an odd function since f(-x) = -f(x) for all values of x (e.g. f(2) = 2 and f(-2) = -2).
Given two odd functions, which of the four combination from Part 1 are odd functions, which are even functions, and which are neither.  Give arguments to support your answers.

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2 Responses to Graphs and symmetric functions

  1. What do you use to make your formulas, Ethan?

  2. esivel says:

    Maria,
    It was an on-line Latex line editor that’s at http://www.hamline.edu/~arundquist/equationeditor/.
    I found it thanks to the ning forum you pointed out to the class.

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