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Symmetry of Graphs: Odd and Even Functions - Concept

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

There are special types of functions that have graph symmetry. The most notable types are even and odd functions. Even functions have graph symmetry across the y-axis, and if they are reflected, will give us the same function. Odd functions have 180 rotational graph symmetry, if they are rotated 180 about the origin we will get the same function. There are algebraic ways to compute if a function is even or odd.

I want to talk about even and odd functions. First the definition. A function f is even if f of -x equals f of x for all x in the domain of f. That means that you can switch x for -x and get the same value. Now what kind of symmetry does that give us? Well the graph of an even function's always going to be symmetric with respect to y axis. Why is that?
Well, if you remember our discussion of symmetry, of reflections, the graph of y equals f of -x. y equals f of -x is a reflection about the y axis, if the reflection of about the y axis of a function is exactly same the same as the function itself then it's symmetric about the y axis. Now let's look at two examples from our parent functions. There's y=x squared and there's y equals the absolute value of x.
Now odd functions. Function f is odd if f of -x equals the opposite of f of x. This means that opposite inputs give opposite outputs. Now, if this is true, the graph of an odd function would be symmetrical with respect to the origin. That means is you could take the the graph, rotate it 180 degrees and it will look exactly the same. So it's 180 degrees symmetry about the origin. Now some examples from our parent functions are y=x, y equals x cubed and also y equals 1 over x.
So remember odd functions: opposite inputs have opposite outputs. Even functions: opposite inputs have the same input. Even functions are symmetric about the y axis, odd functions are symmetric about the origin.

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