# Even and Odd Functions 'Spot patterns and describe the shapes of graphs'

This is an investigation about a particular type of function: even functions and odd functions. You should be able to describe their properties using reflections and rotations, but can you find out the underlying secrets of their graphs connecting their x and y coordinates. Once done, there are some interesting properties to discover when you add, multiply and compose these functions.

### Resources & Description

• Below you should see two graphs. You can drag the blue points labelled x to see more of the graphs.
• For each of the graphs describe how the curves for values of x<0 is related to those of x>0.
• Can you describe them without using the words reflection, flip, mirror or rotate?
• Can you describe them using function notation?

### Investigation

Teachers may wish to print out the following investigation to help students discover the properties of even and odd functions.

In this investigation you should discover how to describe even and odd functions. Once this is done you will find out some interesting properties of even and odd functions by considering the following.

• Is it possible for a function to be odd AND even?
• It is possible for a function to be neither odd nor even?
• If you add two even functions together what type of function do you get (odd, even, …)?
• If you multiply two even functions together what type of function do you get (odd, even, …)?
• If you add two odd functions together what type of function do you get (odd, even, …)
• If you multiply two odd functions together what type of function do you get (odd, even, …)
• Is a(x) = Sinx odd, even or neither?
• Is b(x) = cosx odd, even or neither?Is c(x) = x² odd, even or neither?
• Is d(x) = x3 odd, even or neither?
• The composition function a(c(x)) = sin(x²). Is this function odd, even or neither?
• Try different compositions of odd and even functions like the ones above. Can you find any general rules.
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