1. Even and odd functions

2. Even functions
If a function is such that f(–x) = f(x) for all values of x, it is called an even function.
For example, show that f(x) = 3x4 – x2 + 2 is an even function.
f(–x) = 3(–x)4 – (–x)2 + 2
= 3x4 – x2 + 2
Mathematical Practices
7) Look for and make use of structure.
Students should notice that f(–x) = f(x) i.e. a function is even, if the variables in that function are all raised to even powers. They should use this knowledge to write another example of an even function.
= f(x)
f(–x) = f(x), so the function is even.
Can you write down another even function? 2

3. Graphs of even functions
Here are the graphs of some even functions: y y y x x x
Mathematical Practices
7) Look for and make use of structure.
Students should notice that all of these graphs are symmetric about the y-axis. They should be able to relate this to the algebraic expression describing an even function, f(–x) = f(x); the y-values for positive values of x are the same for the corresponding negative values of x.
What do you notice about these graphs?
The graphs of all even functions are symmetric about the y-axis. 3

4. Odd functions
If a function is such that f(–x) = –f(x) for all values of x, it is called an odd function.
For example, show that f(x) = x3 + 4x is an odd function.
f(–x) = (–x)3 + 4(–x)
= –x3 – 4x
= –(x3 + 4x)
Mathematical Practices
7) Look for and make use of structure.
Students should notice that f(–x) = –f(x) i.e. a function is odd, if the variables in that function are all raised to odd powers. They should use this knowledge to write another example of an odd function.
= –f(x)
f(–x) = –f(x), so the function is odd.
Can you write down another odd function? 4

5. Graphs of odd functions
Here are the graphs of some odd functions: y y y x x x
Teacher notes
Remind students that if an object has order 2 rotational symmetry, it will fit exactly onto itself twice during a 360° rotation.
Mathematical Practices
7) Look for and make use of structure.
Students should notice that all of these graphs have rotational symmetry of order 2 about the origin. They should be able to relate this to the algebraic expression describing an odd function, f(–x) = –f(x).
What do you notice about these graphs?
The graphs of all odd functions have order 2 rotational symmetry about the origin. 5

6. Periodic functions
A function with a pattern that repeats at regular intervals is called a periodic function.
The interval over which a periodic function repeats is called the period of the function.
For example:
y = sin x y x
–2π –π π 2π y –π π x
y = tan x
This graph has a period of 2π.
This graph has a period of π. 6