Even and odd functions

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

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

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

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

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