This topic examines the graphs of functions of the type y = x a, where a is a constant.

Revision of the rules of exponents, also called powers may be useful and are covered in Year 12, Topic 3
Some power functions, such as y = x2 and y = x3 and y = 1 / x = x -1 have been well covered in previous years.

Because many of these graphs have few intercepts and features the best method is to learn the basic types for different values of and then plot a few points to get the shape.

The table below shows examples of power functions for different values of a.


a is a positive integer
a = 2
a = 3
a = 4
a is a negative integer
a = − 1
a = − 2
a = − 4
a is not an integer

a < 0

a = − 0.9

0 ≤ a ≤ 1

a = 0.5

a > 1

a = 1.8


Notes for the power function y = xa

  • If a is a whole number then x can have any value.
  • If a is a negative integer then x can have any value except 0.
  • If a is a any other type of number then x almost always has to be positive.

These properties are all illustrated on the graphs in the table above.