The graphs of y=a^x

atothex

a>0

You can sketch the graph of y=a^x, for example when a=2, by considering the y coordinates that correspond to various x values. See Example 1 below. For any number a>1, the graph will have a very similar shape and will cross at y=a^0=1.

For a=1, you are calculating 1 to any power, which is always 1, and so the graph would be the horizontal line y=1.


atothex2

a<0


The graph of y=a^x for 0<a<1 will have a shape like y=a^x but reflected in the y-axis. This is because when you multiply a number less than 1 by itself, it becomes smaller.

The graph of y=0.5^x, for example, will be the graph of y=2^x reflected in the y-axis since y=0.5^x=\left(2^{-1}\right)^x=2^{-x}. See Transformations.




The diagram shows the graph of y=e^x where e, sometimes known as Euler’s number is given by e=2.718281828459.

e=2.718281828459 is special because everywhere on this graph, the gradient is the same as the y-coordinate.

See Differentiating e to the x


Example 1

The following table shows coordinates for the graph y=2^x for x taking integer values between -3 and 3 (inclusive).

x -3 -2 -1 0 1 2 3
y 0.125 0.25 0.5 1 2 4 8


Example 2

The following table shows coordinates for the graph y=0.5^x for x taking integer values between -2 and 3 (inclusive).

x -3 -2 -1 0 1 2 3
y 8 4 2 1 0.5 0.25 0.125

In the second year of A-Level Maths, you may also be expected to sketch graphs of the form y=e^{ax+b}+c.