# a to the x

The diagram below 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.

The graph of $y=a^x$, however, for any a>1 is of a similar shape. If a=1, then you are calculating 1 to any power, which is always 1, and so this would be the graph of y=1. For a<1, the graph will be reflected in the y-axis. This is because when you multiply a number less than 1 by itself, it become smaller; that is, of course, unless the power is negative.

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