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 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.

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 $y=e^{ax+b}+c$.