# The graphs of

# a>0

You can sketch the graph of , 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 .

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.

# a<0

The graph of for will have a shape like 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 , for example, will be the graph of reflected in the y-axis since . See Transformations.

The diagram shows the graph of 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 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 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 .