#### Exponential graphs

The values of y = 2\(^x\) for a number of values of x were obtained using a calculator, and are shown in the table below.

We can then plot the ordered pairs (x, 2\(^x\)) to produce a graph of y = 2\(^x\).

\(x\) | −3 | −2.5 | −2 | −1.5 | −1 | −0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

\(y\) | 0.125 | 0.177 | 0.25 | 0.354 | 0.5 | 0.707 | 1 | 1.414 | 2 | 2.828 | 4 | 5.657 | 8 |

Note that although we have 'joined the dots' to form a smooth curve, we have not given any meaning at this stage to 2\(^x\) when it is an irrational number; thus we cannot deal with this problem at this stage.

We note the following features of the graph:

- The graph is increasing.
- The values increase quite rapidly as we move along the axis.
- On the left-hand side, the graph approaches, but never reaches, the axis.

### How much rice?

The old man had requested 36 893 488 147 419 103 230 grains of rice.