## Exponential Functions

A function of the form y = ax is called an exponential function.

a is the base and x is the exponent. Exponential functions are sometimes known as growth or decay curves.

A spreadsheet can be used to investigate this type of function.
e.g. For the function y = 2x

 Note • As x gets smaller, y gets smaller and closer to the x-axis.The x-axis is an asymptote. • As x gets larger, y gets larger. • This is the shape of the graph of an exponential function. • Functions of the typey = ax always pass through the point (0, 1).

### Transformations of Exponential Graphs

 Example 1   y = 2x This is a basic exponential function. y = 2-x (is same as y = 0.5x) The (-x) exponent makes a reflection in the y-axis. y = -2x The negative sign makes each value negative and gives a reflection in the x-axis. Example 2   y = 10x The bigger the base number the steeper the graph. y = 10x-2 The graph is moved 2 units forward. y = 10x − 2 The graph is moved down 2 units.

### Graph of y = ex

y = ex is a special exponential function.
The value of e is 2.718281 (to 7 sig.fig.).
It is special because at any point on the graph the gradient of the curve is equal to the y-value at that point.

The gradient of the graph at the point (0, 1) is 1.
The gradient of the graph at the point (1, 2) is 2, etc, etc.