# When an exponential function has a coefficient

- Author:
- Simona Riva

Sometimes, especially in applications, it's more useful to consider an exponential function multiplied by a coefficient, that is of the form:

with and

The applet below allows you to interact with the graph of this type of exponential functions.

- Use the slider that defines the coefficient and correlate the sign of the coefficient to the sign of the exponential function
- Use the slider that defines the base and explore the shape of the graph when or .
- Select the
*Show table*checkbox to view a table of values for the displayed function: three of these values are already defined, that is , (the*y*-intercept) and . Choose the fourth*x*value at which you want to evaluate the function by dragging the point on the*x*-axis. (All the values in the table are approximated to 2 decimal places).

**Note**: you can zoom in or out the*Graphics View*using the mouse wheel or the predefined gestures of your mobile device.## Main characteristics of this form of exponential functions

Given an exponential function , with , and :

- the domain of the function is
- the range of the function is if , and if
- the function has a horizontal asymptote at
- the
*y*-intercept of the graph is at - the function is increasing if , and decreasing if
- the function is concave up if and concave down if