# When an exponential function has a coefficient

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.

1. Use the slider that defines the coefficient and correlate the sign of the coefficient to the sign of the exponential function
2. Use the slider that defines the base and explore the shape of the graph when or .
3. 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