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.
- 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).
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 concave up if and concave down if