Sometimes, especially in applications, it's more useful to consider an exponential function multiplied by a coefficient, that is of the form:[br][center][math]f\left(x\right)=ab^x[/math] with [math]b>0\wedge b\ne1[/math] and [math]a\in\mathbb{R}[/math][/center][br][justify]The applet below allows you to interact with the graph of this type of exponential functions.[br][/justify][list=1][*]Use the slider that defines the coefficient [math]a[/math] and correlate the sign of the coefficient to the sign of the exponential function[/*][br][*]Use the slider that defines the base [math]b[/math] and explore the shape of the graph when [math]b>1[/math] or [math]b<1[/math].[/*][br][*]Select the [i]Show table[/i] checkbox to view a table of values for the displayed function: three of these values are already defined, that is [math]f\left(-1\right)[/math], [math]f\left(0\right)[/math] (the [i]y[/i]-intercept) and [math]f\left(1\right)[/math]. Choose the fourth [i]x[/i] value at which you want to evaluate the function by dragging the point on the [i]x[/i]-axis. (All the values in the table are approximated to 2 decimal places).[br][/*][/list][br][b]Note[/b]: you can zoom in or out the [i]Graphics View[/i] using the mouse wheel or the predefined gestures of your mobile device.

Given an exponential function [math]f\left(x\right)=ab^x[/math], with [math]b>0[/math], [math]b\ne1[/math] and [math]a\in\mathbb{R}[/math]:[br][list][*]the domain of the function is [math]\mathbb{R}=\left(-\infty,+\infty\right)[/math][/*][*]the range of the function is [math]\left(0,+\infty\right)[/math] if [math]a>0[/math], and [math]\left(-\infty,0\right)[/math] if [math]a<0[/math][/*][*]the function has a horizontal asymptote at [math]y=0[/math][/*][*]the [i]y[/i]-intercept of the graph is at [math]y=a[/math][br][/*][*]the function is concave up if [math]a>0[/math] and concave down if [math]a<0[/math][br][/*][/list]