[list][*]Use the input boxes on the left to set the parameters [math]y_0[/math] and [math]a[/math] for the exponential function of the form [math]f(x)=y_0\cdot a^x[/math]. Use the slider tools to adjust these values and see how they effect the graph/behavior of the function. [/*][*]Use the input box for [math]x_0[/math] to set the location of the point [math]P[/math]. Use the slider tool for [math]h[/math] to set the location for the point [math]Q[/math]. [/*][*]The text displays information to show how the [math]y[/math]-values of [math]P[/math] and [math]Q[/math] are related through the growth/decay factor. [/*][/list]

An [b]exponential function [/b]has the form [math]f(x)=y_0\cdot a^x[/math], where [math]y_0[/math] and [math]a[/math] are constants and [math]x[/math] is the independent variable. Notice that the [b]coefficient [/b][math]y_0[/math] determines the vertical intercept of the graph and that the [b]growth factor[/b] [math]a[/math] determines the "steepness" of the graph. [br][br]The fundamental characteristic of an exponential function is that changes in the input correspond to [i]repeated multiplication[/i] in the output. If you move 2 units in the [math]x[/math] direction (i.e., [math]h=2[/math]), then you have to multiply the [math]y[/math]-coordinate by the growth/decay factor 2 times. [br][br]Based on the values of [math]y_0[/math] and [math]a[/math], you should be able to predict whether the graph will be increasing or decreasing. Based on the shape of the graph, what can you say about the concavity?