[list][*]Use the input box for [math]f(x)[/math] to define a function and use the input boxes for [math]a[/math] and [math]b[/math] to define the endpoints of the domain interval. [/*][*]Use the input box or slider tool for [math]c[/math] to move the point of interest across the domain of [math]f(x)[/math], that is, from [math]x=a[/math] to [math]x=b[/math]. [/*][*]Use the slider tool for [math]h[/math] to define how far away the second point will be chosen to calculate the average rate of change. Observe the corresponding [math]y[/math]-values and [math]\Delta y[/math]. [/*][*]Use the three buttons to turn on the [i]trace [/i](leaves behind an image) for the net change in [math]y[/math] and the slope. After clicking the button, use the slider tool for [math]c[/math] to leave a trace at each value of [math]c[/math]. Click the "Clear" button to remove all traces and to turn the traces off. [/*][*]Use the checkbox "Plot Slope" to show the slope plotted as its own graph. Click the "Trace Slope" button and use the slider tool for [math]c[/math] to generate a graph of the (average) slope function. [/*][/list]

When we refer to the "[b]behavior[/b]" of a function, our goal is to describe important characteristics about how changes in the input (x) correspond to changes in the output (y). Rate of change plays an important role in describing a function's behavior. [br][br][b]Monotonicity [/b]refers to the direction in which a function is changing. If outputs are getting bigger, we say the function is increasing. If outputs are getting smaller, we say the function is decreasing. The [b]net change[/b] in outputs is calculated by:[br][br][math]\Delta y=y_2-y_1[/math][br][br]If the final value y_2 is bigger than the initial value y_1, then this difference is [b]positive[/b]. If the final value y_2 is smaller than the initial value y_1, then this difference is [b]negative[/b]. Therefore:[br][list][*]A function is increasing when its slope is positive. [/*][*]A function is decreasing when its slope is negative. [/*][/list][br][b]Concavity [/b]refers to the direction in which a curve is bending. This concept is related to how the slopes are getting steeper or less steep; in other words, this is about whether the slopes are increasing or decreasing. [br][list][*]A function is concave up when its slopes are increasing. [/*][*]A function is concave down when its slopes are decreasing. [/*][/list]