1D. Function Behavior
Instructions:
- Use the input box for to define a function and use the input boxes for and to define the endpoints of the domain interval.
- Use the input box or slider tool for to move the point of interest across the domain of , that is, from to .
- Use the slider tool for to define how far away the second point will be chosen to calculate the average rate of change. Observe the corresponding -values and .
- Use the three buttons to turn on the trace (leaves behind an image) for the net change in and the slope. After clicking the button, use the slider tool for to leave a trace at each value of . 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 to generate a graph of the (average) slope function.
Function Behavior
When we refer to the "behavior" 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.
Monotonicity 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 net change in outputs is calculated by:
If the final value y_2 is bigger than the initial value y_1, then this difference is positive. If the final value y_2 is smaller than the initial value y_1, then this difference is negative. Therefore:
- A function is increasing when its slope is positive.
- A function is decreasing when its slope is negative.
- A function is concave up when its slopes are increasing.
- A function is concave down when its slopes are decreasing.