Area Function Exercise

The purpose of this exercise is to explore the graph of an area function; that is, the graph of a function defined in terms of an integral .

Exercise Part I

1.

Click here to display the area function applet in a separate window.      Smaller applet
Position point a at 4, the left-most point in the domain of the piece-wise function.  If you also position point x at 4 to coincide with a, do you see why the value of  is 0?  The values of x = -4, f(-4), and  have already been entered in the chart below.

2.

Now position x as close as possible to 3.

It may be easier to accurately position points if you enlarge the graph by moving the unit point (1, 0) further away from the origin, then reposition the origin to view important parts of the image..

Enter the values for f(-3) and  in the chart below.  Do you see why the value of  is negative?

Click here to view a printable form to record your answers.

3.

Complete the chart for x = -2, -1, 0, …, 4

4.

Now plot the points (x, A(x) ) on the grid given below, and connect the points with a smooth curve to represent the graph of the area function y = A(x).

5.

Refresh the area function applet if you have enlarged it, then click the button to Show the Area Function Graph.  Does your graph agree with the applet?  Adjust the values you entered in the table and your graph to properly show maximum, minimum and inflection points.  Observe how the black point moves along the area function curve as you drag x along the x-axis of the piecewise function.

Exercise Part II

Here we will explore the effect on the graph of y = A(x) caused by changing the value of a.

6.

Move point a as close as possible to 3 on the x-axis and position x at 4.  Explain why the value of  is positive even though the region lies below the x-axis?

7.

Complete the chart below with f(x) and  for x = -3, -2, … ,4.  Then plot the points (x, A(x)), sketch a smooth curve for y = A(x), and compare your graph with the curve given by the applet for a = -3.

8.

Use the applet to explore what happens to the graph of y = A(x) as a moves along the x-axis.

Explain in your own words why the graph of y = A(x) is translated upward when a moves from 4 to 3.

9.

Explain why the graph of y = A(x) is translated downward as a moves from 3 to +4.

10.

Explain why the graph of y =A(x) is always negative when a = +4 even though the region is sometimes above and sometimes below the x-axis.