In this unit you will investigate the family of functions of the form:
where a and b are real numbers. We are particularly interested in finding values of a and b that will result in a function with a removable discontinutity. The graph below starts with a = 10 and b = 0. You can change the values of these parameters by clicking at any point on the blue graph. You will be able to see the new function on the green graph.
You will notice that as you move the point on the blue graph, discontinuity shows up on the green graph as a vertical line. Try moving your point up, down, left, or right and observe what happens. You can also use the arrows just above or to the right of the blue graph. The smaller arrows use smaller steps. Find values of a and b when the graph of the function seems continuous. Mark the points by pressing the "Mark point" button. Be sure to include points to the left and to the right of b = 0. Do you see a pattern?
If b = 2, then
In general, what is the value of a in terms of b, that will give the function f a removable discontinuity ?
The applet on this page uses the Parameter_Plane applet from Frank Wattenberg's Lite Applet Collection.