Problem:
Explore the graphs of the piecewise defined functions

for values of a between -3 and 3.

Exploration:

Note that the graph of f is, in general, in two pieces: a part of a parabola and a part of a line. At x = a, LiveMath draws an additional line segment from the end of parabola to the beginning of the other line; this additional line is not a part of the graph. The correct graph would have a break instead of the addtional line. In a later section, we'll see why this line is drawn.

With this exploration you should become familiar with graphing a piecewise defined function when you know the graphs of the two functions which are used to define f.

1. Substitute each of the values of -3, -2, -1, 0, 1, 3 for a in the definition of f above. View an animation to see how this can be done.
2. Which of these values for a do the parabola and the line meet without using an additional line?
3. Explain why there are no other values for a for which the parabola and the line meet without using an additional line; in other words, why there are no other values for a which gives you a graph which has no breaks in it.