Assignment 2:

Varying "a" of a Parabolic Equation

by

Kristina Little

Let . Now let "a" vary over all integers. How does "a" effect the graph? What changes?

Watch the equation vary of a = -10 -> 10.

First, let a > 0. Look at only the graphs from the animation (at the left) that show a > 0. Notice how, when the |a| becomes larger that the parabola of our equation becomes a smaller U shape. Oppositely, you can notice how when a gets close to 0 how wide the U shape becomes.

Similarily, let a < 0. The parabola is different. Suddenly, the parabola is "upside down." We can conclude that the sign of "a" effects the direction of the parabola. Again when the |a| becomes larger the U shape becomes narrower, and as |a| approaches 0 the U shape flattens out.

Now, look specifically at the graph where a = 0 like the one shown to the left. Allowing a = 0 removes the initial term to the polynomial. Without this initial term, the equation is no longer parabolic and has reverted to a linear equation once again.

Look at the close up of 7 graphs overlayed below. All the graphs seem to go through a central point. Each separate graph is actually a example of . The next few steps will endeavor to show by algebra how all the samples of this form do go through one point.

Click here to see the math behind the conclusions about a common point of intersection between all graphs of the form of our parabolic equation.

After solving two general forms of our equation set equal to one another, we are able to find that all the equations of this form will go through a single point. The same point that the graph above leads you to believe may be such a point. This point is (0,2).