Assignment 2: Graphical Implications of Varying the Coefficients of a Quadratic

by Shawn Broderick

For this assignment, I have chosen to do the first question.

Here, again, are the directions:

1. Examine graphs for the parabola:

*y* = *ax*^{2} + *bx* + *c*

for different values of *a*, *b*, and *c*. (*a*, *b*, *c* can be any rational numbers).

Try using the GC animation by replacing *a*, *b*, or *c* with an *n* and selecting an appropriate range for *n*.

There are probably a million ways to explore this situation. I want to just show how I thought about this problem. In the first movie, I varied *a* with the *n* slider and then set *b* and *c* equal to one. In the second movie, the *n* was in the *b* position and I set *a* and *c* both equal to one. For the third movie, I set *a* and *b* equal to one and *c* equal to *n*. Here are the movies that show the results of that initial approach (with -2 <= *n* <= 2):

*nx*^{2} + *x* + 1

This video shows what happens to the graphs as you vary *a*, or the coefficient to the *x*^2 term. When the video is played, notice that the *a* determines the position of the focus of the parabola.

*x*^{2} + *nx* + 1

Here, we have varied the *b*, or the coefficient of the *x* term. When the video is played, notice that the *b* determines the horizontal position of the vertex of the parabola.

* x*^{2} + *x* + *n*

Here, we have varied the *c*, or the constant term. When the video is played, notice that the *c* determines the vertical position of the parabola.

Afterward, I wanted to see what happened when I changed signs and set *a*, *b*, and *c* equal to *n*. Here are the movies that show the results of that approach:

*nx*^{2} + *nx* + *n*

The vertex of the parabola starts in Quadrant III. The focus is far from the vertex. As n increases, the vertex does as well. Also, the distance from the focus to the vertex decreases. As n goes from -2 through 0 and positive, the focus passes the directrix. So, the parabola starts by opening down, then gets wider, opens up, and gets skinny.

*nx*^{2} - *nx* + *n*

This video shows that a subraction in the beginning changes the starting point, path, and ending point all to the other side of the plane. This means that the parabola vertex starts in Quadrant IV and ends in Quadrant I. All other behavior is the same.

*nx*^{2} + *nx* - *n*

When we subract as the second operation, this reflects the first equation about the x-axis. Meaning that all the behavior is the same, but the parabola vertex starts in Quadrant II and ends in Quadrant III.

*nx*^{2} -* nx* - *n*

When both operations are subtraction, the parabola behaves as a reflection of the second graph, the one where there is subtraction as the first operation and addition as the second. Therefore, the vertex starts in Quadrant I and ends in Quadrant IV.