(as b varies)

by

This exploration is about the graphs of quadratic equations of the form.

For this exploration, we are going to fix 2 of the values in the quadratic (a, b, c) and see what conclusions can be found.

Before we begin, let's look at the parent graph.

Here is a movie to see the changes in the quadratic equations with a=1, c = 1, and b varies from -10 to 10 in one hundred steps. Click here.

Here is a graph of 9 quadratic equations with a=1, c=1, and b = 0 (red), b = 1 (purple), b = 2 (blue), b = -2 (green), b = -1 (aqua), b = -0.5 (yellow), b = 0.5 (grey), b = -3 (black), b = 3 (purple).

With varying the b value in the quadratic equation, it appears that the graph moves in a parabolic path.

The graph above is the same as the colorful graph above it.  The only thing I did was made all of the equations in the colorful graph black and graph the parabolic path in red,

which is y = -x2 +1.

Interesting!  Here is another graph of different quadratic equations in black and the parabolic path in red.

For the black graphs, a = -2 and c =-1 with b = -3, -2, -1, -0.5, 0, 0.5, 1, 2, 3.

For the red graph, the parabolic path satisfies y = 2x2 -1.

Thus, it appears that the parabolic path will always be the opposite of the a value given, b = 0, and the same c value given.

Question:  Given the quadratic equation, y = -0.5x2 + bx - 4.  What would be the parabolic path?