(as a 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 .

The Constant a

For this exploration n = a.

To see a movie as n varies, click here.

After watching the movie, what are your conclusions?

First for positive n:  As n increases, the graph will increase at a faster rate than the parent graph.  Thus, the graph gets closer to the y-axis (or the graph appears thinner.)  As n decreases, the graph will decrease at a slower rate than the parent graph.  Thus, the graph gets further away from the y-axis (or the graph appears wider.)

For negative n:  As n decreases (obtaining a larger absolute value), the graph again moves closer to the y-axis becoming thinner.   As n increases (obtaining a smaller absolute value), the graph moves away from the y-axis becoming wider.

What about when n = 0?  Then the equation becomes y = 0 (a constant function).  In other words, no quadratic is produced.

Now lets fix b and c and vary a in the equation .

For this exploration b = 1, c = 2, and

a = 1 (red), a = 2 (blue), a = -1 (purple), a = -2 (green), a = -3 (aqua), a = 3 (yellow), a = 5 (grey), a = -5 (black), a = 6 (purple), a = -10 (blue).

The vertices of the graphs above appear to be slightly moving and it appears there is a slant asymptote that the graph is approaching.

After looking at the movie, what is the equation of the slant asymptote?

Here again is the graph:

So, the line has slope of 1 and y-intercept of 2.  So the asymptote is the linear equation:

Or, this equation is the same as the quadratic equation:

with n = 0.