**Exploration of Quadratic**

**(as a **

**
**by

Chad Crumley

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.

Look at this
movie. Click __here__.

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.