Exploration of Quadratic

(as a varies)


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.