Investigating the graph of

Write-up by

Blair T. Dietrich

EMAT 6680

What happens to the graph of as
*a* varies?

This investigation will explore the relationship
between the
quadratic coefficient *a* and the related graph.

(Note that a necessary condition is that ; otherwise, the equation would not be quadratic.)

If we let *a* = 1, we have the equation _{}. We will refer to
this as the "parent" equation.

What happens as *a* gets larger than 1?

Notice that as *a* gets larger, the graph of
the
parabola gets narrower than the graph of the parent equation.

If we start by letting *a* = -1, we have the
reflection
of the parent graph about the x-axis.

As we allow this (negative) value of *a* to
have
greater magnitude (i.e. as |*a*| gets larger), the graph again
becomes
narrower.

Below are some examples for negative values of *a*.

Now let us consider values of *a* in the
open interval
(0,1):

Notice that values of *a* that are closer to
zero
result in a wider graph than that of the parent graph.

Values of *a* in the interval (-1,0) yield a
similar
result, i.e. values closer to zero yield a wider graph.
The only difference is that a negative value
of *a* results in a graph that has been reflected about the
x-axis.

To try your own values of *a*, click here.