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.