Assignment 3:  Further Exploration of Parabolas

by Kristina Dunbar, UGA


In Assignment 2, we explored the graphs of the equation y = ax2 + bx + c while varying the values of a, b, and c one at a time.  In this assignment, we will be exploring roots of this equation.  For simplicity, we will let a = 1 and c = 1 and vary the value of b.

Let's consider the equation

x2 + bx + 1 = 0

First, we'll graph the equation in the xb-plane:

We notice that there are no solutions to this equation when -2 < b < 2.  We also notice that if b equals 2 or -2, there is exactly one solution to the equation, and that when b < -2 or b > 2, there are two solutions.  Here are some further examples:


Let's consider the same equation now, but overlay different values of b.


It is very clear by looking at these graphs that there are no solutions to the equation x2 + bx + 1 = 0 when b is between -2 and 2.

We can look at the same information graphically in a different way, by looking at the parabolas for the equation y = x2 + bx + 1.


The graph has roots where the parabola crosses the x-axis. 


There is one other investigation we can do with the above graph.  Let's graph the vertices of all of the above parabolas.

Notice that this parabola satisfies the equation y = -x2 + 1. 

How do the graphs of these parabolas tie in to what we know about quadratic equations?


Let's consider the general solution to the equation y = ax2 + bx + c. 

The portion of the equation

is considered the discriminant, and the quantity under the square root symbol must be greater than or equal to zero to get real roots.

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