# Exploration of the Quadratic Equation

By Brooke Norman

In this assignment, we are going to look at the quadratic equation of:

I will use Graphing Calculator 3.2 to examine the effects of changing the values for ÔaÕ, ÔbÕ, and Ôc.Õ

First, lets explore the equation by letting ÔbÕ and ÔcÕ equal 1 and varying ÔaÕ from -3 to 3.

As you can see from the above graphs, each equation crosses the y-axis at 1.  The graph when ÔaÕ=0 is a straight line that is tangent to the other graphs.  When ÔaÕ is a positive number, the parabolas are concave up and when ÔaÕ is negative, the parabolas are concave down. As ÔaÕ changes, so do the vertexes. As the value of ÔaÕ increases, the parabolas become more steep and skinnier.

The graph shifts to the left as ÔbÕ increases in a positive direction and the graph shifts to the right as ÔbÕ increases in a negative direction. Notice that the vertex shifts as ÔbÕ is varied.

We are now going to explore the equation but this time lets let ÔaÕ and ÔbÕ equal to 1 and let ÔcÕ vary from -3 to 3.

As you see, when ÔcÕ varies, the vertex is shifting up and down.  Therefore ÔcÕ is changing the y-coordinate of the vertex.

In conclusion:

The position of the vertex is affected by Ôa, b, and c.Õ  The x- and y-coordinates of the vertex are affected by Ôa and bÕ and only the y-coordinate appears to be affected by Ôc.Õ  It also appears that ÔaÕ also determines the steepness and he direction of concavity of the parabola.