Quadratic and Cubic Equations

Assignment 3
by Caralena Luthi

In this assignment we are looking at the quadratic equation in terms of the xb plane.  We use the following equation to start with: 

We are experimenting what would happen to the equation and graph if we change the b value from 5 to -5.  We begin to notice that the graph has a hyberloic tendency.  We also find that the equation will provide 2 roots, one positive and one negative, or there will not be any roots.  To use the graphing calucator program we will use the following equation:

Using the y value, we can change the value from 5 to -5.  We notice that when y is less than 2 but greater than -2 that we have no roots.  This can be seen in the graph below.

We can prove that we have two vertices on this graph at (-1, 2) and (1, -2) by solving the equation when y = 2 or -2.  When you solve for the x value you find that there is only one solution for each section of the graph.  x = 1 or -1.

Using the Graphing calculator program we can trace the y values from 5 to -5 and visual we can see the roots being traced. 

Next we can change the equation to where c = -1 instead of 1.  This will change the hyberolic direction.  We will see this new graph flip over the y-axis and become wider. 

When we changed the c value to -1 we also see that we will have always 2 roots, one positive and one negative.  The original graph once again lacks roots when y is less than 2 but greater than -2. 

We know that the vertices of the black graph are (-1, 2) and (1, -2).  The green graph represents the linear equation of y = -2x.  This linear equation cuts through the graph and is bisecting the graph and acting as a midpoint between the two roots on either side.