Further Investigation of equations of the form ax^{2} + bx + c = y

by

Jackie Ruff

Let's begin this investigation by looking at a group of parabolas, y = x

It appears that the vertices are also following the path of a parabola, opening downward with a vertex at (0,1). |

If we graph this equation with our group of parabolas, it appears that we have found the correct equation. |

We
could confirm our results by finding the equation another way. If
we select three of the vertices and find the quadratic equation through
those points, we find the same equation. To see the work, click here.

Part 2:

Next we will consider the equation x

So, now we see that we can find the roots of any of a group of quadratic equations, by graphing in the xb plane and looking at the x-values for a particular value of b. We can also see for which values of b that group of equations has one or two real roots and when it has only imaginary roots.

So, let's look at one more example now that we know what to look for.