Amberly Roberts

**Assignment 3**: **An Investigation of the equation **

We will begin by fixing b = 1, c= 1, and letting "a" vary. The graphs below demonstrate the behavior of the curve.

Notice that the point (0,1) is on the graph of each parabola.

Next, we will fix a = 1, c = 1, and let b vary. The graphs below demonstrate the behavior of the curve.

It appears that the locus of vertices is parabolic in shape. We will use an algebraic investigation to see if this is true.

We'll conclude Investigation 1 with a graph that is convincing that the equation we worked to obtain is true.

Now, we will examine the roots of the equation and investigate how the value of "b" affects the roots. An animation will give us an idea of the behavior.

It appears that the number of real roots changes as the value of "b" changes.

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