Using Graphing Calculator 3.2 we can explore various patterns of roots and vertices through different graphs of the equation:

Let's examine various graphs of this equation
that are generated for **b**** **= -3, -2, -1, 0, 1, 2,
3.

To see the positive and negative values of
**b** in direct relation to each other, click here.
View the animation of this graph for various positive and negative values of **b**.

**b**, our parabola will always cross
the y-axis at y = 1.

** b** is chosen, our parabola will have a
particular number of roots. The roots of a parabola are defined
as the points on the parabola, where y = 0, where that parabola
crosses the x-axis.

We can clearly see that the vertices of these parabolas share a special relationship. In order to view this relationship more clearly, let's mark each vertex of our graph.

Now, let's remove our parabolas to see the only those vertex points.
Can you see the resulting shape these vertices will take?

It appears as though these vertices form their own parabola. In order to get the equation of the parabola, we must take three of our vertex points and then solve the resulting systems of equations. View the algebraic steps, and proof that this is the equation of the parabola that joins our vertices above.

These points are joined by the parabola of :

Therefore, our equation:

with varying values of **b**, will generate
a parabola that will have ONE, TWO, or NO real roots, and will
have vertices that fall on the parabola:

View the animation of this graph, and see its vertices move along our new parabola.

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