## Explorations of Quadratic Equations

### Marianne Parsons

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

while we explore different values of b.

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.

### What do we notice?

Regardless of the value we choose of b, our parabola will always cross the y-axis at y = 1.

Depending on what value of 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.

When b = 2, the resulting parabola has ONE real root that is negative. When b = -2, the resulting parabolas has ONE real root that is positive.

When b > 2, the resulting parabolas will have TWO real roots, both negative. Similarly, when b < -2 the resulting parabola will also have TWO real roots, but these will both have positive values.

When b < 2 AND b > -2. the resulting parabolas will have NO real roots. As shown in the graph above, these graphs never cross the x-axis.

We also notice the vertices of these graphs have a very interesting relationship that we will examine further.

### Vertices

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.