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,
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.
of the value we choose of b, our parabola will always cross
the y-axis at y = 1.
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.
b = 2, the resulting parabola has ONE real root that is negative.
When b = -2, the resulting parabolas has ONE real root that is
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.
b < 2 AND b > -2. the resulting parabolas will have NO real
roots. As shown in the graph above, these graphs never cross the
also notice the vertices of these graphs have a very interesting
relationship that we will examine further.
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
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.Return to Class Page