It has now become a rather standard exercise, with availble technology,
to construct graphs to consider the
equation
and to overlay several graphs of
for different values of a, b, or c as the other two are held constant. From
these graphs discussion of the
patterns for the roots of
can be followed. For example, if we set
for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture
is obtained.
We can discuss the "movement" of a parabola as b is changed. The
parabola always passes through the
same point on the y-axis ( the point (0,1) with this equation). For b <
-2 the parabola will intersect the
x-axis in two points with positive x values (i.e. the original equation
will have two real roots, both
positive). For b = -2, the parabola is tangent to the x-axis and so the
original equation has one real and
positive root at the point of tangency. For -1 < b < 2, the parabola
does not intersect the x-axis -- the
original equation has no real roots. Similarly for b = 2 the parabola is
tangent to the x-axis (one real
negative root) and for b > 2, the parabola intersets the x-axis twice
to show two negative real roots for
each b.
Now consider the locus of the vertices of the set of parabolas graphed from
To determine the vertices of the set of parabolas we need to take the derivative of the equation. The derivative will determine the slope of the line that is tangent to the parabola. At the vertex of each parabola, we know that the slope of the tangent line must be equal to zero because it is the tangent line is a horizontal line.
The derivative of the parabola is
When we set the derivative equal to zero, we find
We can use this to solve for x
This will allow us to find our x coordinate. To calculate our y coordinates we can use the original equation. Below is a table of values for x and y.
If we plot these points we will be able to see the locus of the vertices is the parabola
The locus of the parabola can help us to determine the vertices of set of original parabolas. In fact, there are many relationships when we are looking at sets of parabolas and the values of a, b, and c.
Consider again the equation
If we graph this relation in the xb plane with the lines b = -3, -2, -2,
1, 2, 3, we can use this to determine the roots of the parabola for the
given values of b.
If we take any particular value of b, say b = 3 and b = -3, and overlay
this equation on the graph we add two lines
parallel to the x-axis. If it intersects the curve in the xb plane the intersection
points correspond to the roots
of the original equation for that value of b. We have the following graph.
For each value of b we select, we get a horizontal line. It is clear on
a single graph that we get two
negative real roots of the original equation when b > 2, one negative
real root when b = 2, no real roots
for -2 < b < 2, One positive real root when b = -2, and two positive
real roots when b < -2.
The same is true for the varying values of a.
Consider the relation
and its graph
You can determine what values of a will give you the number of roots that are real.
The same is true for the values of c and the graph in the xc plane.
You can use these graphs to show the relationship between geometry and algebra.