we may consider the equation
for different values of a, b, or c as the other two are held constant.
If we overlay several of these graphs, we can see how the graph changes
as the chosen coefficient varies.
For example, if we set
for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, we can explore
the relationship between the roots of the equation and b. Due to the possible
confusion caused by six graphs on one axes, we chose to separate the graphs
into -3 <= b <= 0 and 0 < = b <= 3. The following are graphs
for -3 <= b <= 0.
We can discuss the "movement" of a parabola as b changes.
parabola always passes through the same point on the y-axis ( the point
(0,1) with this
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).
b = -2, the parabola is tangent to the x-axis and so the original equation
has one real,
positive root at the point of tangency.
-2 < b < = 0, the parabola does not intersect the x-axis -- the original
equation has no real
The following are graphs for 0 <= b <= 3.
again that each parabola always passes through the same point on the y-
axis ( the point (0,1) with this equation).
b > 2 the parabola will intersect the x-axis in two points with negative
x values (i.e. the original equation will have two real roots, both negative).
b = 2 the parabola is tangent to the x-axis (one real, negative root).
0 < = b < 2, the parabola does not intersect the x-axis -- the original
equation has no real roots.
To summarize, we see from both graphs that, the equation has
real roots when
b <-2 (both roots are positive) or
b > 2 (both roots are negative)
b = -2 (root is postive) or
b = 2 (root is negative)
in the xb plane. We obtain the following graph.
If we take any particular value of b, say b = 3, and overlay this equation
on the graph, i.e., adding a line parallel to the x-axis, we find that b=3
intersects the curve in the xb plane. These points of intersection correspond
to the roots of the original equation for b=3. See the following graph.
For each value of b we obtain a horizontal line. Thus, 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. In other words, from this one graph students can glean the
same information, they obtained from the two previous graphs, with less
work and less visual confusion.
Now let's consider an equation in which we will vary c instead of b. For
If the equation is graphed in the xc plane, the curve will be
a parabola. For each value of c considered, its graph will be a line crossing
the parabola in 0, 1, or 2 points -- the intersections are the roots of
the orignal equation for a particular value of c. In the picture below,
the graph of c = 1 is included.
Thus, the equation
will have two negative roots -- approximately -0.2 and -4.8.
There is one value of c where the equation will have only 1 real root --
at c = 6.25. For c > 6.25 the equation will have no real roots and for
c < 6.25 the equation will have two roots, both negative for 0 < c
< 6.25, one negative and one 0 when c = 0 and one negative and one positive
when c < 0.
For further investigation, we could vary the coefficient a.
In summary, we wish to emphasize that using this type of graph (the xb or
the xc plane, versus the xy plane), we can get the same amount of information
pertaining to the relationship of the roots to the changing coefficent with
less work and less visual confusion!