Some Different Ways to Examine
James W. Wilson and Summer D. Brown
University of Georgia
With availble technology, it has now become a rather standard
exercise to construct graphs to consider the quadratic 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
can be followed.
For example, consider varying the value of b while keeping
the a and c values constant. Let's choose to let a = 1 and c =
1. If we set
for b = -3, -2,
3, and overlay the graphs, the following
picture is obtained.
We can discuss the "movement" of a parabola as b
is changed. In every graph, the parabola always passes through
the same point on the y-axis ( the point (0,1) with this
equation). In general, the parabola will always pass through the
point (0,c) for any value of b.
Using the graphs, we can determine the roots of each equation.
The roots are represented where the graphs intersect the x-axis.
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). From the graph,
we can see that one root will be between 0 and 1 and the other
will be between 2 and 3.
For b = -2, the parabola is tangent
to the x-axis. This means that the original equation has one real,
positive root at the point of tangency. In this example, the point
of tangency and thus the root is at x =1.
For -2 < b < 2 (red, blue, and green graphs),
the parabola does not intersect the x-axis. This means that the
original equation has no real roots.
Similarly for b = 2 the parabola
is tangent to the x-axis. Again, a point of tangency indicates
one real, and in this case, negative root. The root is at x =
Finally, for b > 2, the parabola
intersets the x-axis twice to show two negative real roots for
this value of b.
For every value of b in the equation ,
the shape of the parabola remains the same. However, the location
changes as b changes. Consider the locus of the vertices of the
set of parabolas graphed from
The locus seems to be a downward facing parabola with its
vertex at the point (0,1). We can find an equation for the locus
of the vertices of the parabolas.
For any quadratic equation, the formula for the x-coordinate
of the vertex is x = -b/(2a). In this example, a = 1, so x = -b/2.
Plug x = -b/2 into the original equation
Since , . We
can substitute this into the above equation. Hence, the equation
for the locus of the vertices of the parabolas is:
The graph of this parabola is shown below in black. It is
in fact a downward facing parabola with its vertex at (0,1). Notice
that each vertex of the parabolas passes through the black parabola.
We can generalize our above results using the same
process for the general graph of .
In this case, we can leave . By
substitution into the general quadratic equation, we find that
the locus is represented by the graph of .
Graphs in the xa plane.
Consider the equation:
Here the value of a varies as the values of b and c both remain
fixed at 1.
Graph this relation in the xa plane. To graph in the xa plane,
simply substitute y in for a in the above equation. So, graphing
in the xy plane would produce
the same graph, which is shown below.
If we take any particular value of a and overlay this equation
on the graph we add a line parallel to the x-axis. Intersections
of these horizontal lines and the curve represent roots of the
original equation for that particular value of a. We can test
several values of a and graph them below. Graph a
= 2, a = 0.25, a
= 0.2, a = -2.
If the horizontal line intersects the curve in the xa plane,
the intersection points correspond to the roots of the original
equation for that value of a.
For a > 0.25 (tested by the line a
= 2) no intersections occur. Thus we can conclude that
when a > 0.25, the equation has
no real roots.
When a = 0.25, the horizontal
line intersects the curve at one point of tangency. This
means that when a = 0.25, the original graph will have one real
negative root, since the x-value of the intersection point is
< a < 0.25, there will be two
intersection points for any horizontal line drawn. Thus, there
will be two real roots, which will be negative since the x-coordinate
of the intersection point will be negative.
Last, when a > 0, (tested by the
line a = -2),
there are two points of intersection. These points of intersection
correspond to two real roots, one that will neagtive and one that
will be positive.
Try analyzing the roots of the equation
when c = -1, i.e., graph and analyze in
the xa plane. Here is a picture to get you started. What similarities
and differences do the roots of this equation have with the roots
Graphs in the xb plane.
Consider again the equation
Now graph this relation in the xb plane. To graph in the xb
plane, simply substitute y in for b in the equation of interest.
In this case, graph . We get the following
If we take any particular value of b, say b = 3, and overlay
this equation on the graph we add a line parallel to the x-axis.
We have the following graph. If the horizontal line intersects
the curve in the xb plane, the intersection points correspond
to the roots of the original equation for that value of b. So,
for this example, there will be two real negative roots for the
equation: because the horizontal line intersects
the curve twice with negative x-values.
For each value of b we select, we get a horizontal line. Look
at the graph below to determine when (and if so, how many) the
equation will have real roots. Let's analyze the graph and roots
of by graphing several horizontal
lines to represent various values of b.
Try graphing the lines b = 3,
b = 2, b = 0, b
= -2, and b = -3 over top
of the graph of in the xb plane.
The line b = 3 intersects
the graph twice at negative x-values. The line b = 3 acts as a
"test value." From the graph, we can see that any b-value
greater than 2 will intersect the curve in two distinct points.
Thus, we can conclude that when b > 2, we will get two real
negative roots of the original equation.
The line b
= 2 intersects the graph at a single
point of tangency. This means that when b = 2, the original graph
will have one real negative root, since the x-value of the intersection
point is negative. We can see from the graph that x = -1 is a
The line b = 0 is concurrent with the
x-axis. This line does not intersect the curve at all. In fact,
for -2 < b < 2, no intersections with the curve exist. Thus,
we can conclude that when -2 < b < 2, no real roots exist.
The line b
= -2 intersects the graph at a single
point of tangency. This means that when b = -2, the original graph
will have one real positive root, since the x-value of the intersection
point is positive. We can see from the graph that x = 1 is a root.
Last, the line b
= -3 intersects
the graph twice at positive x-values. From the graph, we can see
that any b-value less than -2 will intersect the curve in two
distinct points. Thus, we can conclude that when b < -2, we
will get two real positive roots of the original equation.
Consider the case when c = - 1 rather than + 1. The graph
below displays both cases when c = -1
and c = 1.
Focus on the graph in the xb plane
by itself to analyze its roots.
When for all values of b, the equation
will have two real roots, one negative and one positive.
In particular, when b = 0, the roots of the equation will
be x = -1 and x = 1.
Graphs in the xc plane.
In the following example the equation
is considered. If the equation is graphed in the xc plane,
it is easy to see that the curve will be a parabola. Consider
various values of c to overlay the graph in the xc plane. For
each value of c considered, its graph will be a line crossing
the parabola in 0, 1, or 2 points -- the intersections being at
the roots of the orignal equation at that value of c. In the graph
below, the horizontal line of c = 1 is shown. The equation
will have two negative roots -- approximately -0.2 and -4.8.
Test other values of c to analyze the roots of .
Try c = 7, 6.25,
1, 0, -1.
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 will be negative for 0 < c <
6.25, one negative and one 0 when c = 0, and one negative
andone positive when c < 0.
The determinant of the quadratic equation is another method
used to determine how many real roots exist. The determinant for
a quadratic equation is:
In this previous example, the values of a and b were fixed,
a = 1 and b = 5. Thus, the determinant becomes 25 - 4c. We can
plug in various values for c, perhaps the same values we graphed
above, to test for roots. If the value of the determinant >
0, then there will be 2 real roots. If the value of the determinant
is = 0, then there will be 1 real root. If the value of the determinant
< 0, then there will be no real roots for the original equation.
For example, test c = 7: the determinant = 25 - 4(7) = -3
< 0. Thus, there will be no real roots when c = 7.
If we test c = 6.25, however, the determinant = 25 - 4(6.25)
= 0. Hence there will be one real root when c = 6.25
Last, if we test real number less than 6.25, the determinant
will be greater than 0. Say for example c = 1, the determinant
= 25 - 4(1) = 21 > 0. Thus, there will be 2 real roots.
Using determinants and analyzing graphs
can be powerful tools in determining the roots of quadratic equations.
The students can visually see the roots of an equation and draw
insightful conclusions about a family of graphs with the aid of
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