Some Different Ways to Examine

by

James W. Wilson, Doug Westmoreland and Brad Simmons
University of Georgia

 

It has now become a rather standard exercise, with available 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.

Let's first take a look at the effect that a has in while we let b and c remain constant. We will let b and c equal 1 and we will set a=-3,-2,-1,1,2,3 and overlay the graphs. The following picture is obtained.

Notice a couple of things. First all of the graphs are going through the point (0,1). Next, when the value of a is zero, then the equation is no longer a quadratic equation. When a=0, the equation reduces to a linear equation. This is shown by the yellow graph above.

For a dynamic presentation of the effects of the a value in the parabola above click here.

 

Graphs in the xa plane.

Consider the equation and the graph of this relation in the xa plane.

 

 

 

If we take any particular value of a, for example a=-2, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xa plane, the intersection points correspond to the roots of the original equation for that value of a. Consider the following graph. We overlay the graph of a=2 (blue horizontal line) and a=-2 (green horizontal line). For a=2, we have no real roots for our original equation. For a=-2, we have two real roots. Furthermore, a=0, (horizontal axis of this graph) yields one real root.

 

Now let's take a look at the effects of the b value in the equation while we let a and c remain constant. We will let a and c equal 1 and we will let b= -3 ,-2 ,-1, 0, 1, 2, 3.

So, if we set for b = -3, -2, -1, 0, 1, 2, 3, and overlay the

graphs, the following picture is obtained.

 

For a dynamic presentation of the effects of the b values in the parabola above click here.

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 -2 < 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

.

Show that the locus is the parabola

Generalize.

 

 

Graphs in the xb plane.


Consider again the equation

Now graph this relation in the xb plane. We get the following graph.


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. 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.



Consider the case when c = - 1 rather than + 1. The image below shows the the graph when c=1 (green), when c=-1 (red) and ovelays the equation b=5 (blue). From the graph below, we can conjecture that when c=-1 the equation will have two real roots for all real values of b.


Finally, let's take a look at the effect the c value has in the equation for c = -3, -2, -1, 0, 1, 2, 3. We will let a and b remain constant with a value of 1. Overlaying the graphs for the different c values, we obtain the following figure:

 

( The bottom purple is when c=-3, red c=-2, blue c=-1, green c=0, lt. blue c=1, yellow c=2, and top purple c=3.)

 

For a dynamic presentation of the effects of the c values in the graph above click here.

 

 

Graphs in the xc plane.

Let's consider the equation .
If the equation is graphed in the xc plane, it is easy to see that 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 being at the roots of the orignal equation at that value of c. In the graph, the graph of c = 1 and c=-2 are shown. The equation

will have have no real roots c=1 (purple line) and will have 2 real roots when c=-2 (red line).

 

There is one value of c where the equation will have only 1 real root -- at c = .25. For c > .25 the equation will have no real roots and for c < .25 the equation will have two roots, both negative for 0 < c < .25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.

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