Some Different Ways to Examine

 

by

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 roots of

 

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, -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. 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 = -1.

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 negative.

When 0 < 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 of ?


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 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. 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 root.

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 graphing technology.

 


Send e-mail to jwilson@coe.uga.edu or to sumtime27@hotmail.com

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