Quadratics in the xb-plane
By Courtney Cody
Consider the following equation:
This equation is the special case of the general quadratic equation , where a=1, c=1, and b is some constant. In this exploration, we will look at the effect b has on the equation by studying the graph of the equation and investigating what happens when the value b is varied. LetŐs start by graphing this equation in the xb-plane and seeing what we get. To graph in the xb-plane, we simply graph and substitute y in for b so that now b becomes a variable. Hence, we get the following graph:
When is graphed in the xb-plane, we notice that it is a hyperbola. If we take any particular value of b, say b = 4, and overlay this equation on the graph we are actually adding a line parallel to the x-axis. This is because we have allowed b to be an axis on our graph. In particular, b is represented as the y-axis in our graphs. If this value of b 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:
If we look at this graph, we see that the red line, which represents b=4, intersects at two different points. What do these two intersections represent? Since our graph is the situation when b=4, then our equation is . Algebraically, if we solve for x using the quadratic formula, we get , or and . These two values are the roots of . Now, if we look at Graph B to see how these two x-values relate, it appears as though these are the x-values where the intersections of and occur. So does this mean that that the intersection of with a value of b represents the real roots of the original equation? LetŐs explore some more values of b to see if this is true!
If we look at Graph B, it looks as though the vertex of the top curve of the hyperbola appears to be at b=2. By letting b=2, we get the following graph:
It appears that the line b=2 touches at one point. If we substitute b=2 into our equation to get and algebraically solve for x, we get , or . Once again, is the root of and is also the x-value where the intersection of the two curves in Graph C occurs.
It is easy to see that if b=-4 and b=-2, then similar results will be found because the bottom curve of the hyperbola will be intersected by b=-4 in two places and by b=-2 in two places. But what about the area between these two curves of the hyperbola? What would happen if we let b=-1? LetŐs explore the graph of this situation!
Clearly, does not intersect at any points. So what does this mean algebraically? Using the quadratic formula, we get , which has only non-real roots. Hence, when there are no real roots.
It appears that for each value of b we select, we get a horizontal line that either does or does not intersect our equation . From Graphs A-D, it is clear that on a single graph we get the following:
á Two negative real roots of the original equation when b > 2
á One negative real root when b = 2
á No real roots for -2 < b < 2
á One positive real root when b = -2
á Two positive real roots when b < -2
Consider the case when c = - 1 rather than + 1 in the equation . LetŐs graph other values of c on the xb-axis and explore the graph! Click the picture below for a movie!
The moving red line in Graph E represents all the different values of b in the domain. If we look at the relationship between the purple curve and the red line, we see that the red line always intersects the purple curve in exactly two places. Since in the previous graphs we discovered that the intersections represented real roots of the quadratic equation, then we can check to see if this is true. Using the quadratic formula, we get , which will always have two real solutions since is always positive. Thus, when , the equation always has two real roots for all values of b.
Why is there a difference in the number of solutions for when and ? LetŐs graph for other values of c.
It appears that for each value of c we select, we get a hyperbola or a line. From Graph F, it is clear that on a single graph we get the following:
á When , there exists a b value such that there are exactly two real roots, exactly one root, or no roots.
á When , the equation is a straight line and there are always exactly two real roots for all values of b.
á When , the equation is a hyperbola and there are always exactly two real roots for all values of b.