Problems addressed: 1,2,3,5,6
We explore the patterns of the roots of: by looking at the family of parabolas:
We remind the reader that the roots of a quadratic equation are given by:
Figure 1 shows three parabolas of the family
for b = 1, 2, 3.
Each parabola intersects the x axis (y=0) in 0, 1 or 2 points, which are the roots for the corresponding quadratic equation. For this family of parabolas, of which three are shown in Figure 1, we can say that:
If we plot the equation in the plane (x,b), we obtain a hyperbola with equation: , shown in Figure 2.
Figure 2 allows us to understand the existence of roots as b (horizontal line) changes its value:
Check this movie where the line b moves up and down and the roots are shown as black dots.
The line 2x+y=0 was plotted in Figure 2. It is the axis of the hyperbola. This line crosses the two vertices of the hyperbola, and for a given b the intersection points with the hyperbola are located at equal distances from the axis. In general, the intersection point in the axis will have coordinates (-b/2a, b) and is located at an equal distance from the roots, at a distance equal to:
If c = -1, the family of parabolas is represented by: .
Decreasing the c value from 1 to -1 makes the parabolas in Figure 1 move down the y axis without changing its shape. They cut now the y axis at y = -1. This means that for a = 1 there will always be two roots, for all b. This will be the case whenever a and c are positive and negative respectively, or viceversa, in such a way that the term is always > 0.
Figure 3 shows the function f(x,b) for c = -1. For any b value there are two roots represented as black dots.
Next let's look at the plane (x,a). Let's start with b=0 and c=1, which makes the parabolas have as axis the y axis, and vertex at (0,1). In this case, if a is positive there are no roots; if a is negative there are two roots. Figure 4 shows: the parabola for a = -3 on the right; f(x,a) on the left; and the coordinates of one of the roots for a = -3 in both sides.
With this file you can play with different values of b. For example, which value of b gives you two roots for a = 3? This next file is similar to the previous one but it also allows you to change a.
Finally let's consider the cubic equation: .
In general for any a value, b is going to open the curve upwards/downwards, c is going to move it from right to left, and d is going to be the intersection with y axis. This file will allow you to play with the four parameters a,b,c,d and see the shape the curve takes. We observe that the number of roots can be 1, 2 or 3. Why never 0?
Let's explore the pattern of roots in the plane (x,b). Figure 5 shows f(x,b) for a=c=d=1.
We observe that:
Figure 6 is a snapshot for a=c=1, b = -1 and d=0, from this file that shows the cubic function on the right side, and the function f(x,b) on the left side. The red line represents b equal to -1.
On the left side it looks like there are no roots for b = -1.... but we see the cubic curve is intersecting the x axis in (0,0) !!! What happens is that f(x,b) includes a line that coincides with the y axis, and it is difficult to see.
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