The following equation is the quadratic equation in the xb plane:
The graph of the quadratic equation where a = 1 and c = 1 is below:
We can vary the constant c by setting c = 1, 3, 5, 7. This creates positive and negative values on the graph as seen below:
As c increases, we see that the graph is getting further away from the origin. What happens when c is negative? Let's compare the graphs when c = 1 and c = -1.
Let's put all of these graphs together to create a family of hyperbolas. Notice that when c = 0, the graph is a straight line on the xb plane.
In the xy plane, we know how many roots an equation has by looking at the discriminant. Let's review this rule. If the discriminant is positive, then there are two roots to the equation. If the discriminant is 0, then there is 1 root to the equation. If the discriminant is negative, then there are no real roots to the equation. We want to find a way to determine how many roots each of the equations we have been investigating have just as easily in the xb plane.
Consider the line y = m (m is a constant). This line is parallel to the x-axis and will either cross the hyperbola 0, 1, or 2 times. As it turns out, This line will show how many roots a graph has at y = m. The number of times the horizontal line crosses the hyperbola is the number of roots the graph has.
For example, let's graph y=4, 3, 2, 1, 0, -1, -2, -3, -4 and see how many roots our original graph when c = 1 has.
Observations....we can see in the graph that:
- |y| > 2, there are exactly 2 roots of the graph because the graph touches the horizontal line twice
- |y| = 2, there is exactly 1 root of the graph because the graph touches the horizontal line exactly 1 time
- |y|< 2, there are exactly 0 roots of the graph because the graph touches the horizontal line 0 times
This lines up with what we know about the discriminant. In the quadratic equation where a = 1 and c = 1, |b| must be greater than 2 for the discriminant to be positive. The discriminant will equal 0 when |b| = 2. Finally, the discriminant will be negative when |b| < 2.