During the assignment "Life of a Parabola" we explored how varying the constants a, b and c in the equation affects the graph of the respective parabola. Now are we going to explore how varying just one of the constants, b, can affect the solutions to . These solutions are also known as roots of .
Since we are exploring the effect the b has on the roots of a quadratic equation, begin by setting a and c equal to 1. Then we are looking at the solutions of . The following at the graphs of when be is equal to -2, -1, 0, 1 and 2. Pay attention to where the parabola crosses the x-axis because that is where y equals 0 and therefore where the roots of the equation reside.
From the graph it appears that the equation only has a single root when b equals 2 and -2, because each graph only crosses the x-axis in one place, and no roots for the other values of b. From "Life of a Parabola" we know that if we continue to increase the value of b the parabolas will move in a curved downward direction. Let's observe when b equals -5, -2, 0, 2 and 5.
When b is either larger than 2 or less than -2 it appears that the equation has two roots since each graph crosses the axis in two places. A few natural questions about the effect of changing b are 1) how can we predict how many roots a quadratic equation has? and 2) how can we predict the x-values of the roots of a quadratic equation?
To discover how changing the value of b affects the solutions to we are going to plot this equation in the x-b plane. What that means is every point of the graph below is an x value and a b value that gives a solution to .
The way we determine how a value of b affects the roots of the equation is by choosing a value for b, say 3, and adding to the graph the line b=3.
Observe where the two graphs intersect. The x-value of those intersection points represent roots of the original equation. What you are finding is a x value such that when it is paired with your chosen b value becomes a true statement. Earlier it appeared that had two roots when b was less than -2 or greater than 2, one root when b equaled 2 or -2 and no roots when b was between -2 and 2. Let's verify that by looking at our graph in the x-b plane. In this graph we have b=-3, -2, 1, 2 and 3.
We have explored well, but what happens if we change c from 1 to -1. Below is the graph of , which represents this change, along with the graph of so that we can see what exactly has changed.
We see that when b is between -2 and 2 the change in c makes a very big difference in how many roots are possible. In fact, there are now always two roots for every value of b. Interestingly, as b grows up towards infinity and down towards negative infinity there seem to be a very small difference in the roots when c equals 1 and when c equals -1. Let's observe the graphs for c=-5, -3, 0, 3 and 5.
It appears that as b gets very large in the positive and in the negative direction, the roots of limit to the same same roots when c equals zero, .