Assignment 3

By: Olamide Alli

Exploration: Graphs in th XB Plane.

Look at . This graph will show the relationship in the (x, b) plane.

Let b = y

If we add a horizontal line to the graph in the (x,b) plane, the intersection of the two graphs produced represents the roots of the equation for whatever value we choose for b.
Let’s use b = 2 as an example.

As we can see from the graph, the horizontal line for x = 2 touches the graph of    at the point (-1, 2). Let’s prove that x = -1 is a factor of the equation  when b =2 , therefore the equation is . By factoring we get the following

We have now shown that the intersection of the 2 graphs is the root of the equation , the graph intersects at a point where x = -1, which is the root of the equation. I’d like to consider the following cases:
Case 1: b = 2
Case 2:  b = -2
Case 3: b < -2
Case 4: b > 2
Case 5: -2 <b <2
We’ve already explored and examined the case of b = 2. Let’s explore and determine the roots for cases 2 through 5.

Case 2:
Let b = -2

Let us solve the equation using factoring:

There is one root at x = 1, which is where the graphs of y=-2 and  intersect.

Case 3:
Let b < - 2….we will choose b = -3

Let’s solve the equation using the quadratic formula:. So for the equation  when b = -3, we get  Now let’s solve for x.

So there are two positive real roots. From the image above, the two graphs intersect at two points.

Case 4:
Let b > 2…we will choose b = 3

Let’s solve the equation using the quadratic formula: . So for the equation  when b = 3, we get  Now let’s solve for x.

So there are two negative real roots. From the image above, the two graphs intersect at two points.

Case 5:
Let -2 < b < 2…we will choose b = 1

Let’s solve the equation using the quadratic formula: . So for the equation  when b = 1, we get  Now let’s solve for x.

So there are two imaginary roots. From the image above, we can see that graph for y = 1 and do not intersect at any points.

Summary of the Case Results
When b = 2, there is one root at x = -1.
When b = -2, there is one root at x = 1
When b < -2, there are two positive real roots
When b > 2, there are two negative real roots.
When -2 < b < 2, there are only imaginary roots.

Now let’s consider the case when c = -1

When we change the value of c from to -1 to 1, the graph transforms into a hyperbola.

Let’s graph other values of c on the same axes.
Let c = -2, -1, 0, 1, 2

From the image above, notice that the value of c changes the graph at three separate instances. The options for the images are two different types of hyperbolas or a line. Please keep in mind that the hyperbolas are restricted to different areas of the (x,b) plane when altering the variable c.
Case 1 is when c = 0, the equation produce a line (y = -x)
Case 2: When c < 0, the equation produces a hyperbola, which s asymptotic to y = -x and x = 0
Case 3: When c > 0, the equation produces a hyperbola which asymptotic to y = -x and x = 0