Assignment 2
Problems and Explorations with Second Degree Equations
By: Nicolina Scarpelli
Graph x2 + y2 = 9. Now, on the same axes, graph the equation x2 + xy + y2 = 9. Describe the new graph. Try different coefficients for the xy term. What kinds of graphs do you generate? What coefficients mark the boundaries between the different types of graphs? How do we know these are the boundaries? Describe what happens to the graph when the coefficient of the xy term is close to the boundaries.
For this assignment I will be using Graphing Calculator Lite software to graph second degree equations and examine the graphs. I will be exploring the effects of different coefficients for the xy term, and observing the graphs that are generated. Thus, to do this in Graphing Calculator Lite, use the parameter n, as the coefficient for the xy- term and investigate what happens to the graphs with various n values in the equation x2 + nxy + y2 = 9. How does the graph change when n is positive or negative? Does n being odd or even change anything?
Let's begin by graphing the equation x2 + y2 = 9 where n is equal to zero.
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From the graph, we see that x2 + y2 = 9 is a circle centered at the origin (0,0) with radius equal to 3.
Next, on the same axes, graph x2 + xy + y2 = 9, thus n is equal to one in this case.
Observe the new graph(shown in blue). Notice it is a graph of an ellipse with x-intercepts at (-3,0) and (3,0) and y-intercepts at (0,3) and (0,-3). However, as you may notice, this is not a graph of a normal ellipse. The formal definition for an ellipse is the set of all points such that the sum of the distances from the foci and any point on the ellipse is constant.
The general equation for an ellipse centered at (0,0) with a horizontal major axis is
where a is the distance from the center of the point of intersection between the major axis and the ellipse, and b is the distance from the center of the point of intersections between the minor axis and the ellipse.
The general equation for an ellipse centered at (0,0) with a vertical major axis is
where a is the distance from the center to the point of intersection between the major axis and the ellipse, and b is the distance from the center to the point of intersections between the minor axis and the ellipse. For both equations listed above, a > b. However, in our example we have the equation x2 + xy + y2 = 9.
When simplified, it equals
. As you can see, the a and b values in this equation are the same. They both equal 3, thus resulting in a "tilted" ellipse, rather than a normal ellipse. The graph of this ellipse is rotated by 45 degrees counterclockwise about the origin.
Let's continue investigating the quadratic equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. How does the Bxy term or (nxy term in our case) effect the graph of the conic section? As stated above, we can see that a positive Bxy term causes a counterclockwise rotation about the origin; however, a negative Bxy term causes a clockwise rotation about the origin. This is shown below.
After discovering this "tilted" ellipse, I continued trying different coefficients for the xy-term. I observed what happened when I let n = 2.
As you can see, this graph is of 2 parallel lines. This is because the equation x2 + 2xy + y2 = 9 can be factored into (x+y)(x+y) = 9 which is equivalent to (x+y)2 = 9.
Then, to simplify this equation, take the square root of both sides and you get two different equations:
x + y = -3 and x + y = 3.
Then, when these equation are put into slope-intercept form y = mx + b, observe that
y = -x - 3 and y = -x + 3.
Both equations have a slope m = -1 which makes them parallel lines.
Now let's graph more values of n between 2 and 3 to see the changes in the graphs.
As you can see from this graph, as the parameter n gets larger and larger, the graph transforms more and more from an ellipse to a hyperbola. Next, let's observe the graph of the equation x2 + 3xy + y2 = 9.
You can see that this appears to be a graph of a hyperbola oriented around the origin with x-intercepts at (3,0) and (-3,0) and y-intercepts of (0,3) and (0,-3). We can prove that this graph is in fact a hyperbola by using the Discriminant Test. The conic section is a hyperbola if the discriminant, B2 - 4AC , is greater than zero. In this case we have A = 1, B = 3, and C = 1. Substituting these values, we get 32 _ 4(1)(1) = 5 as our discriminant, which is indeed greater than zero. Therefore, x2 + 3xy + y2 = 9 is a hyperbola.
Below, I have graphed all of the equations in the same xy plane to give a visualization of how the graph is changing when the parameter n is varied.
From this graph, you can see as the parameter n gets larger and larger, it goes from being a circle centered around the origin to a hyperbola oriented around the origin. This is also shown in the animated movie below.
This animation show the effects of the parameter n on the equation x2 + nxy + y2. In this animation, the parameter n goes from negative 10 to positive 10. From this movie we can see how all the graphs of the equations change over the parameter n. The graph begins as a circle when n = 0. As n increases, it becomes a tilted ellipse rotated in the counterclockwise direction and stretched along the line y = -x, until n = 2, where it becomes two parallel lines, y = -x +3 and y = -x -3. Then n becomes greater than positive 2 and the graph becomes a hyperbola.
Now, let's observe what happens when we let n be negative(thus resulting in a negative Bxy term). Observe the image below:
As you can see, between the values of 0 and -2 the graph is a tilted ellipse(shown in purple) being stretched along the line y = x. It is tilted in the clockwise direction. At n = -2 the graph changes into two parallel lines(shown in red) y = x + 3 and y = x - 3 with the slope m being equal to 1. When n becomes less than negative 2 the graph changes into a hyperbola.
This assignment would be great for a high school class to show students how parameters can affect the shape of graphs. These explorations can be engaging and interactive with students. This could be assigned as a project to complete to know if your students have a strong understanding of the properties of certain graphs.
