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Assignment #2 - 10

The effects of on a circle when you create and manipulate the n varible

(1) When n = 0, (2) When n is positive,

(3) When n is negative, (4) Summary for all values for n

(5) What might be the next investigation

(1) Baseline case of when n = 0, a circle

WHEN n = 0

When n = 0 the middle term of the equation simplifies to:

This basic equation will serve as our reference shape and equation for the entire investigation. We will compare the new findings to this original shape. Also note that the circle intersects at (3,0), (-3,0) and (0,3), (0,-3).

(2) The positive values for n

WHEN

The original circle (red) becomes an ellipic shape as n approaches +1.

Notice that in quadrant 1 & 3 the new shapes are formed inside the circle while in quadrants 2 & 4 the ellipse stretches outside the original shape.

The y = -x line has been places in the diagram to show a line of symmetry for the newly created shapes.

WHEN

The two graphs above show the pattern created by the increased positive value of n . These ellipse shaped figures continue on stretching into quadrant 2 & 4 as long as the value for n is less than 2. As the value for n closes in on the value of 2 the ellipse shapes stretch along the y = -x line. The second graph clearly demonstrates the "flattening" process to the y = -x line.

What happens when n = 2? First we can investigate this question graphically.

To our suprise we find that the ellipses have turned into two linear equations.The lines still go through the 4 intercepts and happen to be parallel to each other and the y = -x line.

Algebraically we can now look at what took place to create this result.

2xy is the exact value such that the right hand side of the equation simplifies into a perfect square. Given that the right hand side is a perfect square and that the left hand side will always be the square of some radius value, we will be able to factor the equation as a difference of squares. The result is two factors that can be solved for zero. The two linear equations represent all the value such the two factors can be solved for zero.

WHEN

Beginning the diagram with the original circle (blue) and the two linear equations y = -x + 3 and y = -x -3 when then manipulated n to approach positive infinity.

We see a hyperbolic shape appear. It is interesting to note that these shapes all still go through the intercepts. The location of the hyperbolic shape is also very defined. In quadrants 1 & 3, it is bound by the two linear equations and the x and y axises. While in quadrant 2 & 4 they are bound by the linear equations and the axises.

WHEN

All Possible Values

The blue area represents the envelope of all possible values of the releationship such n is greater than or equal to 0.

I do wish to mention that the graph gives is a bit misleading in that it appears that (0,0) is a point on one of the graphs. This is not true, the graph approaches the origin but it never touches it, similarily the x and y axis.

(3) The negative values for n

The investigation for the negative values followed a similar pattern as to the positives. Instead of walking through the step by step process, I will simply say that the same phenonenon occured for the negative values for n except that the quadrants in which the grpahs occured was the switched. Also instead of all shapes being symmetrical on the y =-x line, the shapes were symmetrical to the y = x line. I will diagram below the basic finding for the negative values.

These four graphs display the basic makeup for the negative values of n. The last graph like the one for the positves, displays all possible points when n is less than or equal to 0.

(4) Summary for all values for n

Thus the result of all values for n is the union of these two disjoint sets. Again I note that not all points on this plane can be represented in the equation, for example (0,0).

Other interesting shaped graphs occured when we use additive inverses for the values for n.

I think it is interesting to look at the combination of both positive and negative values for n. The graph on the left has a square as its center boundry while the one on the right doesn't extend into the regions between the linear equations.

What might be next to investigate?

The effects of onthe ellipse when you create and manipulate the n varible

Here are a few diagrams from that extension.

The end

WHEN

The two graphs above show the pattern created by the increased positive value of n . These ellipse shaped figures continue on stretching into quadrant 2 & 4 as long as the value for n is less than 2. As the value for n closes in on the value of 2 the ellipse shapes stretch along the y = -x line. The second graph clearly demonstrates the "flattening" process to the y = -x line.



These four graphs display the basic makeup for the negative values of n. The last graph like the one for the positves, displays all possible points when n is less than or equal to 0.


Thus the result of all values for n is the union of these two disjoint sets. Again I note that not all points on this plane can be represented in the equation, for example (0,0).

Other interesting shaped graphs occured when we use additive inverses for the values for n.

I think it is interesting to look at the combination of both positive and negative values for n. The graph on the left has a square as its center boundry while the one on the right doesn't extend into the regions between the linear equations.