Final Examination

II. A New Look

Equations in the form

When exploring the effects of changing multiple numbers in a relationship, a relationship grapher eliminates the repetitiveness of doing calculations. Algebra Xpresser will be used to examine the effects of changing the exponents in the following equation.

The values of n that will be examined will be natural numbers including 2, 3, 4, and 5. By looking at the graphs as n increases, the future graph shapes will be predicted for higher values of n. When n = 2, a circle centered at the origin with radius one is the resulting graph. Throughout the rest of the discussion, the red circle will be the graph of

.

When the preceding equation is graphed, the shape of the graph is a line with a bump in it. When the graph is far away from the bump, it looks like a straight line. When 0< x< 1, the graph behaviors similiarily to the first equation graphed except that a "corner" has been introduced. The graph is being "stretched" towards the point (1,1). The green graph below is the graph with cubic exponents.

When the degree of the equation is raised to four, the shape of the graph is similar to the graph when the equation had degree two, except that the graph has four "corners". These corners are the result of the graph being stretched towards the points (1,1), (1,-1),(-1,1),(-1,-1). The blue line is the newest graph. If the blue and green line are compared, the "corners" are more pronounced in the graphwith a higher exponent.

When we considere the preceding equation, we would expect it to be similiar to the equation with degree three except that the corner woul be more pronounced. Once again the graph has been stretched towards the point (1,1) Based on the pevious results, we expect the corner to be more pronounced than any of the pervious corners. The brown line is the graph when n=5, and its corners are more stretched out than any others.

From these results, the shape of the graph will continue to change. The sides will continue to become flatter, and the corners more pronounced. As n gets larger and is even, the shape of the graph will approach a square. However, the graph does not become a square. If the graph for n = 24 is examined, it is almost a square. When n is increased even higher, to 90 for example, the graph appears to be a square. When a corner region is expanded, what appeared to be a perpendicular corner again takes on a rounded shape.

When n is 25, the corners should become much more pronounced compared to smaller values of n. The graph below shows that the corners become more pronounced. As n continues to increase, and is odd, the corners that exist on the graph should become more pronounced. These results for n = 89 are displayed. Once again, when the corner is expanded, the corner becomes more rounded.

No matter how large n becomes, we should always be able to zoom into a corner region to show that it is still curved, if only slightlty.

When the equation no longer equals 1, the results are not as easy to interpret. 1 is a "nice" number for the equations to equal because one is a power of all values of n. The original equation being examined could be rewritten as

with a = 1. The graphs for other values of a will be briefly examined, as their behavior should be similar to the behavior of the original graph, with the size of the graphs being expanded or contracted depending upon the value of a. The colors on the graphs are the same as before, with red having a power of two, green having n=3, blue has n=4 and brown has n=5. The different values of a produced similair graphs. If the scale was not included on the axis, it would be hard to tell the graphs apart. The three values of a that were chosen, in order, were a = 1/3, a=5 and a = 2.

a = 1/3 a = 5
a=2

 By examining the graphs, it is apparent that the graphs all behave similiarily except for their scale. The lager the a value is, the greater the size of the circle and othe graphs.