Assignment #1

Problem #9

by

Megan Dickerson

Graphs

This graph is a circle with radius 1 and center (0,0).

This graph starts as a line with a negative slope and looks as if it will hit the origin, but then at y=1 the line loops up as if it is trying to go around a circle. It comes back down at x=1 and turns back into a line. It is as if the circle opened up where it should have connected in the 3rd quadrant and extended to infinity in the 2nd and 4th quadrant.

.

This graph is similar to the first graph, but instead of a circle, 4 corners are beginning to assemble.

This graph looks more like a square than a circle but the edges of the "square" are rounded out

This graph is similar to the second graph but the loop that emerges at y=1 is not quite as rounded. This loop is more pointed but still pretty rounded off. It looks as if it is trying to go around a square.

MYPREDICTIONS:
• The graphs with even numbers as the exponents are similar.
• The graphs with odd numbers as the exponents are similar.
• As the even exponents grow larger. the graph goes from a circle to the form of a square.
• As the odd exponents grow larger, the loop in the line goes from being rounded to pointed.
• The curves with odd integers are asymptotic to the line x+y = 0.
• The higher the exponent the more pointed and exact the pointes become in the graph.
• Therefore I think that the graph of will be a very distinct square with area 4, center at the origin, and vertices at (-1,1), (1,1). (1, -1), and (-1,-1).
• I also think that the graph of will be a line with a negative slope, but at (-1, 1) the line will go horizontal until (1,1), then turn vertical through x=1 and turn back into a line with a negative slope at around (1, -1).
• The domain and range of the even exponent graphs is {-1<x<1}.
• The domain and range of the odd exponent graphs is {all Reals}.

My predictions held to be true, except the corners of the "square" in are not as pointed and exact as I had thought they would be. So, I examined the graph of x^100 +y^100 = 1 and was satisfied that the corners of this square are more exact points.

Conclusion:

x^even number + y^even number = 1 is a graph that goes from a circle to a square as the exponent increases in size. The domain is -1 to 1.

x^odd number + y^odd number = 1 is a grpah of a line with negative slope that loops out and becomes more pointed towards (1,1) as the exponent increases in size. The domain is all values of x.

I have always known the equation for a circle. ; where the squareroot of r is the radius, and (h,k) is the center. However, I did not know that if you increase the exponents of this equation you begin to turn it into a square. When I thought of an equation of a square I think about four seperate linear equations being pieced togethor at their intersections. This assignment has created a connection between circles and squares for me.

Further Explorations

What will happen if I change not only the exponents but the center?

It is a square- like shape with center at (5,2) .

Also, what will happen if I change the r^2.?

If I used the equation but then replace the 1 with 25, I would assume that like the equation of a circle this would make the square have a "radius" of 5, or side length of 10.

However, this does not occur. The intercepts are not exactly 1 and -1 anymore. They expanded just a little bit, but not what I was expecting. I was expecting the intercepts to be at 5 and -5. This equation then, does not behave just like the equation of a circle.

Another thing to examine is graphs where the exponent is a negative integer.

Look at

And look at

Now look at 2 positve and negative exponent graphs togethor: