Exploring Exponents

By

Princess Browne

Graph        x2 + y2 = 1

x3 + y3 = 1

x4 + y4 = 1

x5 + y5 = 1

What do you expect for the graph of

X24 + y24 = 1

Or

X25 + y25 = 1

I will start by graphing the first four equations above to find a pattern. From the given equations we will notice that the equations are always equal to 1 and the exponents of x and y are equal in each equation. These factors may be useful when we compare the different equations.

Graph 1

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Graph 2

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Graph 3

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Graph 4

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Graph number 1 represent the equation x2 + y2 = 1. As we can see this graph forms a circle. Graph number 2 is the graph of x2 + y2 = 1 and x4 + y4 = 1. As we compare the two graphs we will see that the even number exponents create a shape that starts of as a circle and as the exponents increases the corners of the circle are extended out into a shape of a square. On the other hand, graph number 3 represent the equation  x3 + y3 = 1. This graph forms a line with a curve in the middle. Graph number 4 is the graph of  x3 + y3 = 1 and x5 + y5 = 1. When we compare the two graphs we will notice that as the odd exponent increases the curve in the middle of the graph extends to form a square like shape. From the following equations and graphs I will guess that the equations with the even exponents starts of as a circle and as the exponents increases the graph will end up looking like a square shape.

Next, we will look at xn+yn=1. Let n equal all even numbers, starting with 2 and increasing by 2.  We will graph n=2, n=4, n=6, and n=8. If we compare graph number 5 with our previous graphs we will see that our hypothesis is true.

Graph 5

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Graph number 5 confirms our prediction that  when n is an even exponent the graph will  form a circle and as n increases the corners of the circle are extended to form a square shape. Next we will look at the equation when the exponents are odd. If we look at xo+yo=1, where o stands for all odd numbers starting with 3 and increasing by 2. We will graph the functions of o=3, o=5, o=7, and o=9. Using the observation from our previous graphs, we will hypothesize that the equations with odd exponents will start of as a line with a curve or hump in the middle.  As the exponent increases the curve would extend to from a square shape.

Graph 6

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Graph number 6 shows that as the odd exponent increases the middle of the line changes from a curve shape to a square shape. If we compare the odd exponents graphs with the even exponents graphs we can conclude that as the exponents increases both graphs ends up with square like corners.

Now we will look at x24+y24 = 1 and x25+y25 = 1 to see if the following conclusions are true. Graph number 7 shows the graph of x24+y24 = 1.

Graph 7

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This graph shows that our hypothesis is correct and that as the exponent increases the corner of the graph extends to from the shape of square. Next, we will look at the graph of x25+y25=1 to also prove our hypothesis about the odd exponents.

Graph 8

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Graph number 8 also shows that our assumption is correct. As the odd exponent increases, the portion of the graph that is located in quadrant I forms a square like shape.

Lets try x100 + y100  = 1 and x101 + y101  = 1

x100 + y100  = 1

Graph looks like a square!

x101 + y101  = 1

Middle looks like a square also!