Michelle E. Chung

* EMAT6680 Assignment 1: Graphing Functions and Relations

 When a is even The graph of The graph of Movie

6. Graph

.

What do you expect for the graph of   or   ?

Before thinking about the graphs of ,

let's seperate these into two different cases, which is when n is an even number and when n is an odd number.

 The graphs of Conclusion when a is ODD When a=1: The graph is a straight line because x+y=1 is same as y= -x+1. The graph of x+y=1 is a straight line that its slope is -1 and its y-intercept is 1.   When a=3: The graph is NOT a straight line. It looks like a top of a hat, and the coordinates of the top of the hat is (0.793701, 0.793701). Also, it is symmetric to y=x.   When a=5: The graph is NOT a straight line, either. It looks like a top of a hat, too, but the coordinates of the top of the hat is (0.870551, 0.870551). It is symmetric to y=x, too.   When a=7: The graph is NOT a straight line, either. It looks like a top of a hat, too, but the coordinates of the top of the hat is (0.905724, 0.905724). It is symmetric to y=x, too.   When a=9: The graph is NOT a straight line, either. It looks like a top of a hat, too, but the coordinates of the top of the hat is (0.925875, 0.925875). It is symmetric to y=x, too. As you see, the graph of when a is odd is a straight line with a curve part, which looks like a top of a hat, except when a=1(it is a straight line). It is symmetric to y=x. The curve part is getting pointier when the value of a is greater, and we actually can notice that the coordinates of the top of it is getting greater.

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 Conclusion The graphs of when n is EVEN When a=2: The graph is a circle that its center is (0,0) and its radius is 1 because it is a equation of a circle. It is also symmetric to y=x because (0.707107, 0.707107) is on the circle.   When a=4: The graph is NOT a circle anymore. It looks like a square with some curve corners, or it looks like a flatted circle that has (0,0) as its center and radius 1, too, but one of the coordinates of the corner is (0.840896, 0.840896). It is also symmetric to y=x, too.   When a=6: The graph is NOT a circle anymore, either. It looks like a square with some curve corners, or it looks like a flatted circle that has (0,0) as its center and radius 1, too, but one of the coordinates of the corner is (0.890899, 0.890899). It is also symmetric to y=x, too.   When a=8: The graph is NOT a circle anymore, either. It looks like a square with some curve corners, or it looks like a flatted circle that has (0,0) as its center and radius 1, but one of the coordinates of the corner is (0.917004, 0.917004). It is also symmetric to y=x, too.   When a=10: The graph is NOT a circle anymore, either. It looks like a square with some curve corners, or it looks like a flatted circle that has (0,0) as its center and radius 1, too, but one of the coordinates of the corner is (0.933033, 0.933033). It is also symmetric to y=x, too. As you see, the graph of when a is even is a flatted circle that has (0,0) as its center and radius 1, except when a=2(it is a circle). It is symmetric to y=x. The corners of the flatted circle are getting pointier when the value of a is greater, and we actually can notice that the coordinates of the top of it are getting greater.

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* The graph of

 The graph of Guess on the graph of Because the graph of when a is even is a flatted circle, the graph of would be also a flatted circle with (0,0) as its center and radius 1; however, its corners would be pointer than others we saw previously because its value of a is greater than them. Also, it would be symmetric to y=x. As you see, it IS a flatted circle with (0,0) as its center and radius 1. Its corners are pointer than others, and one of the coordinates is (0.971532, 0.971532). It is also symmetric to y=x.

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* The graph of

 Guess on the graph of The graph of Because the graph of when a is odd is a straight line with a curve part, which looks like a top of a hat, the graph of would be also a straight line with a curve part, which looks like a top of a hat; however, the top of the hat would be pointer than others we saw previously because its value of a is greater than them. Also, it would be symmetric to y=x. As you see, it IS a straight line with a curve part, which looks like a top of a hat. The top of the hat is pointer than others, and the coordinates of it are (0.972655, 0.972655). It is also symmetric to y=x.

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