
Michelle E. Chung 
*
EMAT6680 Assignment 1: Graphing Functions and Relations 

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
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 yintercept 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.

Conclusion

 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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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.

Conclusion

 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
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.

The graph of

 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.

Back to the Top 
* The graph of
Guess on 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.

The graph of

 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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