*by Soo Jin
Lee*

2. For various **a **and **b**, investigate
(t can be any degree between 0 to 360)

Before we do this investigation, let us first see the definition of the parametric equation.

*A parametric curve in the
plane is a pair of functions*

*where the two continuous
functions define ordered pairs (x,y). The two equations
are usually called the parametric equations of a curve. The extent
of the curve will depend on the range of t and your work
with parametric equations should pay close attention the range
of t . In many applications, we think of x and y
"varying with time t " or the angle of rotation
that some line makes from an initial location.*

let's see what happens when the value of a and b are the same.

assume that a=b=1

then the graph will be the circle.

,

as we can see from the above picture, when a=b, the graph will always turn out to be the circle.

let's see when 'b' is fixed as 1 and the value of 'a'moves.

some interesting things are found from this graph!

First, as the value of 'a' increases, the number of curves also increase.

Secondly when 'a' is odd, the graph always passes through (0,1) & (0,-1)

and all of the graphs pass the point (1,0)

Let's see what happens when we fix the value of 'a' as 1, and changes the value of 'b'

we have more interesting graphs this time,

we can see the curves increase as the value of 'b' increase; the number of oval shape is same as the value of 'b'. In assition, I found that when the graph always pass the point (0,1),(0,-1) when b=odd number. When b= even number, the graph always passes the origin.

To see more differences between the graph when 'b' is odd number and 'b' is even number, see the below picture; you will see the distincetion between two graphs easier.

when b=odd number

(picture 1)

when the value of 'b' is even nunber,

(picture2)

futher investigation gave me another interesting point.

All the graphs are similar to the sine graphs when I see the picture 1

and all the graph are similar to the cosine graphs when I see the picture 2.