Rayen Antillanca.
Assignment 10
For various a and b, investigate 

For 
To investigate, I am going to
consider the next parametric equation 
First
Investigation
a=b 









As you can see, the equation gives a
circle of radius 1 
Consider the system (1) Taking squares of both equations in (1),
we obtain (2) Adding both equations in (2) we get ,
which is reduced to the unitary circle’s Cartesian equation 
Second
Investigation
When
a>b
and 
and 




When
,
this graph looks like a part of a parabola. To prove this consider
the system (1) We might proceed as follow: we use the
double angle formula for the first equation in (1), and we square the second
equation in (1) to get (2) Adding the equations in (2), we obtain (3) The latter represents a parabola, so that
the parametric system (1) is a part of the parabola (3). 

Now, what happens
when a and b increase? 

and 
and 
and 





and 
and 




When the curve has the same shape. But when the
value of b increases the curve looks thicker 

and 
and 
and 





and 
and 




When they have the same shape. But when the value
of b increases the curve looks thicker 

and 
and 
and 





Third
Investigation
When
a<b
and 
and 




When they have the same shape. But when the value
of b increases the curve looks thicker 

and 
and 
and 






and 
and 




When the curve has the same shape. But when the
value of b increases the curve looks thicker 

and 
and 
and 





and 
and 




When
the curve has the same
shape. But when the value of b increases the curve look like
thicker 

and 
and 
and 





Fourth
Investigation
As
you can see in the parametric equation
When
a and b are large, the curve looks thicker. However,
if you zoom in a part of the curve, you can see a net. For example the next
parametric equation
The graph of 

The next graph is a zoom of the curve 

This
net is related to the value of a and b. The next set
of figure show how the net is made. The relationship between a and b is .
All these figure are zoom in.
and 
and 
and 
and 




1 line 
2 lines 
3 lines 
4 lines 
So,
the parametric equation has 30 lines as .
There are as many lines as the value of a.
However,
when or the parametric equations have a different
shape, for example
When
,
the graph has the same shape. But when the value of b increases the curve is
thicker 

and 
and 
and 



This
graph forms a net, but for some value of and this net looks like losing lines. I am going
to explore to find at which value the net reduces its number of lines. The net
behaves similar to what was shown in the previous investigation, the number of
lines is given by the value of a. The the next set of images shows this.
and 
and 
and 



6 lines 
18 lines 
24 lines 
Now,
observe the next set of images.
and 
and 
and 
and 




This investigation suggests that when
a=600 and b=300 the graph of these parametric equations reaches its maximum
number of lines. Then, when a increases the graph looks thinner 
What
happens when a=300 and b=600?
Summary 

Considering
the next parametric equation When
,
the graph is a circle with radius 1 When
a>b or a<b the investigation suggests six different graph shapes. The
six graph shapes are given for the follow relationship.
I
would like to add that there exist more than these six graph shapes, actually
I think there are a lots of them. When
the value of and increases, the graph of the parametric equation
forms a net which has as many line as the value of a. When
and ,
the net reaches its maximum number of lines. If a and
b increase more, then the number of lines begins to decrease again, therefore
the graph looks thinner. 
Note: All graphs of this webpage were made with
Graphing Calculator 4.0 