http://jwilson.coe.uga.edu/EMAT6680Fa11/Antillanca/EMAT6680.gif

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

ab.gif

ab.gif

ab.gif

ab.gif

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

a2b1.gif

a10b5.gif

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

a1000b500.gif

a300b150.gif

a1000b500.gif

 

 

 and

 and

a3b1.gif

a15b5.gif

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

 and

 and

 and

a90b30.gif

a300b100.gif

a900b300.gif

 

 

 and

 and

a5b3.gif

a10b6.gif

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

 and

 and

 and

a75b45.gif

a250b150.gif

a500b300.gif

 

 

Third Investigation

When a<b

 

 and

 and

a1b2.gif

a5b10.gif

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

 and

 and

 and

a50b100.gif

a150b300.gif

a500b1000.gif

 

 

 and

 and

a1b3.gif

a5b15.gif

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

 and

 and

 and

a30b90.gif

a100b300.gif

a300b900.gif

 

 

 and

 and

a3b5.gif

a6b10.gif

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

 and

 and

 and

a90b150.gif

a120b200.gif

a450b750.gif

 

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

 

net2.gif

 

The next graph is a zoom of the curve

 

zoom_net2.gif

 

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

zoom1.gif

zoom2.gif

zoom3.gif

zoom4.gif

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

a1000b500.gif

a300b150.gif

a1000b500.gif

 

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

zoom3.1.gif

zoom3.2.gif

zoom3.3.gif

6 lines

18 lines

24 lines

 

Now, observe the next set of images.

 

 and

 and

 and

 and

zoom4.1.gif

zoom4.2.gif

zoom4.3.gif

zoom4.5.gif

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

 

 

 

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