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