Write-up 10 deals with parametric curves. A parametric curve in the plane is a pair of functions x=f(t) and y= g(t) where two continuous functions defined order pairs (x,y).

**Problem**: Investigate each of
the following for zero less than or equal to t and t is less than or equal
to two pi. Describe each when a=b, a<b, and a>b. The parametric equations
we are looking at are the following.

First I looked at the graphs of degree one. Which is the first two equations above.

I. The following graph shows when a=b, we will always get a circle with the radius of a=b. When -a=-b we get the absolute value and therefore, get the same graph.

II. When a>b, we get a graph of an ellipse with the major axis being the x-axis such that the length is 2a and the minor axis is the y-axis what is equal to b.

III. When a<b, we get a graph of an ellipse, but this time the major axis is the y-axis which is 2b. And the minor axis is the x-axis which is the length of a.

The next set of graphs that I looked at were of degree 2.

I. When a=b, we get a graph of a line segment with the endpoints being a=b. This holds true even when -a=-b. The only difference is the quadrant that you are in. When a=b, you are in the first quadrant and when -a=-b, you are in the third quadrant.

Conclusion: Therefore, whether a>b or a<b, you will always get a line segment with the endpoints being a and b. The only difference is what quadrant you are in.

The last set of parametric equations that I looked were of degree three.

I. When a=b, we get an "inverted" ellipse with the length equal to a and b.

II. When a>b, it stretches out the x-axis of the length of a.

III. When a<b, it stretches out the y-axis the length of b.

**Generalization:**

I. **Even Powers**:

For even powers, your graph will be a line segment with a and b as endpoints, but as the power increases for even powers the graph still has endpoints of a and b. The only difference is that as the even power increases in multiples of two, the graph appears to become asymptotic to the x and y axis.

II. **Odd Powers**:

For odd powers, the endpoints will still be the value of a and b. The graph of odd powers will be our picture of an "inverted" ellipse with it becoming more asymptotic to the x and y axis.

If I was doing this with my students the following is what I would do.

**Step I**: Graph the equations
x=a(cos(t)) and y=b(sin(t)). For your first exploration
let a and b equal each other.

a.Use a set of negative numbers for a and b.

b. Use a set of positive numbers for a and b.

What observation can you make? How would you describe the shape of the parametric equations to someone who does not know what a parametric equation is?

**Step II**: Graph the equations
x=a(cos(t)) and y=b(sin(t)). Now let graph the two parametric equations,
but this time let a>b.

a. negative values?

b. positive values?

What observations can you make about these graphs. What are some observations you can make from Step I and II?

**Step III**: Graph the equations
x=a(cos(t)) and y=b(sin(t)). Now let a<b.

a. negative values?

b. positive values?

What observations can you make about these set of graphs? Are they the same, different, or not related at all to the graphs in Steps I and II?

**Step IV**: Follow the same step
as you did for Steps I, II, and III. Now use x=a(cos(t))^2 and y=b(cos(t))^2.
Answer the same questions as above.

**Step V**: Follow the same steps
as in Steps I, II, and III. This time use the graphs x=a(cos(t))^3 and y=b(sin(t))^3.
Answer the same questions as above.

**Step VI**: What are some generalizations,
if any, you can make about increasing the degree without graphing?

**Step VII**: Write about one other
degree for a set of parametric equations. Please have pictures to support
your findings.

**Step VIII**: Please prepare writ-up
that contains all of your work and explanations of what you discovered with
working with parametric equations.