In this assignment we will explore parametric curves in the form:

Here we will change the values of a and b to where a = b, a > b and a <b and we will also change the exponent of sin and cos. Lets begin a graph of the equation above where a = b:

This is the statndard equation of a circle. Yet if a > b, then the graph becomes:

Here you have constructed an ellipse where a = 2 and b = 1. This makes sense since the ellipse is constructed 1 unit along the y-axis and 2 units along the x-axis. Therefore I will make the assumption that when a < b the ellipse will be b units in hieght and a units in length, hence the ellipse above will rotate 90 degrees when a = 1 and b = 2.

Lests explore the effects of changing the signs of a and b. Therefore we would have

So we can assume that there is no effect on this function by changing the signs of a and b. Now lets examine the equation form

from the interval [ 0, 2pi] for the three seperate instances a =b, a < b and a > b. Here when a = b then the graph looks like this:

This graph looks like a linear equation, more specifically a line segment given the interval for t, so lets explore if a < b, specifically a = 1 and b = 2.

The shape of the graph does not change but the vaue of the y-intercept increase to 2. Therefore I conjucture that if for whatever the values of a and b, the x and y intercepts will change accordingly. Next I will show a = 1 and b = 3, a = 2 and b = 1, and finally a = 3 and b = 1.

This is very interesting as when the function is to the first power we have an ellipse or circle in a specific instance and when the power is 2 then it becomes a segment in the first quadrant with x and y intercepts a and b respectivley. now lets look when the a and b values are negative, hence we have:

So we can see that the signs of a and b determine what quadrant the segment is in given that each quadrant has specific signs that make up every point in that quadrant for x and y.

Therefore we will examine the next function

As I did before, I will show the graphs of this function where a = b, a < b and a > b on the interval [ 0, 2pi]. So when a = b we have the graph

Notice that the new shape of the function is star or four curves, one in each quadrant, extending from some x value to some y value. Again the x and y intercepts depend on the values of a and b respectively. Therefore when a < b the hieght is greater than the length and when a > b then the length is greater than the hieght. It seems to me that the thrid power function is some combination of the first two graphs. This graph is in all four quadrants and is constructed of curves liek the ellipse but it also extends from the x intercept to the y intercept like the segment graph. Now when the values of x and y are negative we have the same graph as above:

Therefore the signs of a and b have no effect on this function in the range. Lets explore the next power of the equation as we have above:

As we can see that this graph looks very similar to the function to the second power excpet that the segment becomes a curve fromx axis to y axis.

This function also takes on some of the characteristics as the second power because the signs of a and b determine what quadrant the curve is located.

Another similar characteristic is that the x-intercept is the a value and the y-intercept is the b value, hence this is why the signs change the location of the curve and why the intercepts are the same as the values. Now lets explore the function a last time where the power is the fifth for all the same conditions as above.

This graph is the same as the function to the third power except that the curvesare steeper than in the previous problem.

Just like in the thrid power the signs of a and b do not matter in determining the location of the graph because it is already located in all four quadrants, not just one quadrant like the even power functions.

This is an interesting function as the graphs are different for even and odd powers of the same function. Even functions will always be located in one quadrant, depend on the signs of a and b for location and intercepts. Odd functions will be located in all four quadrants and depend on the a and b values for intercepts only.