Assignment 10:  Parametric Equations

By:

Jonathan Sabo

Investigate each of the following for:
.

Descirbe each when a=b, a < b, and a > b.

First we will look at these parametric equations.

In the first graph we will look at when a = b.  In this problem we will set both a and b equal to 1.  Using Graphing calculator 3.5 we will graph the given parametric equation.

Observe, when we have a = b the graph will create a circle.

For the second graph we will look at when a < b.  In this problem we will set a equal to 1 and b equal to 3.

Observe, our graph is no longer a circle.  It has been stretched vertically.  We can see that our x intercepts are exactly the same they were at (-1,0) and (1,0) in the previous graph and they are in the same place now.  Our y-intercepts however have been changed, they are now at (0,3) and (0,-3).

For the third graph we will look at when a > b.  In this problem we will set a equal to 3 and b equal to 1.

Observe that this also creates a graph that is not a circle.  This graph has been stretched horizontally.  We can see that our y-intercepts are exactly the same they were at (0,-1) and (0,1) in the original graph and they are in the same place now.  Our x-intercepts however have been changed, they are now at (-3,0) and (3,0).

Now we will look at the parametric equations:

In the first graph we will look at when a = b.  In this problem we will set both a and b equal to 1.  Using Graphing calculator 3.5 we will graph the given parametric equation.

Similar to our first parametric equation we set both a and b equal to 1.  We have a x-intercept at (1,0) and a y-intercept at (0-1).  The graph results in a straight line that is connecting these 2 intercepts.  We do not have any negative intercepts because the cos and sin equations are both  being squared.

For the second graph we will look at when a < b.  In this problem we will set a equal to 1 and b equal to 3.

Observe, our graph is now a longer line.  It has been stretched vertically.  We can see that our x intercept is exactly the same it was at (1,0) in the previous graph and it is in the same place now.  Our y-intercept however has been changed, it is now at (0,3).

For the third graph we will look at when a > b.  In this problem we will set a equal to 3 and b equal to 1.

Observe that this also creates a graph that is a longer line.  This graph has been stretched horizontally.  We can see that our y-intercept is exactly the same as it was at (0,1) in the original graph and it is in the same place now.  Our x-intercept however has been changed, it is now at (3,0)

Now we will look at the parametric equations:

In the first graph we will look at when a = b.  In this problem we will set both a and b equal to 1.  Using Graphing calculator 3.5 we will graph the given parametric equation.

In this graph we have again have both positive and negative values for x and y.  Now we have x-intercepts at (-1,0) and (1,0) and y-intercepts at (0,-1) and (0,1).  After these observations we can see that when we have an odd exponent we have graph that has both negative and positive values for x and y.  When we have an even exponent we would only have either positive exponents or negative exponents.

For the second graph we will look at when a < b.  In this problem we will set a equal to 1 and b equal to 3.

Observe, our graph has been stretched vertically.  We can see that our x intercepts are exactly the same they were at (-1,0) and (1,0) in the previous graph and they are in the same place now.  Our y-intercepts however have been changed, they are now at (0,3) and (0,-3).

For the third graph we will look at when a > b.  In this problem we will set a equal to 3 and b equal to 1.

Observe that this creates a graph that has been stretched horizontally.  We can see that our y-intercepts are exactly the same they were at (0,-1) and (0,1) in the original graph and they are in the same place now.  Our x-intercepts however have been changed, they are now at (-3,0) and (3,0).