Assignment 10

By: Olamide Alli

Exploration: Look at the parametric equations of the form:

Questions that need answers during this exploration:

- What types of graphs will the equations produce?
- What will changing the values of the variables a and b do the graph?

Let’s start with the latter question:

From the graph we can deduce that when a > b our value is stretched horizontally. When b > a, our graph is stretched vertically, which better known as an ellipse. Also notice that as the difference of the variables a and b increase, the graph produced from the equation, the ellipse becomes elongated.

Parametric equations provide accurate graphs of ellipses as well as shows us the boundaries of the ellipses based on the values for a and b. For Example if a = 1 and b = 2, where a corresponds to the x-axis and b corresponds to the y-axis. The width of the ellipse on the x-axis would be from -1 to 1 and the length would be from -2 to 2.

One last exploration:

Let’s examine one of the previously constructed graphs again but let’s add a twist.

The graph of:

And let’s change b = 4 to b = -4

The graph appears to be identical but I am willing to state that it is impossible for both graphs to be identical. Let’s try changing the parameter of t.

So let 0 < t < pi

and

Here is the “a-ha!!” moment. Now we can see that changing the value of b does indeed affect the graph. Change the value of b to a negative number changes the trace of the graph.