Assignment 10 :: Parametric Equations

 

By Jamie K. York

 

 

Let's explore the parametric equations below for 0 ² t ³ 2¹.

 

x = a cos(t)

y = b sin(t)

 

As we saw with the quadratic equations, modifications of the variables resulting in various patterns in the resulting graphs. We will start with a value of 1 for both a and b. See the resulting graph below.

 

 

 

Note that this configuration results in a circle. To explore additional configurations where a = b, review the animated graph below.

 

 

 

What happens when these variables are not equal? Let's investigate further to see.

 

 

 

For each graph, one variable has been set constant at a value of 1. As the additional variable changes between 2 and 4, we notice that these modifications form the shape called an ellipse. This differs from the circle resulting from a = b. However, with either confirguration, it is clear that the values of a and b stretch or shrink the figure.

 

Note that for a t range of 0 ² t ³ 2¹, the values of +a and -a result in the same graph. The same applies to the b variable.

 

 

 

 

 

 

 

 

Does the result change with a modification to the range of t? Explore the same graphs below with various t values to find out.

 

  0 ² t ³ ¹

  ¹ ² t ³ 2¹

 

 

 

 

 

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