Practically Perfect Parametric Curves

Assignment 10

By Amber Candela

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Parametric Curves

A parametric curve is a set of continuous functions of where the parameter of each equation is "t". The curve will depend on the values of "t". An example of a parameterize function is taking the function y =x^{2} and using the parameter "t" to create two new functions, x = t and y = t^{2}. The value of "t" can be any range. If you graph t from -10 < t < 10 you will get a parabola with a domain from -10 to 10. The range of the function will be from 0 to 100.

**Investigating Parametric Functions**

Trigonometric FunctionsGraph x = sin(t) and y = cos (t) from 0 ≤ t≤2π. What graph do you think you will get? If you conjectured the unit circle you would be correct. The graph below shows the parametric curve.

In order to further investigate the trigonometric functions we need to change parametric equation. Let us now look at x=cos(at) and y=sin(bt) for different values of "a" and "b". If a = 1 and b = 2 we get the following graph.

Now look at a =1 and b = 3.

If "a" were to stay equal to 1 and we kept increasing "b", what would happen? Look at the animation below to see what happens as 1 ≤ b ≤ 10.

After looking at the animation, use the Graphing Calculator file below to change the values of "b" where b is a whole number in the range 1 ≤ b ≤ 10.

Linear Parametric Curves

Other functions can be parameterized. Take a look at the linear parametric curve, x = a + t and y = b +kt. What happens as we change the values of t?

First lets look at "t" from -1 to 1 where a, b and k all equal 1.

Then look at "t" from -2 to 2 where a, b and k all equal 1.

What is happening to the line when "t" is doubled? The graph is doubled. When "t" is from -1 to 1, the domain of the graph is from 0 to 2. When "t" is from -2 to 2, the domain of the graph is from -1 to 3. As the range of "t" changes, what happens to the domain of the graph? Use the Graphing Calculator file below to change different values of "t" to see what happens to the domain as "t" changes.

**Now Change "k"**

What if the range of "t" values is set and the value of "k" changes? The following image is where "k" is in the set {-3,-2,-1,-,1,2,3} and t ranges from -2 to 2.

What colors do you think are the different values of k? Check below to see if you are correct!

What else can explore with parametric equations? Use Graphing Calculator to investigate!