Exploring Parametric Curves

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Objective: To show the various forms of a parametric equation.

A parametric equation is an equation defined using parameters such as x = f(t) and y = f(t). The most commonly used parameter is t. This exploration is going to focus on trigonometric functions and specifically sine and cosine functions.

We begin by graphing the equations x = cos (t) and y = sin (t) for 0 < t < 2pi

Figure 1: Graph of x = cos(t) and y = sin(t)

Notice that the graph of the equation has radius 1. Recall that the equation of a circle is

Therefore, the equation of our circle becomes

hence the radius being equal to 1

Next lets investigate the equations x = cos (at) and y = sin (bt) for 0 < t < 2pi

1st case: Let a = b. Notice that you still get a circle. Click here to investigate for different equal values

2nd case: Make a < b. Let a = 2b

Figure 2: Graph of cos (t) and sin (2t)

Is the graph always the same as figure 2 when the ratio is 1/2? Click here to investigate. How about if the ratio is 2/3? After investigating, you should notice that equations with the same ratio of a/b produces the same graph.

3rd case: Make a > b

Figure 3: Graph of cos (2t) and sin (t)

Is the graph always the same as figure 3 when the ratio is 2? Click here to investigate. How about if the ratio is 3? After investigating, you should notice that equations with the same ratio of a/b produces the same graph.

Now, graph the equation with a=3 and b=1 and then with a=1 and b=3

Notice that the graph is just a reflection across the x-axis. Does this work is a=2 and b=1? Click here to investigate. You should discover that this only works when the number not equal to 1 is an odd number.

Finally, lets investigate the equations x = a cos (t) and y = b sin (t) for 0 < t < 2pi

1st case: Let a = b = 3.

Notice that you still get a circle with a ratio of 3. Do you always get the same ratio as your value for a and b? Click here to investigate for different equal values

What happens when you make a=b=1 and then do the opposite? Do you still get the same reflection across the x-axis as previously stated? The answer is yes

Also notice that a always controls the radius while b controls the height. And finally, notice that when a is not equal to b, you get an elipse