Parametric equations are pairs of equations, usually in the form of Òx=Ó and Òy=Ó. The two equations together define the points of a graph in terms of a third parameter, usually ÒtÓ.

LetÕs see what happens to the graphs when various coefficients are changed are a basic pair of equations. In the graph below the purple circle has coefficient Ô1Õ. Changing it to Ô2Õ increases the size of the circle.

Graph 1

This graph gets a little more interesting. The purple and red circles are created when the constants of the cos and sin expressions are equal. The ellipses are created when they are unequal. When the constant for the sin expression is larger, the ellipse is longer along the y-axis. When the constant for cos is larger, the ellipse is longer along the x-axis.

Graph 2

LetÕs stop a moment and think about what we are seeing. We are looking at graphs that represent the values that ÔsolveÕ the two equations. LetÕs look at a cosine curveÉ.

(purple) (red)

Graph 3

and a sine curveÉ..

(purple) (red)

Graph 4

We see curves that continually repeat along the x and y axes (in the example IÕve chosen). The extreme values of the curves never exceed 1 on the scale.

*Review:*

*The sine and cosine
functions are periodic*

Hey, wait a minute! We saw that happen in Graphs 1 and 2. When we multiplied sine and cosine by a number the circles and ellipses got bigger. We increased their amplitude. Notice another similarity. While the sine and cosine graphs are bounded by 1 (in our example) , the parametric equations are bounded also. We called that amplitude in our review. Our parametric equations are bounded all the way around, like they are contained in a rectangle.

Now that weÕve refreshed our memories on sine and cosine, letÕs look at some more parametric equations and see what else we can learn.

Notice in Graph 5 that although the value of ÔbÕ is 2, the graph is contained with a Ô1 by 1Õ area. (We didnÕt change the amplitude.) There are two peaks on each side of the x-axis. Is that significant?

Now b=3 and the graph has changed so that there are 3 peaks above and below the x-axis.

Four peaks and b=4É

b=4

Six peaks and b=6É

If we increase the values of a and let b=1, hereÕs what we getÉ.

Very similar to b> a . but oriented around the y-axis.

So what happens when a and b are greater than 1 and a not equal to b?

Wow! Change around 2 numbers and things changed completely. But look below where I change two numbers around and the graphs are very similarÉ.

WeÕve really seen the power of numbers here. Depending where in an expression you make a change, you can increase the size, orientation, and the appearance of a graph, sometimes in unexpected ways. So letÕs do just a little more investigation and see what happens when we raise the cosine and sine functions to powers greater than 1.

If we look at this parametric pair where a=b and their value ranges from 1 to 2.5, we get thisÉ.

If we set a and b less than 0É..

LetÕs see what happens when we increase to a power of 3É.

ThatÕs a big change from the line segments in the previous two graphs.

If we increase to a power of 4 and a and b both positive and negative É.

To a power of 5É.

If we look back at this lesson we can see a wide range of curves that are all produced by cosine and sine expressions both singly and in parametric pairs. Singly, the curves are continuous, in pairs they are bounded.