parametric equations and curvestill to this point

Question

Parametric Equations and Curves

Till to this point we have looked almost completely at functions in the form y = f (x) or x = h (y) and approximately all of the formulas that we've developed needs that functions be in one of these two forms.  The complexity is that not all curves or equations that we'd like to come across at fall easily into this form.

Take, for instance, a circle. It is very easy to write down the equation of a circle centered at the origin with radius r.

x² + y² = r²

Though, we will never be capable to write the equation of a circle down as a single equation in either of the forms as illustrated above. Make sure that we can solve for x or y as the following two formulas show

y = + √ (r² - x²)

x = + √ (r² - y²)

But actually there are two functions in each of these. Each formula illustrates a portion of the circle.

y = √ (r² - x²)  (top)

x = √ (r² - y²) (right side)

y = - √ (r² - x²) (bottom)

x = - √ (r² - y²) (left side)

Unfortunately we generally are working on the whole circle, or just can't say that we're going to be working just only on one portion of it.  Although, if we can narrow things down to just only one of these portions the function is still frequently fairly unpleasant to work with.

There are as well a great several curves out there that we can't even write down as a single equation in terms of just only x and y.  Thus, to deal along with some of these problems we introduce parametric equations.

In place of defining y in terms of x (y= f (x)) or x in terms of y (x = h (y)) we describe both x and y in terms of a third variable known as a parameter as follows,

 x = f (t)

y = g (t)

This third variable is generally represented by t (as we did here) but doesn't have to be of course. Occasionally we will restrict the values of t that we'll make use of and at other times we won't. This will frequently be dependent on the problem and just what we are attempting to do.

Every value of t represents a point (x, y) = (f (t) , g (t)) that we can plot. The collection of points which we get by letting t be all possible values is the graph of the parametric equations and is termed as the parametric curve.

Sketching a parametric curve is not all time an easy thing to do.  Let us take a look at an instance to see one way of sketching a parametric curve. This instance will also demonstrate why this method is generally not the best.