Parametric Equations Dr. Dillon Calculus II Spring 2000

Introduction Some curves in the plane can be described as functions.

Others... cannot be described as functions.

Ways to Describe a Curve in the Plane An equation in two variables This equation describes a circle. Example:

Parametric Equations Example: The “parameter’’ is t. It does not appear in the graph of the curve!

Why? The x coordinates of points on the curve are given by a function. The y coordinates of points on the curve are given by a function.

Two Functions, One Curve? Yes. then in the xy-plane the curve looks like this, for values of t from 0 to If

Why use parametric equations? Use them to describe curves in the plane when one function won’t do. Use them to describe paths.

Paths? A path is a curve, together with a journey traced along the curve.

Huh? When we write we might think of x as the x-coordinate of the position on the path at time t and y as the y-coordinate of the position on the path at time t.

From that point of view... The path described by is a particular route along the curve.

As t increases from 0, x first decreases, Path moves left! then increases. Path moves right!

More Paths To designate one route around the unit circle use

counterclockwise from (1,0). That Takes Us...

Where do you get that? Think of t as an angle. If it starts at zero, and increases to 360 degrees then the path starts at t=0, where

To start at (0,1)... Use

That Gives Us...

How Do You Find The Path Plot points for various values of t, being careful to notice what range of values t should assume Eliminate the parameter and find one equation relating x and y Use the TI82/83 in parametric mode

Plotting Points Note the direction the path takes Use calculus to find –maximum points –minimum points –points where the path changes direction Example: Consider the curve given by

Consider The parameter t ranges from -5 to 5 so the first point on the path is (26, -10) and the last point on the path is (26, 10) x decreases on the t interval (-5,0) and increases on the t interval (0,5). (How can we tell that?) y is increasing on the entire t interval (-5,5). (How can we tell that?)

Note Further x has a minimum when t=0 so the point farthest to the left on the path is (1,0). x is maximal at the endpoints of the interval [-5,5], so the points on the path farthest to the right are the starting and ending points, (26, -10) and (26,10). The lowest point on the path is (26,-10) and the highest point is (26,10).

Eliminate the Parameter Still use Solve one of the equations for t Here we get t=y/2 Substitute into the other equation Here we get

Some Questions What could you do in the last example to reverse the direction of the path? What could you do to restrict or to enlarge the path in the last example? How can you cook up parametric equations that will describe a path along a given curve? (See the cycloid on the Web.)cycloid

Web Resources MathView Notebook on your instructor’s site (Use Internet Explorer to avoid glitches!)MathView Notebook IES Web

Summary Use parametric equations for a curve not given by a function. Use parametric equations to describe paths. Each coordinate requires one function. The parameter may be time, angle, or something else altogether...