Parametric Equations

Introduction

Some curves in the plane can be described as functions.

)(xfy

Others...

cannot be described as functions.

Ways to Describe a Curve in the Plane

An equation in two variables

This equation describes a

circle.

086222 yxyxExample:

A Polar Equation

r

This polar equation describes a double spiral.

We’ll study polar curves later.

Parametric Equations

Example:

1

22

ty

ttx

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.

ttx 22 The y coordinates of points on the

curve are given by a function.

1ty

Two Functions, One Curve?

Yes.

then in the xy-plane the curve looks like this, for values of t from 0 to 10...

1 and 22 tyttxIf

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

1

22

ty

ttx

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

1

22

ty

ttx

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

)sin(

)cos(

ty

tx

counterclockwise from (1,0).

That Takes Us...

Where do you get that?

Think of t as an angle.

2 ,If it starts at zero, and increases to

then the path starts at t=0, where

cos(0) 1, and sin(0) 0.x y

To start at (0,1)...

Use

)cos(

)sin(

ty

tx

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

2 1, 2 , 5 5x t y t t

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?)

2 1, 2 , 5 5x t y t t

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).

2 1, 2 , 5 5x t y t t

Eliminate the Parameter

2 2( / 2) 1 or ( / 4) 1x y x y

2 1, 2 , 5 5x t y t t Still use

Solve one of the equations for t

Here we get t=y/2

Substitute into the other equation

Here we get

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...