10.6 Parametrics

10.6 Parametrics

Objective

• To evaluate sets of parametric equations for given values of the parameter.

• To sketch curves that are represented by sets of parametric equations

• To rewrite sets of parametric equations as single rectangular equations by eliminating the parameter.

• Suppose you were running around an elliptical shaped track.

• You might be following the elliptical path modeled by the equation

2 2

125 9

x y

• This equation only shows you where you are, it doesn’t show you when you are at a given point (x, y) on the track. To determine this time, we introduce a third variable t, called a parameter. We can write both x and y as functions of t to obtain parametric equations.

Definition of a Plane Curve

• If f and g are continuous functions of t on an interval I, the set of ordered pairs

• (f(t), g(t)) is a plane curve C. The equations x = f(t) and y = g(t)

are parameter equations for C, and t is the parameter.

A parameterization of a curve consists of the parametric equation and the interval of t-values.

Time is often the parameter in a problem situation, which is why we normally use t for the parameter

• Sometimes parametric equations are used by companies in their design plans. It is easier for the company to make larger and smaller objects efficiently by simply changing the parameter t.

Sketching a Plane Curve

• When sketching a curve represented by a pair of parametric equations, you still plot points in the xy-plane.

• Each set of coordinates (x, y) is determined from a value chosen for the parameter t.

Example 1:Sketch the curve given by

x = t + 2 and y = t2, – 3 t 3.

t – 3 – 2 – 1 0 1 2 3

x – 1 0 1 2 3 4 5

y 9 4 1 0 1 4 9 y

x-4 4

4

8

orientation of the curve

Graphing Utility: Sketch the curve given by x = t + 2 and y = t2, – 3 t 3.

Mode Menu:

Set to parametric mode.

Window Graph Table

Eliminating the parameter is a process for finding the rectangular equation (in x and y) of a curve represented by parametric equations.

x = t + 2 y = t2

Parametric equations

t = x – 2 Solve for t in one equation.

y = (x –2)2 Substitute into the second equation.

y = (x –2)2 Equation of a parabola with the vertex at (2, 0)

Solve for t in one equation.

Substitute into the second equation.

Example 2:2y t Identify the curve represented by x = 2t and

by eliminating the parameter.

2xt

22y x

y

x-4 4

4

8

22xy

Parametric equation for x.

Substitute into the original rectangular equation.

Example 3:Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = x.

x = t

y = 4t – 3

x

y

-4 4

4

-4

8 y = 4t – 3

Example

• Use the parameter t = 2 – x in the previous example.

Parametric Conics

• The use of two of the three Pythagorean Trigonometric Identities allow for easy parametric representation on ellipses, hyperbolas, and circles.

Pythagorean Identities

2 2cos sin 1 2 2sec tan 1

Circles

Compare the standard form of a circle with the 1st Pythagorean Identity

Standard form:

2 2 2( ) ( )x h y k r

Change the equation so that it equals one:

2 2

2 2

( ) ( )1

x h y k

r r

Pythagorean Identity

2 2cos sin 1t t

Using simple substitutions:2

22

22

2

( )cos

( )sin

x ht

r

y kt

r

Solving for x and y

cos

sin

x r t h

y r t k

Example 4

• Graph2 2( 4) ( 1) 16x y

• Set calculator: Mode: Parametric

Window

[0, 2 ], ,[ 15,15],1,[ 10,10],136

Examples 5

• Graph

2 2( 3) ( 5) 9x y

Example 7

• Sketch the curve represented by

• Eliminating the parameter.

cos and 2sin , 0 2x y

Example 8

• The motion of a projectile at time t (in seconds) is given by the parametric equations:

• Where x(t) gives the horizontal position of the projectile in feet and y(t) gives the vertical position of the projectile in feet.

2

( ) 25

( ) 16 30 10

x t t

y t t t

a. Find the vertical and horizontal position of the projectile when t = 2

• x = 50, y = 6

b. At what time will the projectile hit the ground?

The ball will hit the ground between t = 2.16 and t = 2.18. Notice y goes from positive to negative.

Example

• The parametric equations below represent the hawk and dove populations at time t, where t is measured in years.

( ) 10cos 202

( ) 100sin 1502

th t

td t

a. Use your calculator in function mode to graph the hawk and dove

populations over time.

Dove

Hawk

b. Find the maximum and minimum values for each population.

• Hawk minimum 10 maximum 30

• Dove minimum 50 maximum 250

c. Now using Parametric mode on your calculator, graph the hawk

population versus the dove

As the hawk population increases, the dove populations decreases, followed by a decrease in hawk population and a decrease in the dove population.

d. Using the parametric graph, find the population of hawks and doves

after one year.

• Dove population is 250, hawk population is 20

e. When will the population of hawks reach its maximum value and

what is that value?

Hawk population will be 30 at year 2.

Example 9The complete graph of the parametric equations x =

2cos t and y = 2 sin t is the circle of radius 2 centered at the origin. Find an interval of values for t so that the

graph is the given portion of the circle.

• A) the portion in the first quadrant. (0, π/2)

• B) the portion above the x-axis. (0, π)

• C) the portion to the left of the y-axis – (π/2, 3π/2)

Example 10 Ron is on a Ferris wheel of radius 35 ft that turns

councterclockwise at the rate of one revolution every 12 seconds. The lowest point of the Ferris sheel is 15 feet above ground level at the point, (0, 15) on a

rectangular coordinate system. Find parametric equations for the position of Ron as a function of

time t in seconds if the Ferris wheel starts with Ron at the point (35, 50)

Example 11 Al and Betty are on a Ferris wheel. The wheel has a radius of 15 feet and its center is 20 feet above the ground. How high

are Al and Betty ath the 3 o’clock position? At the 12 o’clock position? At the 9 o’clock position?

Example 12A dart is thrown upward with an initial velocity of 58 ft/sec at

an angle of elevation of 41°. Find the parametric equations that model the problem situation. Whne will the dart hit the ground? Find the maximum height of the dart. When will

this occur?

The dart will hit the ground at about 2.51 seconds. The maximum height of the dart is 26.6 feet. This will occur at 1.22 seconds.