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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

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Page 1: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

10.6 Parametrics

Page 2: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.

Page 3: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

• 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

Page 4: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

• 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.

Page 5: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.

Page 6: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 7: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

• 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.

Page 8: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.

Page 9: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 10: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 11: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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)

Page 12: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 13: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 14: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Example

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

Page 15: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Parametric Conics

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

Page 16: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Pythagorean Identities

2 2cos sin 1 2 2sec tan 1

Page 17: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Circles

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

Standard form:

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

Page 18: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Change the equation so that it equals one:

2 2

2 2

( ) ( )1

x h y k

r r

Page 19: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Pythagorean Identity

2 2cos sin 1t t

Page 20: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Using simple substitutions:2

22

22

2

( )cos

( )sin

x ht

r

y kt

r

Page 21: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Solving for x and y

cos

sin

x r t h

y r t k

Page 22: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Example 4

• Graph2 2( 4) ( 1) 16x y

Page 23: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

• Set calculator: Mode: Parametric

Page 24: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Window

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

Page 25: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 26: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 27: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 28: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Examples 5

• Graph

2 2( 3) ( 5) 9x y

Page 29: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 30: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 31: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

Example 7

• Sketch the curve represented by

• Eliminating the parameter.

cos and 2sin , 0 2x y

Page 32: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 33: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

• x = 50, y = 6

Page 34: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.

Page 35: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

Page 36: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

populations over time.

Dove

Hawk

Page 37: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

• Hawk minimum 10 maximum 30

• Dove minimum 50 maximum 250

Page 38: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.

Page 39: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

after one year.

• Dove population is 250, hawk population is 20

Page 40: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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

what is that value?

Hawk population will be 30 at year 2.

Page 41: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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)

Page 42: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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)

Page 43: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 44: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 45: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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?

Page 46: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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?

Page 47: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets
Page 48: 10.6 Parametrics. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets

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.