9.5Parametric Equations

2015

Calculator

Ships in the Fog

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.

Parametric Equations

Definition of a Plane Curve

• If f and g are continuous functions of t on an interval, the set of ordered pairs (f(t), g(t)) graphs out a plane curve.

• The equations x = f(t) and y = g(t) are parametric equations, and t is the parameter.

• Any parameter can be used, but t is commonly used as a parameter to represent time.

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: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

Ex: sketch the curve given by the parametric equations, then eliminate the parameter and find the rectangular equation: for 2 4

2

x t

ty

32 t

t – 2 – 1 0 1 2 3

x 0 – 3 – 4 – 3 0 5

y – 1 –.5 0 .5 1 1.5

y

x-4 4

4

8

Parametric equation for x.

Substitute into the original rectangular equation.

Example:

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

8t – 2 – 1 0 1 2

x –2 –3 0 1 2

y – 11 –7 –3 1 5

Homework

• P.673

• 1-8 all.

Example: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

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)

You try:

Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = 2 – x.

t – 2 – 1 0 1 2

x 4 3 2 1 0

y 13 9 5 1 -3

y

x-4 4

4

8

x = 2 – t

y = 4(2-t )– 3 = 8– 4t – 3 =5– 4t

Identify the curve represented by the equations by eliminating the parameter:

1

11

tx and y

tt

2Rectangular equation : 1 ,y x

Defined only for t >-1, so x>0.

Eliminate the parameter to identify the curve represented by the parametric equations:

2

2

1 3 6 4

1 2 1

t tx y

t t t

2 3y x

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

y

x

-4 4

4

8

2 2 2( ) ( )x h y k r The set of all points satisfying the equation gives the circle with center (h,k).

The location of any x coordinate in a circle is: cos

The location of any y coordinate in a circle is: sin

x r h

y r k

cos

Base = x-h

height = y-k

h

k Pythagorean Theorem?

r

y k

r

sin

x h

r

cos

sin

x r h

y r k

To write the equation of a circle in parametric form use:

Find a set of parametric equations to represent the conic, then graph in parametric mode:

2 2( 4) ( 1) 16x y

cos sinx r t h y r t k

4cos 4 4sin 1x t y t

Ex:

Write and Graph a set of parametric equations for the function:

2 2( 3) ( 5) 9x y

3cos 3

3sin 5

x

y

Ex:

Sketch the graph.

3cos 3sinx y

equation that represents the graph.

Eliminate the parameter and write the corresponding rectangular

cos sin3 3

x y

2 2cos sin 1 2 2

13 3

x y

Circle with center at origin and radius of 3

2 2

19 9

x y

D: 3 3x

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

2 2 9x y

Ex:

Horizontal Axis Ellipse: Vertical Axis Ellipse :

cos

sin

x a h

y b k

sec

tan

x a h

y b k

Parametric Equations for ellipse and hyperbola:

Horizontal Axis Hyperbola: Vertical Axis Hyperbola:

tan

sec

x b h

y a k

cos

sin

x b h

y a k

Eliminate the parameter. Describe the graph of the conic.

cos and 2sin , 0 2x y

2 2

11 4

x y

Ellipse

Ex:

Eliminate the parameter. Describe the graph of the conic.

2

14 4

x y

Parabola

Ex:

24cos , 2sinx y

You try:

Sketch the curve represented by

by eliminating the parameter. Describe the

graph of the conic.

2sec 1, 3 tan 2, 0 2x y

2 21 2

14 9

x y

Find a set of parametric equations for the ellipse with vertices and foci . 5,0 4,0

You try:

5cos

3sin

x

y

Parametric equations of a line separate the horizontal and vertical components of points on the line.

Find parametric equations to represent the line that passes through

1,4 5, 2and

1 4

4 6

x t

y t

• The x equation describes the horizontal position for some change in the parameter.

• The y equation describes the vertical position for some change in the parameter.

Parametric equations of a line

1 2 1

1 2 1

x x t x x

y y t y y

You try:

Find the parametric equations to represent the line that passes through .

3, 7 2,5and

3 5

7 12

x t

y t

Example:

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

Homework

• P.673 9-29 odd, 33-36 all, 37-41 odd.

Extras Optional Additional Probs,

• 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: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 wheel 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: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:A 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.