Ecuaciones Parametricas 13 14

8/17/2019 Ecuaciones Parametricas 13 14

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10. Parametric and Polar Equations - 1 - www.mastermathmentor.com - Stu Schwartz

Unit 10 – Parametric and Polar Equations - Classwork

Until now, we have been representing graphs by single equations involving variables x and y. We will nowstudy problems with which 3 variables are used to represent curves. Consider the path followed by an object

that is propelled into the air at an angle of 45o. If the initial velocity of the object is 48 feet per second, the

object follows the parabolic path given by

y =" x

2

72+ x

However, although you have the path of the object, you do not know when the object is at a given time. In orderto do this, we introduce a third variable t , called a parameter. By writing both x and y as a function of t , you

obtain the parametric equations:

x = 24t 2 and y = "16t 2+ 24t 2

From this set of equations, we can determine that at the time t = 0, the object is at the point (0, 0). Similarly at

the time t = 1, the object is at the point 24 2,24 2 "15( ) .

Definition of a Plane Curve

If f and g are continuous functions of t on an interval I , then the equations x = f t ( ) and y = g t ( )

are called parametric equations and t is called the parameter. The set of points x, y( ) obtained as t varies over the interval I is called the graph of the parametric equations.

Taken together, the parametric equations and the graph are called a plane curve.

When sketching a curve by hand represented by parametric equations, you use increasingvalues of t . Thus the curve will be traced out in a specific direction. This is called the

orientation of the curve. You use arrows to show the orientation.

Example 1) Sketch the curve described by the parametric equations:

x = t 2- 4 and y =

t

2 " 2# t # 3

t

x

y

3-2 -1 0 1 2


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Example 2) Sketch the curve described by the parametric equations:

x = 4t 2

- 4 and y = t "1# t # 32

t

x

y

0 1-1 -.5 .5 1.5

Note that both examples trace out the exact same graph. But the speed is different. Example 2’s graph is traced

out more rapidly. Thus in applications, different parametric equations can be used to represent various speeds atwhich objects travel along paths.

Note that the TI-84 calculator can graph in parametric mode. Go to MODE and switch to Parametric mode as

shown below. Your Y= button now gives you the screen below. TheX,T," ,n

button now gives a T when pressed. The equation in example 1 can now be generated. You set the T values by going to your WINDOW

and placing them in Tmin and Tmax.

Tstep controls the accuracy and speed of your graph. Large values of Tstep give speed but not little accuracy.

Small values of Tstep give a lot of accuracy at the cost of speed. Xmin, xmax, Ymin, Ymax work as before. Note that the arrow showing orientation does not display when you graph a parametric on the calculator. If you

are asked to draw a parametric on an exam, you must include it.

Finding a rectangular equation that represents the graph of a set of parametric equations is called eliminating

the parameter. Here is a simple example of eliminating the parameter.

Example 3) Eliminate the parameter in x = t 2- 4 and y =

t

2

• Solve for t in the second equations • Substitute in the second equations and simplify

Note that both equations give the same graph although they are not plotted in the same direction.


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Example 4) Eliminate the parameter in the following parametrics. In each problem, it will usually be easier tosolve for t in one equation than the other. Then graph, showing that the 2 equations graph the same

curve.

a) x = t " 3 and y = t 2 + t " 2 b) x = 3t + 2 and y =1

2t "1

c) x =t

2 and y = sin t +1( ) "1 d) x =

1

t +1 and y =

t

t +1 t > -1

Example 5) Sketch the curve represented by x = 3cos" and y = 3sin" 0 #" < 2$

• Solve for cos" and sin" in both equations.

• Use the fact that sin2" +cos

2" =1 to form an equation using only x and y.

• This is a graph of an circle centered at (0, 0) with diameter endpoints at (3, 0), (-3, 0), (0, 3), and

(0, -3). Note that the circle is traced counterclockwise as! goes from 0 to 2!.

Example 6) Finding parametric equations for a given function is easier. Simply let t = x and then replace your y

with t . Find parametric equations for

a) y = x 2 " 2 x + 3 b) y =2 x " 4

3 x "1

Note that there are many ways of finding parametric equations for a given function. For the problems above, let x = t + 2 and find the resulting parametric equations.

a. b.


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Example 7) At any time t with 0 " t "10 , the coordinates of P are given by the parametric equations: x = t " 2sint and y = 2" 2cost

Sketch this using your calculator.

t=0

t=1

t=2

t=3

t=4

t=5

t=6

t=7

t=8

t=9

t=10

• Points corresponding to integer values of t are

shown. At t = 1, P has coordinates of (-.68,. 92); at

this instant, P is heading almost due north.

• The full curve is not the graph of a function; some

x values have more than one y value.

• The picture shows the x- and y- axes but no t-axis.

• The bullets on the graph appear at equal time

intervals but not at equal distances from each other

because P s eeds u and slows down as it moves.

Example 8) Parametric curves may have loops, cusps, vertical tangents and other peculiar features. Parametriccurves are not necessarily functions. Trying to graph the curves below in function mode would

necessitate very complex piecewise functions.

Graph a) x = 2cost + 2cos 4t ( ) and y =sin t +sin 4t ( ) 0 " t "5 b) x = sin 5t ( ) and y = sin 6t ( ) 0" t " 2#

a.

t=0t=5

b. This is called a Lissajou curve.

Projectile Motion

If a projectile is launched at a height of h feet above the ground at an angle of " (measured in degrees or radians) with the horizontal. If the initial velocity is v

0 feet per

second, the path of the projectile is modeled by the parametric equations:

x = v0t cos" y = h + v

0t sin" #16t

2

Example 9) For each problem, use the calculator to graph two parametric equations for a projectile fired at the

given angle at the given initial speed at ground level. Then use the calculator’s trace ability toestimate the maximum height of the object as well as its range.

a." = 30°,v

0= 90 ft sec

" = 30°,v0=120 ft sec

b." = 30°,v

0= 90 ft sec

" = 70°,v0= 90 ft sec

Par. equation 1 __________________________ Par. equation 1 __________________________

Par. equation 2 __________________________ Par .equation 2 __________________________


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Max height 1: _____ Range 1: _____ Max height 1: _____ Range 1: _____

Max height 2: _____ Range 2: _____ Max height 2: _____ Range 2: _____

Example 10) A baseball player is at bat and makes contact with the ball at a height of 3 ft. The ball leaves the

bat at 110 miles per hour towards the center field fence, 425 feet away which is 12 feet high. Ifthe ball leaves the bat at the following angles of elevation, determine whether or not the ball will

be a home run. Show your equations and explain your answers.

a. " =17° b. " =18°

Polar Coordinates

We are used to the coordinate system where coordinates are x and y. The parametric equations we just saw use a

3rd

variable t , but still graphs points in the form of x, y( ). Thus the system is called the rectangular system orsometimes referred to the Cartesian coordinate system. This system is based on straight lines – thus a rectangle

The polar coordinate system is based on a circle.

To form the polar coordinate system, we fix a point O called the pole or the origin. Each point P in the plane

can be assigned polar coordinates r ,! ( ) as follows: r is the directed distance from O to P and ! is the directedangled, counterclockwise from polar axis to segment OP . The diagram below shows three points on the polar

coordinate system. It is convenient to locate points with respect to a grid of concentric circles intersected by

radial lines through the pole. The line ! = 0 is called the polar axis.

0

!/2

!

3!/2

(3,!/6)

(4, -!/4) or (-4,3!/4)

(4,2!/3)

Depending on whether you want to measure angles as radians or degrees, the coordinate changes but still refers

to the same point. Thus the point 3,"

6

#

$ %

&

' ( is the same point as 3,30°( ).


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Note that in the rectangular system, there is only one way to label a point. In the polar system, there are severalways to label a point, actually an infinite number of ways.

Example1) For each polar point, label it in two other ways:

a. 4,60°( ) b. "5,315°( ) c. 2,"90°( )

d. 1,5"

6

#

$

%

&

'

(

e. "8,#

6

$

%

&

'

(

)

f. " 3

2,"

5#

3

$

%

&

'

(

)

To convert to and from the polar system to the coordinate system, you must know the following relationships.

x

y

!

r

x = r cos! tan! = y

x

y = r sin! r 2 = x2 + y2

r ,!

( ) or x, y

( )

Example 2) Convert the following polar points to rectangular coordinates.

a. 6,90°( ) b. 4,60°( ) c. 10,225°( )

d. (5, !) e. 2 3,"

6

#

$

%

&

'

(

f.5

2,5"

3

#

$

%

&

'

(

Example 3) Convert the following rectangular points to polar coordinates.

a. (-5, -5) b. (0, -2) c. 1," 3( )

d. "7,0( ) e. 5,12( ) f. 6,"3( )


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The TI-84 calculator is capable of making these conversions although it is slightly

cumbersome. The commands are located in the ANGLE menus.

To convert the rectangular point 4, 4 3( ) to polar form, you use5 : R"Pr( and 6: R" P# .( These commands ask for the value of r and the value

of " as two separate statements. The value of " will depend on whether you are indegree or radian mode.

To convert the polar point 2 2,225°( ) to rectangular form, you use7 : P"Rx( and 8: P"Ry(. These commands ask for the values of x and the value of

y as two separate statements. Again, when you input " , be sure it matches the form

you specified in Mode – radian or degree.

Example 4) Convert the following rectangular equations to polar equations. Confirm by calculator.

a) x2

+ y

2

= 25 b) x + 2( )

2

+ y

2

= 4

c. y=

3

d. x = 3 e. xy =1 f. 2 x " 3 y " 2 = 0

Example 5) Convert the following polar equations to rectangular equations. Confirm by calculator.

a) r = 2 b) " =2#

3

c. r = 4sec"


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d) r = "2csc# e) r =12

3sin" # 4cos" f) r =

3

1+sin"

Example 6) Plot the points and sketch the graph of the polar equation r = 3cos" . (1 decimal place)

" 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° r

You can graph polar equations on your calculators. You must switch to polar mode

using your MODE button as shown to your right. You can graph in either degree or

radian mode although you will gain more control with degree mode. However, there aregraphs that MUST be graphed in radian mode. Any polar equation using " not

involving trig functions must be in radian mode.

You enter the equation as usual, through the Y= button.

" is now generated through the X,T," ,n button. You

have to control the values " , X and Y through the

WINDOW button. " step controls the speed and

accuracy of the graph. Large values of " step generate

curves quickly but with less accuracy while small values of " step give a lot of accuracy but graph slower. As

usual, you have to ZSquare to have an accurate graph.


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Example 7) Plot the points and sketch the graph of the polar equation r = 3 + 2sin" . (1 decimal place)

" 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° r

Classifying Polar Graphs

Just as was true with rectangular graphs, there are graphs in polar form that occur all the time and students

should be able to recognize them by their equations.

r = a r = asin" r = acos" r = a"

Circles – a is the diameter Spiral of Archimides

Sine curves are symmetric to the y-axis a controls the widthCosine curves are symmetric to the x-axis must be in radian mode


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Limaçons are in the form r = a ± bsin" symmetric to y - axis( ) or r = a ± bcos" symmetric to x - axis( ).

a

b


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Example 8) Match the polar equations with their graphs below.

___1) r = 3 " cos# ___2) r = 2 "2sin# ___3) r = 5cos3" ___4) r = 2+ 2cos"

___5) r = 3+1.5sin" ___6) r = 3.5cos2" ___7) r = 5sin3" ___8) r 2= "16cos2#

___9) r = 2 " 3cos# ___10) r = 3cos4" ___11) r = "4cos# ___12) r = 3.5sin2"

a. b. c.

d. e. f.

g. h. i.

j. k. l.


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Unit 10 – Parametric and Polar Equations - Homework

1. Consider the parametric equations x = t and y = 2t "1

a) Complete the table

t

0

1

2

3

4 x

y

b) Plot the points ( x, y) in the table and sketch a graph of the parametric

equations. Indicate the orientation of the graph.

c) Find the rectangular equation by eliminating the parameter.

2. Consider the parametric equations x = 4cos2" and y = 2sin"

a) Complete the table

t "# 2 "# 4 0 " 4 " 2

x

y

b) Plot the points ( x, y) in the table and sketch a graph of the parametric

equations. Indicate the orientation of the graph.

c) Find the rectangular equation by eliminating the parameter.

3. In the following exercises, eliminate the parameter and confirm graphically that the rectangular equations

yield the same graph as the parametrics. Be sure you take domain and range of the parametric into account.

a. x = 4t "1 and y = 2t +3 b. x = t +3 and y = t 2

c. x = t 3 and y = 3" t 2 d. x = t 2 "1 and y = t 2 + t


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e. x = t " 2 and y =t

t " 2 f. x = t " 3 and y = t +3

g. x = sec2" and y = tan2" h. x = cos" and y = 4sin"

(hint: think trig identities)

i. x = e t and y = e"t j. x = t 5 and y = 5ln t

4. For each rectangular equation, find 2 sets of parametrics, the first by letting x = t and the second by setting xequal to the given expression and finding the y-component.

a. y = 2 x 2 " 3 x " 2, x = t "1 b. y =2 x " 5

x2" x " 2

, x = t + 2

5. Use your calculators to graph the curve represented by the parametric equations. Indicate the orientation of

the curve. Identify any points at which the curve is not smooth. Do not take these problems lightly. Yourtask is to come up with an appropriate window to view them. Let your t run from 0 to 2!, 4!, 8!, etc.

a. Cycloid: The curve traced by a point on the b. Prolate Cycloid: Same as a) except the point

circumference of a circle as it rolls on a straight line. goes below the line (railroad track)

x= 2 " # sin"

( ) and

y= 2 1# cos"

( ) x= 2" # 4sin" and

y= 2# 4cos"

Scale

[ , ][ , ]

Scale

[ , ][ , ]


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c. Hypocycloid: x = 3cos3" and y = 3sin

3" d. Curtate cycloid: x = 2" # sin" and y = 2#cos"

Scale

[ , ][ , ]

Scale

[ , ][ , ]

e. Witch of Agnesi: x = 2cot" and y = 2sin2" f. Folium: x =

3t

1+ t 3 and y =

3t 2

1+ t 3

Scale

[ , ][ , ]

Scale

[ , ][ , ]

6. A dart is thrown upward from 6 ft. high with an initial velocity of 18 feet/sec at an angle of elevation of 41°.

a. Write a parametric equation that describes the position of the dart at time t.

x t ( ) = ___________________ y t ( ) = ___________________

b. Approximately how long will it take for the dart to hit the ground?

c. Find the approximate maximum height of the dart.

d. How long will it take for the dart to reach maximum height?

7. An arrow is shot from a platform 20 feet off the ground with an initial velocity of 150 feet/sec at an angle of

elevation of 23°.

a. Write a parametric equation that describes the position of the arrow at time t.

b. Find the approximate maximum height of the arrow.

c. Approximately how long will it take for the arrow to reach maximum height?

d. There is a wall 30 feet high 500 feet from the archer. Will the arrow hit it?

If so, how long will it take to hit it? If not, when will the arrow hit the ground beyond the walland how far away will it land?


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8. A golfer hits a ball with an initial velocity of 90 mph at angle of elevation of 64o.

a. Write a parametric equation that describes the position of the ball at time t.

x t ( ) = ___________________ y t ( ) = ___________________

b. Approximately how long will it take for the ball to hit the ground?

c. Find the approximate maximum height of the ball.

d. The green is 150 yards away. Will the ball reach the green? Explain.

9. An NFL kicker at the 33-yard line attempts a field goal. The ball leaves his foot at 69 feet/sec at an angle of

elevation of 38°.

a. Write a parametric equation that describes the position of the ball at time t.

x t ( ) = ___________________ y t ( ) = ___________________

b. How high does the ball get above the field?

c. The goal posts are 10 feet high and are 45 yards away from him. If the kick is straight, is the field goal

good? Explain.

10. Jack and Jill are standing 60 feet apart. At the same time, they each throw a softball from an initial height of

two feet towards each other. Jack throws the softball at an initial velocity of 45 ft/sec at an angle of

elevation of 44°. Jill throws her ball with an initial velocity of 41 ft/sec with an angle of elevation of 37°o.

a. Write 2 parametric equations that describes the position of the ball at time t. Remember they are

throwing the balls toward each other.

x1 t ( ) = ___________________ y1 t ( ) = ___________________

x2 t ( ) = ___________________ y2 t ( ) = ___________________

b. Find the heights of each ball. c. About how far does each ball travel?

d. When does each ball hit the ground?

e. By trial and error, find the time when you think the balls are closest together?


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11. Convert the following polar points to rectangular coordinates.

a. 6,"

2

#

$ %

&

' (

b. "1,7#

4

$

% &

'

( )

c. "4,"#

3

$

%

&

'

(

)

d. 3,120°( ) e. 8,210°( ) f. 10,72°( )

12. Convert the following rectangular points to polar coordinates.

a. "3,3( ) b.1

2," 3

2

#

$ %

&

' ( c. 0,"4( )

d. "5 2,"5 2( ) e. "2,1( ) f. 7,"24( )

13. For each of the following rectangular equations, change it to polar form and confirm on your calculator.

a. x 2 " y2= 4 b. xy =12

c. 5 x " y = 7

d. x "1( )2

+ y2

=1

e. y = x 3 f. x2+ y

2+ 4 x = 0


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14. For each of the following polar equations, change it to rectangular form and confirm on your calculator.

a. r = 4 b. tan2

" = 9

c. r = 8csc"

d. r = 8cos"

e.r =

5

2sin" # cos" f.r =

1

1+ cos"

15. Plot the points and sketch the graph of the polar equation r = 2 " 2sin# . (1 decimal place)

" 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°

r


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16. Plot the points and sketch the graph of the polar equation r = 2+ 4 cos" . (1 decimal place)

" 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° r

17. Plot the points and sketch the graph of the polar equation r = 5sin3" . (1 decimal place)

" 0° 15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° 180° r


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10 P i d P l E i 19 h S S h

18. Match the polar equations with their graphs below.

___1) r = 2.5+ 2.5sin" ___2) r = 3 ___3) r = 3.5sin3" ___4) r = 4.5sin2"

___5) r = 4.5cos2" ___6) r =1.5+ 2cos" ___7) r = "3sin# ___8) r = 2 " sin#

___9) r 2=16sin2" ___10) r = 4 cos5" ___11) r = 3.5cos3" ___12) r = 2.5 " 2.5cos#

___13) r = 3cos" ___14) r =1+ 4 sin" ___15) r = 4.5sin6" ___16) r = .5"

a. b. c. d.

e. f. g. h.

i. j k. l.

m. n. o. p.

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Ecuaciones Parametricas 13 14