Unit 10 – Parametric and Polar Equations - Classwork
Until now, we have been representing graphs by single equations involving variables x and y. We will now study 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
x2 y = + x 72
However, although you have the path of the object, you do not know when the object is at a given time. In order to 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 . ! ( )
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 a s 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 increasing values 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: t x = t2 - 4 and y = " 2 # t # 3 2
t -2 -1 0 1 2 3 x y
10. Parametric and Polar Equations - 1 - www.mastermathmentor.com - Stu Schwartz
!
! Example 2) Sketch the curve described by the parametric equations: 2 3 x = 4t - 4 and y = t 1 t 2
t -1 -.5 0 .5 1 1.5 x y
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 at which 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. The X,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! va lues 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. t Example 3) Eliminate the parameter in x = t2 - 4 and y = 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 to solve for t in one equation than the other. Then graph, showing that the 2 equations graph the same curve. 1 a) x = t " 3 and y = t 2 + t " 2 b) x = 3t + 2 and y = 2t "1
! !
t 1 t c) x = and y = sin(t +1) "1 d) x = and y = t > -1 2 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 2π.
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
2x " 4 a) y = x 2 " 2x + 3 b) y = 3x "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.
• Points corresponding to integer values of t are t=3 t=9 shown. At t = 1, P has coordinates of (-.68,. 92); at this instant, P is heading almost due north. t=4 t=10 t=2 • The full curve is not the graph of a function; some t=8 x values have more than one y value. t=5 t=1 • The picture shows the x- and y- axes but no t-axis. t=7 • The bullets on the graph appear at equal time intervals but not at equal distances from each other t=0 t=6 because P speeds up and slows down as it moves. We will soon see how to calculate the speed of a Example 8) Parametric curves may have loops, c usparametricps, vertic acurvel tange atnt as point.and other peculiar features. Parametric curves 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# ( ) ( )
t=5 t=0
a. 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 v0 feet per second, the path of the projectile is modeled by the parametric equations:
! x = v t cos" y = h + v t sin" #16t 2 0 0 !
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 to estimate the m!a ximum height of the object as well as its range.
" = 30°,v = 90 ft sec " = 30°,v = 90 ft sec a. 0 b. 0 " = 30°,v0 =120 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. If the 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 or sometimes 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! fi x a point O called the pole or the origin. Each point P in the plane can be assigned polar coordinates as follows: r is the directed distance from O to P and is the directed ( r,! ) ! angled, 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. !/2 ! (4,2!/3)
(3,!/6) ! 0
(4, -!/4) or (-4,3!/4)
3!/2
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, ( is the same point as (3,30°). $ 6'
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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 several ways 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°)
# 5"& $ #' $ 3 5#' ! d. %1 , ( ! e. & "8, ) ! f. & " ," ) $ 6 ' % 6( % 2 3 (
! ! !
To convert to and from the polar system to the coordinate system, you must know the following relationships.
r,! or x, y ( ) ( ) y x = rcos! tan! = r x y 2 2 2 y = rsin! r = x + y ! x
Example 2) Convert the following polar points to rectangular coordinates.
a. (6,90°) b. (4,60°) c. (10,225°)
! ! # "& ! # 5 5"& d. (5, π) e. % 2 3, ( f. % , ( $ 6' $ 2 3 '
Example 3) Convert the following rec!ta ngular points to polar coordinates. !
a. (-5, -5) b. (0, -2) c. (1," 3)
! " d. ("7,0) e. (5,12) f. (6,"3)
! ! ! " 10. Parametric and Polar Equations - 6 - www.mastermathmentor.com - Stu Schwartz 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 use 5 : 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 in degree or radian mode. ! ! ! To convert the polar point (2 2,225!°) to rectangular form, you use 7 : 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 rectangul!a r equations to polar equations. Confirm by calculator. 2 a) x 2 + y 2 = 25 b) (x + 2) + y 2 = 4 c. y = 3
! ! !
d. x = 3 e. xy =1 f. 2x " 3y " 2 = 0
! ! !
Example 5) Convert the following polar equations to rectangular equations. Confirm by calculator. 2# a) r = 2 b) " = c. r = 4sec" 3
!
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! 12 3 d) r = "2csc# e) r = f) r = 3sin" # 4cos" 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 are graphs 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. ! 10. Parametric and Polar Equation!s - 8 - www.mastermathmentor.com - Stu Schwartz ! 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 width ! ! Cosine curves are sy!m metric to the x-axis ! must be in radian mode
10. Parametric and Polar Equations - 9 - www.mastermathmentor.com - Stu Schwartz Limaçons are in the form r = a ± bsin" (symmetric to y - axis) or r = a ± bcos" (symmetric to x - axis).
!
a a a a <1 =1 1< < 2 " 2 b b b b limaçon with Cardioid Dimpled limaçon Convex limaçon inner loop (heart shaped) (one side is flattened)
!R ose curves are in the for!m r = asinn" or r = acos!n " . The maximum diam!et er of the petal is controlled by a. If n is even, the rose curve will have 2n petals. If n is odd, the rose curve will have n petals. Interesting patters can be formed if n is a decimal and the curve is viewed with " starting at 0 and going out to very large numbers. ! !
r = asin2" r = acos3" r = acos4" r = asin5"
Lemniscates look like infinity signs and are in the form r2 = a2 sin2" which is symmetric to the origin and r2 = a2 cos2" which is symmetric to the x-axis. If the coefficient of " is a number other than 1 or 2, you get a ! deformed lemniscate. ! ! !
! ! !
r2 = a2 sin2" r2 = a2 cos2" r2 = a2 cos1.3"
! ! ! 10. Parametric and Polar Equations - 10 - www.mastermathmentor.com - Stu Schwartz 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) r2 = "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.
10. Parametric and Polar Equations - 11 - www.mastermathmentor.com - Stu Schwartz 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 2. Consider the parametric equations x = 4cos " and y = 2sin"
a) Complete the table
t "# 2 "# 4 0 " 4 " 2 x y
! b) P!lot the points (x, y)! in the ta!bl e 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.
2 a. x = 4t "1 and y = 2t +3 b. x = t +3 and y = t
c. x = 3 t and y = 3" t 2 d. x = t 2 "1 and y = t 2 + t
! ! !
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! ! t e. x = t " 2 and y = f. x = t "3 and y = t +3 t " 2
2 2 g. x = sec " and y = tan " h. x = cos" and y = 4sin" (hint: think trig identities)
i. x = et and y = e"t j. x = t 5 and y = 5lnt
! !
4. For each rectangular equation, find 2 sets of parametrics, the first by letting x = t and the second by setting x equal to the given expression and finding the y-component.
2x " 5 a. y = 2x 2 " 3x " 2, x = t "1 b. y = , x = t + 2 x 2 " x " 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. Your ! ! task is to come up with an appropriate window to view them. Let your t run from 0 to 2π, 4π, 8π, 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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! ! 3 3 c. Hypocycloid: x = 3cos " and y = 3sin " d. Curtate cycloid: x = 2" #sin" and y = 2 #cos"
Scale Scale
[ , ] [ , ] [ , ] [ , ]
3t 3t 2 e. Witch of Agnesi: 2 f. Folium: x = 2cot" and y = 2sin " x = 3 and y = 3 1+ t 1+t 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 wall and how far away will it land?
10. Parametric and Polar Equations - 14 - www.mastermathmentor.com - Stu Schwartz 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?
10. Parametric and Polar Equations - 15 - www.mastermathmentor.com - Stu Schwartz 11. Convert the following polar points to rectangular coordinates. # "& $ 7#' $ "#' a. % 6, ( b. & "1, ) c. & "4, ) $ 2' % 4 ( % 3 (
! ! ! d. (3,120°) e. (8,210°) f. (10,72°)
! ! ! 12. Convert the following rectangular points to polar coordinates. # 1 " 3& a. ("3,3) b. % , ( c. (0,"4) $ 2 2 '
! ! ! 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.
2 2 a. x " y = 4 b. x y =12
!
2 c. 5x y 7 d. x "1 + y2 =1 " = ( )
2 2 e. y = x 3 f. x + y + 4x = 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. t an2 " = 9
c. r = 8csc" d. r = 8cos"
! !
5 1 e. r = f. r = 2sin" # cos" 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 !
! ! ! ! ! ! ! ! ! ! ! ! ! !
10. Parametric and Polar Equations - 17 - www.mastermathmentor.com - Stu Schwartz 16. Plot the points and sketch the graph of the polar equation r = 2 + 4cos" . (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 !
! ! ! ! ! ! ! ! ! ! ! ! ! !
10. Parametric and Polar Equations - 18 - www.mastermathmentor.com - Stu Schwartz 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) r2 =16sin2" ___10) r = 4cos5" ___11) r = 3.5cos3" ___12) r = 2.5 " 2.5cos# ! ! ! ! ___13) r = 3cos" ___14) r =1+ 4sin" ___15) r = 4.5sin6" ___16) r = .5" ! ! ! !
! ! ! ! ! ! ! !
a. b. c. d.
e. f. g. h.
i. j k. l.
m. n. o. p. 10. Parametric and Polar Equations - 19 - www.mastermathmentor.com - Stu Schwartz