Download - Polar Graphing
Polar GraphingMiss Hayley Summers
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• High school students (9th or 10th graders) in Algebra II or Pre-calculus
• Requires previous math knowledge (up to Algebra II)
• Students generally interested in learning• Any socioeconomic level• Ability to complete assignment with study
materials
Target Audience
Learning Environment
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• Access to a computer• Access to Internet, class notes, book, etc.• Quiet or noisy setting depending on learner’s
preference• Work is individual• Lesson moves at learner’s own pace
Learning Environment
Target Audience Objectives
• Given a PowerPoint presentation of information and review and practice, students should:– Be able to recognize different types of graphs and
draw graphs on polar coordinate planes in 100% accuracy on the quiz.
– Be able to plot points and find the function to double check their work and receive 100% accuracy on the quiz.
– Be able to compare and contrast the different graphs in an “A” essay given Word processing.
Objectives
Learning Environment
History
CirclesSpirals
Lemnis-cates
Limacons Roses
Quiz
Review
Main Menu
Practice
Modern
Use
http://www.conmishijos.com/dibujos/Iglu_1_g.gif
Do you remember the Polar Coordinate System??
Review!
pole polar axis
Θ (polar angle)radi
us
point
More Review
Review!•Circular grid based off a central fixed origin and ray•A point is graphed based on the length (r) from the origin and bond angle theta (θ) in relation to fixed ray•(r,θ) exists as coordinates and location of the point
(r, θ)
More ReviewReview
• Symmetry(r, -θ) = (-r, -πθ)Sine: symmetric to vertical axisCosine: symmetric to horizontal axis
Review!
• Graphing on calculator!**Only to be used in emergencies**1. 2nd FORMAT (ZOOM)
RectGC PolarGC2. MODE
Func Pol3. Y=
r1= (enter equation) HistoryReview
• Pythagoras: octave ratio 2:1, chord• Archimedes: spiral (r=a+bθ)• Hipparchus: Worked off Archimedes spiral and
Pythagoras’ theorems to create a table of chord, to determine given length of a chord for each angle
History
Modern UseReview
Modern Use
• Calculus! (Differential and Integral)• Finding Arc length• Flight Navigation• Surveying• Physics• Spirals : Parker spiral of solar wind, Catherine’s
wheel of fireworks
SpiralsHistory
Spirals
• r= aθ• For smaller values a and b, the spiral is tighter.
For larger values a and b, the spiral is wider.
CirclesModern
Use
• r= asinθ or r= acosθ• r= diameter• Remember!– Sin: symmetric to y– Cos: symmetric to x
Circles
r= 3sinθ
LimaconsSpirals
• r= a+bcosθ1. a>2b: convex Limacon2. a>b: Limacon w/ dimple3. a=b: Cardioid (heart shape)4. a<b: Limacon w/ loop
Limacons
1 2 3 4
For cosine: Length left of y
axis: a-b Length right of
y axis: a+b
Lemnis-catesCircles
• r2= a2cos2θ– a= length of each loop– cosθ indicates symmetry
around x-axis– sinθ indicates symmetry
around y-axis
Lemniscates
RosesLimacons
• r= asin (nθ)• a= length of petals• n= determines # of petals
n=even 2n petalsn=odd n petals
• Cos: aligns on x-axis, or all axes when n is even
• Sin: aligns on y-axis, or between axes when n is even
Roses
r=cos4θ
r= -4.5 sinθPracticeLemniscates
Practice ProblemsHere are 3 problems for you to try on your own!1. Draw the polar coordinate graph (a picture is
given on the next slide) on a piece of paper.2. Analyze the different parts of the function
and decide what each tells you about the graph.
3. Draw the graph!
Proceed to Practice Problems!Roses
1. Graph r= 2cosθ
Practice- #1
S#1Instructions
Solution- #1
• Watch me work out Problem #1 here!– Please note this link will take you out of the
presentation. After viewing the solution, please click back into the presentation and continue.
P#2P#1
Practice- #2
• Graph r= 2cos(3θ)
S#2S#1
Solution- #2
• Watch me work out Problem #2 here!– Please note this link will take you out of the
presentation. After viewing the solution, please click back into the presentation and continue.
P#3P#2
Practice- #3
• Graph r= 2- 2sinθ
S#3S#2
Solution- #3
• Watch me work out Problem #3 here!– Please note this link will take you out of the
presentation. After viewing the solution, please click back into the presentation and continue.
QUIZP#3
Quiz! Are you ready?
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QuizPractice
Quiz- #1
• What is the polar graph of r= 2cosθ?
Circle of radius _____ centered at _____.
ABCD
2, x axis1, y axis4, x axis2, y axis
Try Again!• What does cos(θ) indicate?• What does the value “a” represent in the
equation r= a cosθ ?
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Quiz- #1
Correct!
The answer is A:• Cos (θ) indicates the equation lies on the x axis• A= length (diameter)= 2
Next Question!
Quiz- #1
Quiz- #2
• What is correct about the number of petals on a rose?
ABCD
n petals if n is even, 2n if n is odd2n petals if n is even, n if n is odd2n petals if n is even, 4n if n is odd4n petals if n is even, n if n is odd
Try Again!
• A rose has the equation r= acos(nθ).• What occurs in the graph when n is even or
odd?
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Quiz- #2
Correct!
The answer is B:• A rose has n petals if n is odd and 2n petals if n
is even!
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Quiz- #2
Quiz- #3
• What is the polar graph of r= 2-sinθ ?
A B
C D
Try Again!• Does the negative sign effect the graph in any
way?• Where does θ=0?
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Quiz- #3
Correct!
The answer is D:• Because sinθ has a negative sign, the graph points
down.• The graph intersects the x axis at 3.
Next Question!
Quiz- #3
Quiz- #4
• Which Greek philosopher developed the table of chord?
ABCD
ArchimedesDonatelloHipparchusSocrates
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• Think back to the people discussed in the History section.
• Hint: He’s not a ninja turtle!
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Quiz- #4
Correct!The answer is C:• Hipparchus discovered the table of chord!
– Archimedes discovered the spiral– Socrates was a Greek philosopher.– Donatello was an Italian artist and sculptor (also a ninja turtle!)
Next Question!
Quiz- #4
Quiz- #5
• What shape does the graph r= 6-4cosθ make?
ABCD
LemniscateLimacon with loopCardioidLimacon with dimple
Try Again!
• Limacons have the equation r= a-bcosθ.• What is the relationship between a and b?
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Quiz- #5
Correct!
The answer is D:• a>b, in the equation r= a-bcosθ so the
limacon has a dimple!
Next Question!
Quiz- #5
Quiz- #6
• What is the graph of r=3sin4θ?
A B
C D
Try Again!
• In a rose equation r= asin(nθ), what does the value “a” represent? “n”?
• How does sinθ affect the graph?
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Quiz- #6
Correct!The answer is B:• In the rose equation r=asin(nθ),
– a=3, the length of the petals– n=4, which is even, so there are 2n or 8 petals total
• Sinθ gives symmetry to the y-axis
Next Question!
Quiz- #6
Quiz- #7
• What does the equation r2= a2sin2θ represent?
ABCD
CircleLimaconRoseLemniscate
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• Which graph has an r2 value in its general equation?
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Quiz- #7
Correct!
The answer is D:• Lemniscates are the only polar graphs with an
r2 value in their general equation!
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Quiz- #7
Quiz- #8
• Which is NOT a way polar graphing is used today?
ABCD
Differential/ Integral CalculusPhysics and Arc LengthFlight and NavigationAll of the above are uses of polar graphing.
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• Remember polar graphing has many uses!
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Quiz- #8
Correct!
The answer is D:• Polar graphing has many real world applications,
and that is why we are taking the time to learn it!
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Quiz- #8
Quiz- #9
• In a general spiral equation r=aθ, a spiral is tighter for _______ “a” values and wider for ______ “a” values?
ABCD
larger, smallereven, oddsmaller, largerodd, even
Try Again!
• It is the size of the number “a” that shrinks or widens the spiral.
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Quiz- #9
Correct!
The answer is C:• Just as you would think, smaller values shrink
the graph and larger values widen it!
One More Question!
Quiz- #9
Quiz- #10
• What shape does the graph y=sin(θ)cos(3θ) make?
ABCD
SpiderFishButterflyFlower
*Hint: You may need to use your calculator!
Try Again!
• Did you switch your calculator to polar coordinates?
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Quiz- #10
Correct!The answer is C:• It’s a (sideways) butterfly!!
Congrats! Check out your results!
Quiz- #10
ResultsNumber of Questions Correct
Eskimo Status
0-2Eskimo Faux- You need to brush up on some material and retake the quiz!
3-5Eskimo Slow- You should review the material and retake the quiz!
6-7Average Eskimo Joe- You should still review the material but you’re on your way!
8-10Eskimo Pro- Review the material before the test, but you’re well prepared!
Check out these resources for more information!
Resources• Anderson, Dawn Leigh. “Assignment 11: Polar Equations.” The University of
Georgia. 23 June 1999.<http://jwilson.coe.uga.edu/>
• “Graphing in Polar Coordinates.” Sparknotes. <http://www.sparknotes.com/math/precalc/ parametricequationsandpolarcoordinates/section3.rhtml
• Leathrum, Tom. “Graphing in Polar Coordinates” Java Applet. Addison-Wesley Materials. 2002. Web. 11 Nov 2011. <http://cs.jsu.edu/~leathrum/Mathlets/polar.html
• http://us.123rf.com/400wm/400/400/cthoman/cthoman1110/cthoman111000522/10771155-a-happy-cartoon-polar-bear-jumping-and-smiling.jpg
• http://www.lucyannmoll.com/beautifulwarrior/friday-funnies-write-a-caption-2
Now that you’re done, go take a nice polar bear snooze!