Polar Graphing

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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

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

History

CirclesSpirals

Lemnis-cates

Limacons Roses

Review

Main Menu

Practice

Modern

http://www.conmishijos.com/dibujos/Iglu_1_g.gif

Do you remember the Polar Coordinate System??

Review!

pole polar axis

Θ (polar angle)radi

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

• 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

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

P#3P#2

Practice- #3

• Graph r= 2- 2sinθ

S#3S#2

Solution- #3

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 _____.

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?

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

Try Again!

Review Material!

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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!

Next Question!

Quiz- #2

Quiz- #3

• What is the polar graph of r= 2-sinθ ?

Try Again!• Does the negative sign effect the graph in any

way?• Where does θ=0?

Try Again!

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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?

ArchimedesDonatelloHipparchusSocrates

Try Again!

• Think back to the people discussed in the History section.

• Hint: He’s not a ninja turtle!

Try Again!

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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?

LemniscateLimacon with loopCardioidLimacon with dimple

Try Again!

• Limacons have the equation r= a-bcosθ.• What is the relationship between a and b?

Try Again!

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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?

Try Again!

• In a rose equation r= asin(nθ), what does the value “a” represent? “n”?

• How does sinθ affect the graph?

Try Again!

Review Material!

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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?

CircleLimaconRoseLemniscate

Try Again!

• Which graph has an r2 value in its general equation?

Try Again!

Review Material!

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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!

Next Question!

Quiz- #7

Quiz- #8

• Which is NOT a way polar graphing is used today?

Differential/ Integral CalculusPhysics and Arc LengthFlight and NavigationAll of the above are uses of polar graphing.

Try Again!

• Remember polar graphing has many uses!

Try Again!

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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!

Next Question!

Quiz- #8

Quiz- #9

• In a general spiral equation r=aθ, a spiral is tighter for _______ “a” values and wider for ______ “a” values?

larger, smallereven, oddsmaller, largerodd, even

Try Again!

• It is the size of the number “a” that shrinks or widens the spiral.

Try Again!

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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?

SpiderFishButterflyFlower

*Hint: You may need to use your calculator!

Try Again!

• Did you switch your calculator to polar coordinates?

Try Again!

Review Material!

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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!

Polar Graphing

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