Polar Graphing

1. Polar Graphing Miss Hayley Summers Start Lesson! http://www.free-wallpapers-free.com

2. Action Buttons Go back to the Previous Slide Head “Home” to the Main Menu for other sections or the Quiz! Go ahead to the Next Slide

3. Target Audience • 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 ActionButtons Learning Environment

4. Learning Environment • 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

5. 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 receive100% accuracy on the quiz. • Be able to compare and contrast the different graphs in an “A” essay given Word processing. Learning Environment

6. Main Menu • Review • Modern • Use • History • Spirals • Circles • Limacons • Lemnis- • cates • Roses • Practice • Quiz http://www.conmishijos.com/dibujos/Iglu_1_g.gif

7. Review! Do you remember the Polar Coordinate System?? point radius Θ (polar angle) pole polar axis More Review

8. 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 Review Review

9. Review! • Symmetry (r, -θ) = (-r, -πθ) Sine: symmetric to vertical axis Cosine: symmetric to horizontal axis • Graphing on calculator! • **Only to be used in emergencies** • 1. 2nd FORMAT (ZOOM) • RectGCPolarGC • 2. MODE • FuncPol • 3. Y= • r1= (enter equation) Review History

10. History • 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 Modern Use Review

11. Modern Use • Calculus! (Differential and Integral) • Finding Arc length • Flight Navigation • Surveying • Physics • Spirals : Parker spiral of solar wind, Catherine’s wheel of fireworks History Spirals

12. Spirals • r= aθ • For smaller values a and b, the spiral is tighter. For larger values a and b, the spiral is wider. Modern Use Circles

13. Circles • r= asinθ or r= acosθ • r= diameter • Remember! • Sin: symmetric to y • Cos: symmetric to x r= 3sinθ Limacons Spirals

14. Limacons • For cosine: • Length left of y axis: a-b • Length right of y axis: a+b • r= a+bcosθ • a>2b: convex Limacon • a>b: Limaconw/ dimple • a=b: Cardioid (heart shape) • a<b: Limaconw/ loop 1 2 3 4 Lemnis-cates Circles

15. Lemniscates • r2= a2cos2θ • a= length of each loop • cosθ indicates symmetry around x-axis • sinθ indicates symmetry around y-axis Limacons Roses

16. Roses • r= asin (nθ) • a= length of petals • n= determines # of petals n=even  2n petals n=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 sinθ Practice Lemniscates

17. PracticeProblems Here are 3 problems for you to try on your own! • Draw the polar coordinate graph (a picture is given on the next slide) on a piece of paper. • Analyze the different parts of the function and decide what each tells you about the graph. • Draw the graph! Proceed to Practice Problems! Roses

18. Practice- #1 • Graph r= 2cosθ Instructions S#1

19. 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#2 P#1

20. Practice- #2 • Graph r= 2cos(3θ) S#2 S#1

21. 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#3 P#2

22. Practice- #3 • Graph r= 2- 2sinθ S#3 S#2

23. 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. QUIZ P#3

24. Quiz! Are you ready? Quiz Practice Go home at any time to review material! Warning! Returning Home during quiz will not save your place!

25. Quiz- #1 • What is the polar graph of r= 2cosθ? Circle of radius _____ centered at _____. A 2, x axis 1, y axis 4, x axis 2, y axis B C D

26. Quiz- #1 Try Again! • What does cos(θ) indicate? • What does the value “a” represent in the equation r= a cosθ? Try Again! or Review Material! or Go Home!

27. Quiz- #1 Correct! The answer is A: • Cos (θ) indicates the equation lies on the x axis • A= length (diameter)= 2 Next Question!

28. Quiz- #2 • What is correct about the number of petals on a rose? A n petals if n is even, 2n if n is odd 2n petals if n is even, n if n is odd 2n petals if n is even, 4n if n is odd 4n petals if n is even, n if n is odd B C D

29. Quiz- #2 Try Again! • A rose has the equation r= acos(nθ). • What occurs in the graph when n is even or odd? Try Again! or Review Material! or Go Home!

30. 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!

31. Quiz- #3 • What is the polar graph of r= 2-sinθ ? A B C D

32. Quiz- #3 Try Again! • Does the negative sign effect the graph in any way? • Where does θ=0? Try Again! or Review Material! or Go Home!

33. 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!

34. Quiz- #4 • Which Greek philosopher developed the table of chord? A Archimedes Donatello Hipparchus Socrates B C D

35. Quiz- #4 Try Again! • Think back to the people discussed in the History section. • Hint: He’s not a ninja turtle! Try Again! or Review Material! or Go Home!

36. 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!

37. Quiz- #5 • What shape does the graph r= 6-4cosθ make? A Lemniscate Limacon with loop Cardioid Limacon with dimple B C D

38. Quiz- #5 Try Again! • Limacons have the equation r= a-bcosθ. • What is the relationship between a and b? Try Again! or Review Material! or Go Home!

39. Quiz- #5 Correct! The answer is D: • a>b, in the equation r= a-bcosθ so the limacon has a dimple! Next Question!

40. Quiz- #6 • What is the graph of r=3sin4θ? A B C D

41. Quiz- #6 Try Again! • In a rose equation r= asin(nθ), what does the value “a” represent? “n”? • How does sinθ affect the graph? Try Again! or Review Material! or Go Home!

42. 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!

43. Quiz- #7 • What does the equation r2= a2sin2θ represent? A Circle Limacon Rose Lemniscate B C D

44. Quiz- #7 Try Again! • Which graph has an r2 value in its general equation? Try Again! or Review Material! or Go Home!

45. Quiz- #7 Correct! The answer is D: • Lemniscates are the only polar graphs with an r2 value in their general equation! Next Question!

46. Quiz- #8 • Which is NOT a way polar graphing is used today? A Differential/ Integral Calculus Physics and Arc Length Flight and Navigation All of the above are uses of polar graphing. B C D

47. Quiz- #8 Try Again! • Remember polar graphing has many uses! Try Again! or Review Material! or Go Home!

48. 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!

49. Quiz- #9 • In a general spiral equation r=aθ, a spiral is tighter for _______ “a” values and wider for ______ “a” values? A larger, smaller even, odd smaller, larger odd, even B C D

50. Quiz- #9 Try Again! • It is the size of the number “a” that shrinks or widens the spiral. Try Again! or Review Material! or Go Home!