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Understanding Graphs of Polar Equations: Impact of Parameters

Explore how the sizes of 'a' and 'b' impact the general shape of polar graphs. Learn the effects of trigonometric functions on graph orientation and direction. Discover ways to determine and plot x- and y-intercepts. Use this guide to interpret various types of polar equations.

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Understanding Graphs of Polar Equations: Impact of Parameters

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  1. GRAPHS OF THE POLAR EQUATIONSr = a ± b cos θr = a ± b sin θb ≠ 0

  2. HOW THE SIZES OF a, b IMPACT THE GENERAL SHAPE OF THE GRAPHr = a ± b cos θr = a ± b sin θ

  3. CIRCLEr = a + b cos θ, where |a|=0r = 2 cos θ

  4. LIMACON WITH LOOPr = a + b cos θ, where |a|<|b|r = 1 + 2 cos θ

  5. CARDIOIDr = a + b cos θ, where |a|=|b|r = 2 + 2 cos θ

  6. LIMACON WITH DIMPLEr = a + b cos θ, where |b|<|a|<2|b|r = 3 + 2 cos θ

  7. CONVEX LIMACONr = a + b cos θ, where |a|=2|b|r = 4 + 2 cos θ

  8. CONVEX LIMACONr = a + b cos θ, where |a|>2|b|r = 5 + 2 cos θ

  9. HOW THE TRIGONOMETRIC FUNCTION IMPACTS THE ORIENTATION OF THE GRAPHr = a ± b cos θr = a ± b sin θ

  10. r = a + b cos θ, where a > 0, b > 0r = 1 + 2 cosθ

  11. r = a + b sin θ, where a > 0, b > 0r = 1 + 2 sinθ

  12. HOW THE SIGNS OF a, b IMPACT THE DIRECTION OF THE GRAPHr = a ± b cos θr = a ± b sin θ

  13. r = a + b cos θ, where a > 0, b > 0r = 1 + 2 cos θ

  14. r = a + b cos θ, where a > 0, b < 0r = 1 - 2 cos θ

  15. r = a + b cos θ, where a < 0, b > 0r = -1 + 2 cos θ

  16. r = a + b cos θ, where a < 0, b < 0r = -1 - 2 cos θ

  17. HOW WOULD YOU FINDTHE x- AND y-INTERCEPTSOF A POLAR CURVE r = f(θ) ?

  18. FIND THE VALUES OF rFOR θ = 0, π/2, π AND 3π/2BE CAREFUL OFNEGATIVE VALUES OF r(THE INTERCEPTS WILL APPEAR ON THE “OPPOSITE” SIDE OF THE AXIS)

  19. GRAPHS OF THE POLAR EQUATIONSr = a ± b cos θr = a ± b sin θ 1. Determine the shape of the graph by comparing |a| to |b| and 2|b|2. Determine the axis of symmetry from the trigonometric function3. Determine the direction of the “fat” part from the sign of the trigonometric function4. Find and plot the x- and y-intercepts by finding the values of r for θ = 0, π/2, π AND 3π/25. Connect and form the appropriate shapeCircles, cardioids and limacons with loops pass through the pole; dimpled and convex limacons do not

  20. PUTTING IT TOGETHERGUESS THE GRAPHBY DETERMINING THE SHAPE,THE ORIENTATION,THE DIRECTION & THE INTERCEPTSr = -4 - 3 sin θ

  21. r = -4 - 3 sin θ

  22. PUTTING IT TOGETHERGUESS THE GRAPHBY DETERMINING THE SHAPE,THE ORIENTATION,THE DIRECTION & THE INTERCEPTSr = -4 + 4 cos θ

  23. r = -4 + 4 cos θ

  24. GRAPHS OF THE POLAR EQUATIONSr = a cos nθr = a sin nθ

  25. HOW THE VALUE OF n IMPACTS THE GENERAL SHAPE OF THE GRAPH r = a cos nθr = a sin nθ

  26. r = 3 cos θ

  27. r = 3 cos 2θ

  28. r = 3 cos 3θ

  29. r = 3 cos 4θ

  30. r = 3 cos 5θ

  31. r = 3 cos 6θ

  32. PUTTING IT TOGETHERGUESS THE EQUATIONOF A ROSE CURVE WITH[a] 16 petals[b] 15 petals[c] 14 petals

  33. PUTTING IT TOGETHER[a] r = a sin 8θ orr = a sin 8θ[b] r = a sin 15θ orr = a sin 15θ[c] no such rose curve

  34. HOW THE TRIGONOMETRIC FUNCTION IMPACTS THE ORIENTATION OF THE GRAPHr = a cos nθr = a sin nθ

  35. r = 3 cosθ

  36. r = 3 sinθ

  37. r = 3 cos 2θ

  38. r = 3 sin 2θ

  39. r = 3 cos 3θ

  40. r = 3 sin 3θ

  41. r = 3 cos 4θ

  42. r = 3 sin 4θ

  43. r = 3 cos 5θ

  44. r = 3 sin 5θ

  45. r = 3 cos 6θ

  46. r = 3 sin 6θ

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