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LESSON 9 –2

LESSON 9 –2. Graphs of Polar Equations. Five-Minute Check (over Lesson 9-1) TEKS Then/Now New Vocabulary Example 1: Graph Polar Equations by Plotting Points Key Concept: Symmetry of Polar Graphs Example 2: Real-World Example: Polar Axis Symmetry

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LESSON 9 –2

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  1. LESSON 9–2 Graphs of Polar Equations

  2. Five-Minute Check (over Lesson 9-1) TEKS Then/Now New Vocabulary Example 1: Graph Polar Equations by Plotting Points Key Concept: Symmetry of Polar Graphs Example 2: Real-World Example: Polar Axis Symmetry Key Concept: Quick Tests for Symmetry in Polar Graphs Example 3: Real-World Example: Symmetry with Respect to the Line=. Example 4: Symmetry, Zeros, and Maximum r-Values Concept Summary: Special Types of Polar Graphs Example 5: Identify and Graph Classic Curves Lesson Menu

  3. A. B. C. D. Graph S(−3, 60°) on a polar grid. 5–Minute Check 1

  4. Find three different pairs of polar coordinates that name if –2π ≤ θ ≤ 2π. A. B. C. D. 5–Minute Check 2

  5. Graph A. B. C. D. 5–Minute Check 3

  6. Which of the following represents the distance between (2, 150°) and (4, −45°)? A.2.13 B.4.91 C.5.95 D.35.45 5–Minute Check 4

  7. Targeted TEKS P.3(E) Graph polar equations such as cardiods, limaçons, or lemniscates by plotting points and using technology. Mathematical Processes P.1(F), P.1(G)

  8. You graphed functions in the rectangular coordinate system. (Lesson 1-2) • Graph polar equations. • Identify and graph classical curves. Then/Now

  9. limacon ΄ • cardioid • rose • lemniscate • spiral of Archimedes Vocabulary

  10. Graph Polar Equations by Plotting Points A. Graph r = 3 cos θ. Make a table of values to find the r-values corresponding to various values of  on the interval [0, 2π]. Round each r-value to the nearest tenth. Example 1

  11. Graph Polar Equations by Plotting Points Graph the ordered pairs (r, ) and connect them with a smooth curve. It appears that the graph is a circle with center (1.5, 0) and radius 1.5 units. Answer: Example 1

  12. Graph Polar Equations by Plotting Points B. Graph r = 3 sin θ. Make a table of values to find the r-values corresponding to various values of  on the interval [0, 2π]. Round each r-value to the nearest tenth. Example 1

  13. Graph Polar Equations by Plotting Points Graph the ordered pairs and connect them with a smooth curve. It appears that the graph is a circle with center (0, 1.5) and radius 1.5 units. Answer: Example 1

  14. A. B. C. D. Graph r = –4 cos θ. Example 1

  15. Key Concept 2

  16. Polar Axis Symmetry Use symmetry to graph r = 1 – 3cos θ. Replacing (r, ) with (r, –) yields r = 1 – 3 cos(–). Because cosine is an even function, cos (–) = cos , so this equation simplifies to r = 1 – 3 cos . Because the replacement produced an equation equivalent to the original equation, the graph of this equation is symmetric with respect to the polar axis. Because of this symmetry, you need only make a table of values to find the r-values corresponding to  on the interval [0, π]. Example 2

  17. This curve is called a limacon with an inner loop. ΄ Polar Axis Symmetry Plotting these points and using polar axis symmetry, you obtain the graph shown. Answer: Example 2

  18. A. B. C. D. Use symmetry to graph r = 1 + 2 cos . Example 2

  19. Key Concept 3

  20. Symmetry with Respect to the Line Because this polar equation is a function of the sine function, it is symmetric with respect to the line . Therefore, make a table and calculate the values of r on A. LIGHT TECHNOLOGY The area lit by two lights that shine down on a stage can be represented by the equation r = 1.5 + 1.5 sin θ. Suppose the front of the stage faces due south. Graph the polar pattern of the two lights. Example 3

  21. Symmetry with Respect to the Line Plotting these points and using symmetry with respect to the line , you obtain the graph shown. This curve is called a cardioid. Example 3

  22. Symmetry with Respect to the Line Answer: Example 3

  23. Symmetry with Respect to the Line B. LIGHT TECHNOLOGY The area lit by two lights that shine down on a stage can be represented by the equation r = 1.5 + 1.5 sin θ. Suppose the front of the stage faces due south. Describe what the polar pattern tells you about the two lights. Answer:Sample answer: The polar pattern indicates that the lights will light up a large portion toward the back of the stage but will not light up very much past the edge of the stage into the audience. Example 3

  24. AUDIO TECHNOLOGYA microphone was placed at the front of a stage to capture the sound from the acts performing during the senior talent show. The front of the stage faces due south. The area of sound the microphone captures can be represented by r = 2.5 + 2.5 sin . Describe what the polar pattern tells you about the microphone. Example 3

  25. A. The microphone will pick up a large portion of sound toward the back of the stage but not much from the front edge of the stage and audience. B. The microphone will pick up a large portion of sound toward the front of the stage and the audience but not much from the back of the stage. C. The microphone will pick up a large portion of sound on the right side of the stage and audience but not much from the left side. D. The microphone will pick up a large portion of sound on the left side of the stage and audience but not much from the right side. Example 3

  26. This function is symmetric with respect to the polar axis and the line , so you can find points on the interval and then use symmetry to complete the graph. Sketch the graph of the rectangular function y = 2 sin 2 on the interval Symmetry, Zeros, and Maximum r-Values Use symmetry, zeros, and maximum r-values to graph r = 2 sin 2θ. Example 4

  27. From the graph, you can see that |y| = 2 when and y = 0 when x = 0 and . Symmetry, Zeros, and Maximum r-Values Example 4

  28. Interpreting these results in terms of the polar equation r = 2sin 2, we can say that |r| has a maximum value of 2 when  = and r = 0 when  = 0 and . Symmetry, Zeros, and Maximum r-Values Use these and a few additional points to sketch the graph of the function. Example 4

  29. Notice that polar axis symmetry can be used to complete the graph after plotting points on . This type of curve is called a rose. Symmetry, Zeros, and Maximum r-Values Answer: Example 4

  30. A. symmetric to the line , |r| = 5 when B. symmetric to the polar axis, |r| = 5 when C. symmetric to the line , the polar axis, and the pole, | r | = 5 when D. symmetric to the line , the polar axis, and the pole, | r | = 5 when Determine the symmetry and maximum r-values of r = 5 sin 4for 0 ≤ θ < π. Example 4

  31. Key Concept 5

  32. Key Concept 5

  33. Identify and Graph Classic Curves A. Identify the type of curve given by r2 = 8 sin 2θ. Then use symmetry, zeros, and maximum r-values to graph the function. The equation is of the form r2 = a2 sin 2, so its graph is a lemniscate. Replacing (r, ) with (–r, ) yields (–r)2 = 8 sin 2 or r2 = 8 sin 2. Therefore, the function has symmetry with respect to the pole. Example 5

  34. The equation r2 = 8 sin 2 is equivalent to r = which is undefined when 2 sin 2 < 0. Therefore, the domain of the function is restricted to the intervals Because you can use pole symmetry, you need only graph points in the interval . The function attains a maximum r-value of |a| or when and zero r-value when x = 0 and Identify and Graph Classic Curves Example 5

  35. Identify and Graph Classic Curves Use these points and the indicated symmetry to sketch the graph of the function. Example 5

  36. Identify and Graph Classic Curves Answer:lemniscates; Example 5

  37. The equation is of the form r = a + b, so its graph is a spiral of Archimedes. Replacing (r, ) with (–r, –) yields (–r) = 2(–) or r = 2. Therefore, the function has symmetry with respect to the line However, since  > 0, this function will show no line symmetry. Identify and Graph Classic Curves B. Identify the type of curve given by r = 2θ, θ > 0. Then use symmetry, zeros, and maximum r-values to graph the function. Example 5

  38. Identify and Graph Classic Curves Spirals are unbounded. Therefore, the function has no maximum r-values and only one zero when  = 0. Use points on the interval [0, 4π] to sketch the graph of the function. Example 5

  39. Identify and Graph Classic Curves Answer:spiral of Archimedes; Example 5

  40. A. cardioid B. lemniscate C. limacon D. rose ΄ Identify the type of curve given by r = 4 cos 6θ. Example 5

  41. LESSON 9–2 Graphs of Polar Equations

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