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Study Guides > Precalculus II

Solutions for Polar Coordinates: Graphs

Solutions to Try Its

1. The equation fails the symmetry test with respect to the line θ=π2\theta =\frac{\pi }{2} and with respect to the pole. It passes the polar axis symmetry test. 2. Tests will reveal symmetry about the polar axis. The zero is (0,π2)\left(0,\frac{\pi }{2}\right), and the maximum value is (3,0)\left(3,0\right). 3. Graph of the limaçon r=3-2cos(theta). Extending to the left. 4. The graph is a rose curve, nn even Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4. 5. Rose curve, nn odd Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin. 6.

Solutions to Odd-Numbered Exercises

1. Symmetry with respect to the polar axis is similar to symmetry about the xx -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line θ=π2\theta =\frac{\pi }{2} is similar to symmetry about the yy -axis. 3. Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at θ=0,π2,πand 3π2\theta =0,\frac{\pi }{2},\pi \text{and }\frac{3\pi }{2}, and sketch the graph. 5. The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation. 7. symmetric with respect to the polar axis 9. symmetric with respect to the polar axis, symmetric with respect to the line θ=π2\theta =\frac{\pi }{2}, symmetric with respect to the pole 11. no symmetry 13. no symmetry 15. symmetric with respect to the pole 17. circle Graph of given circle. 19. cardioid Graph of given cardioid. 21. cardioid Graph of given cardioid. 23. one-loop/dimpled limaçon Graph of given one-loop/dimpled limaçon 25. one-loop/dimpled limaçon Graph of given one-loop/dimpled limaçon 27. inner loop/two-loop limaçon Graph of given inner loop/two-loop limaçon 29. inner loop/two-loop limaçon Graph of given inner loop/two-loop limaçon 31. inner loop/two-loop limaçon Graph of given inner loop/two-loop limaçon 33. lemniscate Graph of given lemniscate (along horizontal axis) 35. lemniscate Graph of given lemniscate (along y=x) 37. rose curve Graph of given rose curve - four petals. 39. rose curve Graph of given rose curve - eight petals. 41. Archimedes’ spiral Graph of given Archimedes' spiral 43. Archimedes’ spiral Graph of given Archimedes' spiral 45. Graph of given equation. 47. Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin) 49. Graph of given equation. 51. Graph of given equation. Similar to original Archimedes' spiral. 53. Graph of given equation. 55. They are both spirals, but not quite the same. 57. Both graphs are curves with 2 loops. The equation with a coefficient of θ\theta has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4π4\pi to get a better picture. 59. When the width of the domain is increased, more petals of the flower are visible. 61. The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole. 63. The graphs are spirals. The smaller the coefficient, the tighter the spiral. 65. (4,π3),(4,5π3)\left(4,\frac{\pi }{3}\right),\left(4,\frac{5\pi }{3}\right) 67. (32,π3),(32,5π3)\left(\frac{3}{2},\frac{\pi }{3}\right),\left(\frac{3}{2},\frac{5\pi }{3}\right) 69. (0,π2),(0,π),(0,3π2),(0,2π)\left(0,\frac{\pi }{2}\right),\left(0,\pi \right),\left(0,\frac{3\pi }{2}\right),\left(0,2\pi \right) 71. (842,π4),(842,5π4)\left(\frac{\sqrt[4]{8}}{2},\frac{\pi }{4}\right),\left(\frac{\sqrt[4]{8}}{2},\frac{5\pi }{4}\right) and at θ=3π4,7π4\theta =\frac{3\pi }{4},\frac{7\pi }{4} since rr is squared

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