Solutions to Odd-Numbered Exercises

1. Symmetry with respect to the polar axis is similar to symmetry about the [latex]x[/latex] -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] is similar to symmetry about the [latex]y[/latex] -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 [latex]\theta =0,\frac{\pi }{2},\pi \text{and }\frac{3\pi }{2}[/latex], 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 [latex]\theta =\frac{\pi }{2}[/latex], symmetric with respect to the pole

11. no symmetry

13. no symmetry

15. symmetric with respect to the pole

17. circle

19. cardioid

21. cardioid

23. one-loop/dimpled limaçon

25. one-loop/dimpled limaçon

27. inner loop/two-loop limaçon

29. inner loop/two-loop limaçon

31. inner loop/two-loop limaçon

33. lemniscate

35. lemniscate

37. rose curve

39. rose curve

41. Archimedes’ spiral

43. Archimedes’ spiral

45.

47.

49.

51.

53.

55. They are both spirals, but not quite the same.

57. Both graphs are curves with 2 loops. The equation with a coefficient of [latex]\theta[/latex] has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to [latex]4\pi[/latex] 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. [latex]\left(4,\frac{\pi }{3}\right),\left(4,\frac{5\pi }{3}\right)[/latex]

67. [latex]\left(\frac{3}{2},\frac{\pi }{3}\right),\left(\frac{3}{2},\frac{5\pi }{3}\right)[/latex]

69. [latex]\left(0,\frac{\pi }{2}\right),\left(0,\pi \right),\left(0,\frac{3\pi }{2}\right),\left(0,2\pi \right)[/latex]

71. [latex]\left(\frac{\sqrt[4]{8}}{2},\frac{\pi }{4}\right),\left(\frac{\sqrt[4]{8}}{2},\frac{5\pi }{4}\right)[/latex]
and at [latex]\theta =\frac{3\pi }{4},\frac{7\pi }{4}[/latex] since [latex]r[/latex] is squared