Solutions for Polar Coordinates: Graphs

Solutions to Try Its

1. The equation fails the symmetry test with respect to the line [latex]\theta =\frac{\pi }{2}[/latex] 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 [latex]\left(0,\frac{\pi }{2}\right)[/latex], and the maximum value is [latex]\left(3,0\right)[/latex].

3.
Graph of the limaçon r=3-2cos(theta). Extending to the left.

4. The graph is a rose curve, [latex]n[/latex] even
Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.

5. Rose curve, [latex]n[/latex] 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 [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
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 [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