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.
4. The graph is a rose curve, [latex]n[/latex] even
5. Rose curve, [latex]n[/latex] odd
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
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 squaredLicenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
