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Exploring Polar Equations: Graphs of Circles, Limaçons, and Rose Curves

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lisle
27 diapositives · 462 Views · Jul 30, 2014

This guide explores the various types of polar graphs, including circles, limaçons, and rose curves. We delve into the techniques for graphing these shapes by creating value tables, utilizing symmetry, and identifying maximum values and zeros. The specifics of sketching the graph of r = 6cos(θ) and such polar equations are discussed in detail, illustrating the unique features of each shape. Key concepts like symmetry with the polar axis and petal placement for rose curves are emphasized, offering comprehensive insight into the beauty of polar coordinates.

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  1. Section 10.8 Graphs of Polar Equations by Hand

  2. Types of Polar Graphs

  3. 1. Circle with center not at the pole. 2. Limaçon with and without a loop. 3. Cardioid Rose Curve

  4. Ways to Graph

  5. Making a table of values. Using symmetry. Finding the maximum value of r and the zeros.

  6. Quick Tests for Symmetry in Polar Coordinates

  7. Sketch the graph r = 6cos θ using all ways. Describe the graph in detail. A circle with the center at (3, 0) and a radius of 3. 1. Find the maximum r value. r = 6 cos 0 = 6 (6, 0)

  8. 2. The way to use a table is to find an exact ordered pair on the graph. When is cosine ½? Find r when

  9. 3. What symmetry will the graph have? Symmetry with the polar axis. Now reflect all the points you have across the polar axis.

  10. 5. Finally find the zeros of the graph. r = 6 cos θ 0 = 6 cos θ

  11. Now graph the circle.

  12. Sketch the graph r = 4sin θ.

  13. Sketch the graph r = 2 + 4sin  Describe the graph in detail. This is a limaçon with a loop. Find the maximum r, find the zeros, graph all points possible with exact integer values for r, and finally use symmetry to find other points.

  14. r = 2 + 4sin  Find the maximum r. r = 6 Find the tip of the loop. r = -2

  15. 3. Find the zeros. 0 = 2 + 4sin  Find the point with exact integer values for r. r = 4

  16. 4. r = 2 + 4sin 0p r = 2 (2, 0p) 5. Now use symmetry to find other points on the graph.

  17. Sketch the graph r = 4 – 4cos θ.

  18. Sketch the graph r = 6 – 4sin θ.

  19. Sketch the graph r = 4sin 2. Describe the graph in detail. A rose curve Since n is even, there will be 2n petals. A rose curve with 4 petals. Find the first petal by setting the equation equal to the maximum r then use the angle measure between petals to find the 4 petals.

  20. 4 = 4sin 2θ 1 = sin 2θ 2θ = 90° θ = 45° So the tip of a petal is at (4, 45°). Use this information and the angle measure between petals to find the other 3 petals.

  21. Sketch the graph r = 8cos 3. Since n is odd, there are n petals. So there are 3 petals. These 3 petals are 120° apart. Find the 1st petal the same way as before. The 1st petal is at (8, 0).