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Chapter 6: Problem 31

Use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{3}{-4+2 \cos \theta}\)

Short Answer

Expert verified
The graph of the polar equation \(r = \frac{3}{-4 + 2\cos\theta}\) may vary depending on the exact feature sets of your graphing utility. One possible result could be a type of spiral or a lima莽on shape. Always examine your graph thoroughly for accurate identification.

Step by step solution

01

Use a graphing utility

First, put the given polar equation \(r = \frac{3}{-4 + 2\cos\theta}\) in the graphing utility. Make sure the mode of the calculator or online graphing tool is set to 'polar'.
02

Plot the graph

Make sure to adjust the range of the polar coordinates to ensure the graph can be seen clearly. Once the polar equation has been entered and the range has been adjusted, plot the graph.
03

Identify the graph

Lastly, examine the graph carefully. The shape, orientation and size, along with how the graph behaves when \(\theta\) changes, should give enough information to identify the graph.

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Most popular questions from this chapter

(d) Explain how the result of part (c) can be used as a sketching aid when graphing parabolas. Consider the parabola \(x^{2}=4 p y\) (a) Use a graphing utility to graph the parabola for \(p=1, p=2, p=3,\) and \(p=4\). Describe the effect on the graph when \(p\) increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola?

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(6 x^{2}+2 y^{2}+18 x-10 y+2=0\)

Describe the relationship between circles and ellipses. How are they similar? How do they differ?

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (0,2),(8,2)\(;\) minor axis of length 2

The chord perpendicular to the major axis at the center of the ellipse is called the _________ ____________ of the ellipse.
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