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Chapter 7: Problem 66

Use a graphing utility to graph the polar equation. $$r=2+4 \cos \theta$$

Short Answer

Expert verified
The graph is a circle with a diameter that goes from the pole to the point (4, 0).

Step by step solution

01

Transformation to Cartesian Coordinates

Polar equations are typically expressed in terms of \( r \) and \( \theta \). Cartesian coordinates on the other hand are expressed in terms of \( x \) and \( y \). The given equation is a polar equation, \( r = 2 + 4 \cos \theta \). It is easier to analyze this function when it is presented in Cartesian form. The standard transformations are \( x = r \cos \theta \) and \( y = r \sin \theta \). Plug these into equation, squaring and adding the two results and simplifying, the Cartesian equation is obtained.
02

Sketch the Polar Curve

Use a graphing utility to sketch the Cartesian function resulting from the transformation. Note the interval and labels on the axes. The shape should be a circle with a radius of \( 2 \), centered at \( (2,0) \). Plot plenty of points for accuracy.
03

Analyze the Graphed Function

Analyze the finished graph, focusing on any key points, the shape, and any symmetry.

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Most popular questions from this chapter

Explain how to find the product of two complex numbers in polar form.

Test for symmetry and then graph each polar equation. $$r=\frac{2}{1-\cos \theta}$$

Use a graphing utility to graph the polar equation. $$r=2+4 \sin \theta$$

Exercises \(116-118\) will help you prepare for the material covered in the next section. Use the distance formula to determine if the line segment with endpoints \((-3,-3)\) and \((0,3)\) has the same length as the line segment with endpoints \((0,0)\) and \((3,6)\)

Exercises \(99-101\) will help you prepare for the material covered in the next section. Refer to Section 1.4 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i .\) $$\frac{2+2 i}{1+i}$$
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