Problem 66 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Analyze the Graphed Function

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