Question

identify and graph the polar equation. ( r = 4 + 4 cos \theta ) which graph represents the equation? a. b. c. d.

Explanation:

Step1: Analyze the form of the polar equation

The general form of a limacon in polar coordinates is \(r = a + b\cos\theta\). When \(a = b\), it is a cardioid. Here \(a = 4\), \(b = 4\), so \(r=4 + 4\cos\theta\) is a limacon (specifically a cardioid since \(a = b\)). The cardioid \(r=a + a\cos\theta\) is symmetric about the polar axis (\(\theta = 0\)).

Step2: Find key points

• When \(\theta=0\), \(r=4 + 4\cos(0)=4 + 4\times1 = 8\)
• When \(\theta=\frac{\pi}{2}\), \(r=4+4\cos(\frac{\pi}{2})=4+4\times0 = 4\)
• When \(\theta=\pi\), \(r=4 + 4\cos(\pi)=4+4\times(- 1)=0\)
• When \(\theta=\frac{3\pi}{2}\), \(r=4 + 4\cos(\frac{3\pi}{2})=4+4\times0 = 4\)

The graph of \(r = 4+4\cos\theta\) has a cusp at the pole (when \(r = 0\) at \(\theta=\pi\)) and is symmetric about the polar axis.

Answer:

C.