Problem 36 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=2-4 \cos \theta$$

Step-by-Step Solution

Answer

The polar graph for the equation \(r=2-4 \cos \theta\) is a cardioid starting at \(r = 2\) at \(\theta = 0\), gravitying towards the origin at \(\theta = \frac{\pi}{3}\) and at \(\theta = \frac{5 \pi}{3}\), and extending to \(r = 6\) at \(\theta = \pi\).

Step 1: Determine Symmetry

The equation is not symmetric with respect to the origin because neither \(r = 2 - 4 \cos(-\theta)\) nor \(-r = 2 - 4 \cos(\theta)\) are equivalent to the original equation.

Step 2: Identify Zeros

To locate the zeros of the equation, we set \(r = 0\), which yields \(\cos \theta = \frac{2}{4}\), hence \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5 \pi}{3}\). Therefore, the graph touches the origin at angles of \(\frac{\pi}{3}\) and \(\frac{5 \pi}{3}\).

Step 3: Find Maximum r Values

The maximum \(r\) values are obtained by setting \(\frac{dr}{d\theta} = 0\). This gives us \(\theta = 0\) and \(\theta = \pi\). Substituting these values into the equation, we find \(r =2\) when \(\theta = 0\) and \(r =6\) when \(\theta = \pi\). These are the maximum distances from the origin the graph extends.

Step 4: Plot Additional Points

The equation can yield additional points to be graphed. For instance, at \(\theta = \frac{\pi}{2}\) we get \(r = 2 - 4 \times 0 = 2\), and at \(\theta = \frac{3 \pi}{2}\) we get \(r = 2 + 4 = 6\). These points aid in drawing an accurate graph.

Step 5: Sketch the Graph

The graph starts at \(r = 2\) at \(\theta = 0\), extends to \(r = 6\) at \(\theta = \pi\), and then back to \(r = 2\) at \(\theta = 2 \pi\). It gravitys towards the origin at \(\theta = \frac{\pi}{3}\) and at \(\theta = \frac{5 \pi}{3}\). The overall shape is a cardioid.