consider the polar functions r = f(θ) and r = g(θ), where f(θ)=1 + cos(θ + π/3) and g(θ)=1 + cos θ. which of…

consider the polar functions r = f(θ) and r = g(θ), where f(θ)=1 + cos(θ + π/3) and g(θ)=1 + cos θ. which of the following describes the relationship between the graphs of r = f(θ) and r = g(θ) in the polar coordinate system for 0 ≤ θ ≤ 2π?

consider the polar functions r = f(θ) and r = g(θ), where f(θ)=1 + cos(θ + π/3) and g(θ)=1 + cos θ. which of the following describes the relationship between the graphs of r = f(θ) and r = g(θ) in the polar coordinate system for 0 ≤ θ ≤ 2π?

Answer

Explanation:

Step1: Recall polar - function transformation rules

The general form of a horizontal shift in a polar function $r = h(\theta)$ is $r=h(\theta +\alpha)$. If $r = g(\theta)$ and $r = f(\theta)=g(\theta+\alpha)$, the graph of $r = f(\theta)$ is a horizontal shift of the graph of $r = g(\theta)$. In the given functions, $f(\theta)=1 + \cos(\theta+\frac{\pi}{3})$ and $g(\theta)=1+\cos\theta$.

Step2: Determine the shift amount

For a polar function $r = \cos\theta$ and $r=\cos(\theta+\alpha)$, the graph of $r = \cos(\theta+\alpha)$ is a counter - clockwise rotation of the graph of $r=\cos\theta$ by an angle $\alpha$. Here, $\alpha=\frac{\pi}{3}$. So, the graph of $r = f(\theta)=1+\cos(\theta +\frac{\pi}{3})$ is the graph of $r = g(\theta)=1+\cos\theta$ rotated counter - clockwise by $\frac{\pi}{3}$ radians about the pole.

Answer:

The graph of $r = f(\theta)$ is the graph of $r = g(\theta)$ rotated counter - clockwise by $\frac{\pi}{3}$ radians about the pole.