15 mark for review consider the polar functions r = f(θ) and r = g(θ), where f(θ)=1 + cos(θ + π/3) and…

15 mark for review 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? a the graphs of r = f(θ) and r = g(θ) are identical in shape and location. b the graph of r = f(θ) is a reflection of the graph of r = g(θ) over the positive x - axis (polar axis). c the graph of r = f(θ) is a translation of the graph of r = g(θ). d the graph of r = f(θ) is a rotation of the graph of r = g(θ) about the origin

15 mark for review 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? a the graphs of r = f(θ) and r = g(θ) are identical in shape and location. b the graph of r = f(θ) is a reflection of the graph of r = g(θ) over the positive x - axis (polar axis). c the graph of r = f(θ) is a translation of the graph of r = g(θ). d the graph of r = f(θ) is a rotation of the graph of r = g(θ) about the origin

Answer

Explanation:

Step1: Recall polar - function transformation rules

In polar coordinates, for a function (r = f(\theta)), if we have (r=f(\theta+\alpha)) ((\alpha\neq0)), it represents a rotation of the graph of (r = f(\theta)) about the origin. The function (r = f(\theta)=1+\cos(\theta +\frac{\pi}{3})) and (r = g(\theta)=1+\cos\theta). Here, the argument of the cosine function in (f(\theta)) is (\theta+\frac{\pi}{3}) compared to (\theta) in (g(\theta)).

Step2: Determine the transformation

When we change from (y = \cos\theta) to (y=\cos(\theta+\frac{\pi}{3})) in polar - coordinate functions (or in general, in trigonometric functions), the graph of (y = \cos(\theta+\frac{\pi}{3})) is obtained by rotating the graph of (y=\cos\theta) counter - clockwise by an angle of (\frac{\pi}{3}) about the origin. The same applies to the polar functions (r = 1+\cos(\theta+\frac{\pi}{3})) and (r = 1+\cos\theta).

Answer:

D. The graph of (r = f(\theta)) is a rotation of the graph of (r = g(\theta)) about the origin.