2. sketch a graph of g given by g(θ) = sin(θ) + 3.

Explanation:

Step1: Recall properties of y = sin(θ)

The function $y = \sin(\theta)$ has an amplitude of 1, a period of $2\pi$, and it oscillates between - 1 and 1 with $\sin(0)=0$, $\sin(\frac{\pi}{2}) = 1$, $\sin(\pi)=0$, $\sin(\frac{3\pi}{2})=-1$, $\sin(2\pi)=0$.

Step2: Analyze transformation

The function $g(\theta)=\sin(\theta)+3$ is a vertical - shift of the function $y = \sin(\theta)$ upwards by 3 units.

Step3: Find key - points

For $g(\theta)=\sin(\theta)+3$: When $\theta = 0$, $g(0)=\sin(0)+3=0 + 3=3$. When $\theta=\frac{\pi}{2}$, $g(\frac{\pi}{2})=\sin(\frac{\pi}{2})+3=1 + 3=4$. When $\theta=\pi$, $g(\pi)=\sin(\pi)+3=0 + 3=3$. When $\theta=\frac{3\pi}{2}$, $g(\frac{3\pi}{2})=\sin(\frac{3\pi}{2})+3=-1 + 3=2$. When $\theta = 2\pi$, $g(2\pi)=\sin(2\pi)+3=0 + 3=3$.

Step4: Sketch the graph

Plot the key - points $(0,3),(\frac{\pi}{2},4),(\pi,3),(\frac{3\pi}{2},2),(2\pi,3)$ and draw a smooth curve that oscillates between $y = 2$ and $y = 4$ with a period of $2\pi$.

Answer:

Sketch a sine - wave with key - points $(0,3),(\frac{\pi}{2},4),(\pi,3),(\frac{3\pi}{2},2),(2\pi,3)$ that oscillates between $y = 2$ and $y = 4$ with a period of $2\pi$.