what is the graph of the following function?\n$y = \\sin 2\\theta+1$

Explanation:

Step1: Identify the general form of sine - function

The general form of a sine function is $y = A\sin(Bx - C)+D$. For the function $y=\sin(2\theta)+1$, we have $A = 1$, $B = 2$, $C = 0$, and $D=1$.

Step2: Determine the amplitude

The amplitude of the sine - function is given by $|A|$. Since $A = 1$, the amplitude $|A|=1$.

Step3: Determine the period

The period of a sine function $y = A\sin(Bx - C)+D$ is $T=\frac{2\pi}{|B|}$. Here, $B = 2$, so $T=\frac{2\pi}{2}=\pi$.

Step4: Determine the vertical shift

The vertical shift of the function is given by $D$. Since $D = 1$, the graph of $y=\sin(2\theta)$ is shifted 1 unit up.

Step5: Analyze key - points

For $y=\sin(2\theta)+1$, when $2\theta=0$, $y = 1$; when $2\theta=\frac{\pi}{2}$, $y=2$; when $2\theta=\pi$, $y = 1$; when $2\theta=\frac{3\pi}{2}$, $y=0$; when $2\theta = 2\pi$, $y = 1$.

The graph of $y=\sin(2\theta)+1$ is a sine - wave with amplitude 1, period $\pi$, and vertical shift 1 unit up. Without seeing the specific options clearly, we know it should be a sine - like curve oscillating between $y = 0$ and $y = 2$ with a period of $\pi$ and centered around $y = 1$.

Answer:

(You need to choose the option among the given ones that matches the above - described characteristics. Since the options are not clearly readable, the above analysis helps in making the correct choice.)