State the vertical shift, amplitude, period and phase shift of the function $y=3+2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]$ and then graph the function.

The given function is $y=3+2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]$ which can also be written as $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$.

A function of the form:

$y=a\sin b\left(\theta -h\right)+k,y=a\cos b\left(\theta -h\right)+k\text{and}y=a\tan b\left(\theta -h\right)+k$ has vertical shift $\left(k\right)$. And period $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$ for sine and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\pi }{\left|b\right|}$ for tangent function. The phase shift for the functions is $\left(h\right)$.

The amplitude of tangent and cotangent functions is not defined.

With the help of concept stated above, the vertical shift, amplitude, period and phase shift of the function is evaluated as:

The vertical shift of the function $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$is 3.

The amplitude of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$is $\left|2\right|=2$.

The period of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$ is $\frac{2\pi }{\left|2\right|}=\pi$.

The phase shift of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$ is $-\frac{\pi }{4}$.

The graph of the function $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$ is shown below.

The vertical shift of the function $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$is $3$.

The amplitude of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$is 2.

The period of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$ is $\pi$.

The phase shift of $y=2\sin \left[2\left(\theta +\frac{\pi }{4}\right)\right]+3$ is $-\frac{\pi }{4}$.