91Ó°ÊÓ

The given function is $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$.

A function of the form:

$y=a\sin b\left(\mathrm{θ}-h\right)+k,y=a\cos b\left(\mathrm{θ}-h\right)+k\text{and}y=a\tan b\left(\mathrm{θ}-h\right)+k$

has amplitude of $\left|a\right|$, vertical shift $\left(k\right)$ and period $\frac{360°}{\left|b\right|}\text{or}\frac{2\mathrm{π}}{\left|b\right|}$ for sine and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\mathrm{π}}{\left|b\right|}$ for tangent function. The phase shift for the functions is $\left(h\right)$.

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 $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$is $10$ and a positive vertical shift of the function indicates that the vertical shift is upwards.

The amplitude of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$is $\left|3\right|=3$.

The period of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$ is $\frac{2\mathrm{π}}{\left|2\right|}=\mathrm{π}$.

The phase shift of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$ is $30°$ and a positive value of phase shift indicates that the phase shift is to the right.

The graph of the function $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$ is shown below.

The vertical shift of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$is 10.

The amplitude of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$is 3.

The period of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$ is $\mathrm{π}$.

The phase shift of $y=3\sin \left[2\left(\mathrm{θ}-30°\right)\right]+10$ is $30°$.