Chapter SH, Problem 12.20EP

To determine

To find: The amplitude, period, phase shift, and vertical shift of given function. And then graph the function.

Answer to Problem 12.20EP

The amplitude of the given function is $2$ ,the period is $\pi$ and phase shift is $\left(\frac{\pi }{6}\right)$ and the vertical shift is $3$ and the graph is shown below.

Explanation of Solution

Given:

The given function is:

  $y=2\tan \left(\theta +30°\right)+3$

Consider the tangent function:

  $y=A\tan \left(B\left(x-C\right)\right)+D$

The amplitude and period of the tangent function is given as:

The amplitude is $A$ .

The period of tangent function is $\frac{\pi }{\left|B\right|}$ .

The vertical shift is $D$ .

The phase shift is $\frac{C}{B}$ .

By comparing the given function by the above function the amplitude and period are as given:

The amplitude of given function is $2$ .

The period is $\pi$ .

The phase shift is $\left(-\frac{\pi }{6}\right)$ .

The vertical shift is $3$ .

The graph of the given function is as:

  

Therefore the amplitude of the given function is $2$ ,the period is $\pi$ and phase shift is $\left(\frac{\pi }{6}\right)$ and the vertical shift is $3$ and the graph is shown above.