Question

Given the equation y = 8sin(pi/6 * x + (7pi)/6) + 4

Answer 1

In general, the sine function can be expressed as shown below

$\begin{gathered} y=Asin(B(x-C))+D \\ A\rightarrow amplitude \\ B\rightarrow period=\frac{2\pi}{B} \\ C\rightarrow\text{ horizontal shift} \\ D\rightarrow\text{ vertical shift} \end{gathered}$

Then, in our case,

$\begin{gathered} y=8sin(\frac{\pi}{6}x+\frac{7\pi}{6})+4 \\ \Rightarrow \\ Amplitude=8 \\ Period=\frac{2\pi}{\frac{\pi}{6}} \\ Horizontal\text{ shift}=-\frac{(\frac{7\pi}{6})}{\frac{\pi}{6}} \\ Vertical\text{ shift}=4 \end{gathered}$

Simplifying,

$\begin{gathered} \Rightarrow Amplitude=8 \\ Period=12 \\ Horizontal\text{ shift}=-7 \end{gathered}$

As for the midline, remember that the maximum/minimum of sin(x) is +1/-1; therefore, the midline is

$midline:y=\frac{(8+4)+(-8+4)}{2}=\frac{8}{2}=4$

The answers are

Amplitude=8

Period=12

Horizontal shift: 7 units to the left

Midline y=4