Answer:
B. [tex]f(x)=9\sin (\frac{x}{3})+3[/tex]
Step-by-step explanation:
We are given that,
The frequency of the function is [tex]\frac{1}{6\pi}[/tex]
The maximum and minimum value is 12 and -6.
Also, the y-intercept is 3.
From the options, we have,
Options C and D have minimum value 6. So, they does not represent the given function.
We know, 'If a function has a period P, then the function [tex]a+f(bx+c)[/tex] will have the period [tex]\frac{P}{|b|}[/tex].
Also, 'The frequency is the reciprocal of the period'.
So, function [tex]a+f(bx+c)[/tex] will have the frequency [tex]\frac{|b|}{P}[/tex].
From the options, we see,
Option B have the frequency, [tex]\frac{\frac{1}{3}}{2\pi}[/tex] i.e. [tex]\frac{1}{6\pi}[/tex].
Option A have the frequency, [tex]\frac{6\pi}{2\pi}[/tex] i.e. 3
Thus, option A is not correct.
Hence, option B is the required sinusoidal function.
