Find the amplitude, the period in radians, the phase shift in radians,and the vertical shift. y=9sin(4Theta -(pi )/(6))+3 Select one: a. Amplitude: 9 Period: (pi )/(2) Phase shift: Right (pi )/(24) Vert. shift: Up 3 b. Amplitude: 9 Period: (pi )/(2) Phase shift: Left (pi )/(24) Vert. shift: Down 3 c. Amplitude: 9 Period: (pi )/(2) Phase shift: Right (pi )/(54) Vert. shift: Up 3 d. Amplitude: (1)/(7) Period: 2pi Phase shift: Right (7pi )/(6)

The given function is $y = 9\sin(4\theta - \frac{\pi}{6}) + 3$.<br /><br />* **Amplitude:** The amplitude is the absolute value of the coefficient of the sine function. In this case, the amplitude is $|9| = 9$.<br /><br />* **Period:** The period of a sine function is given by $\frac{2\pi}{B}$, where $B$ is the coefficient of $\theta$. In this case, $B = 4$, so the period is $\frac{2\pi}{4} = \frac{\pi}{2}$.<br /><br />* **Phase Shift:** The phase shift is determined by the term inside the sine function. We rewrite the argument of the sine function as $4(\theta - \frac{\pi}{24})$. The phase shift is given by the value subtracted from $\theta$. In this case, it is $\frac{\pi}{24}$. Since it is subtracted, the phase shift is to the *right* by $\frac{\pi}{24}$.<br /><br />* **Vertical Shift:** The vertical shift is the constant term added to the sine function. In this case, the vertical shift is 3. Since it is positive, the shift is *up* by 3.<br /><br />Therefore, the correct answer is:<br /><br />a. Amplitude: 9<br />Period: $\frac{\pi}{2}$<br />Phase shift: Right $\frac{\pi}{24}$<br />Vert. shift: Up 3<br /><br />Final Answer: The final answer is $\boxed{a}$