Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t). $f ( t ) = 2 + 5 \cos \left[ \frac { \pi } { 3 } ( t + 1 ) \right]$
A) $p = 3 , b = - 5 , d = 1 , a = 2$
B) $p = 6 , b = 5 , d = - 1 , a = 2$
C) $p = 3 , b = 5 , d = 1 , a = 2$
D) $p = 6 , b = - 5 , d = 1 , a = 2$

Question

Quizplus · Accepted Answer

The general form of a cosine function is $f(t) = a + b\cos(k(t + d))$, where $a$ is the vertical shift, $b$ is the amplitude, $k$ is related to the period by $p = \frac{2\pi}{|k|}$, and $d$ is the phase shift. For $f(t) = 2 + 5\cos\left[\frac{\pi}{3}(t + 1)\right]$, $a = 2$, $b = 5$, $k = \frac{\pi}{3}$ so $p = \frac{2\pi}{\pi/3} = 6$, and $d = -1$ (since the phase shift is to the left for $+1$ inside the cosine function).

Determine the Period P,the Amplitude B,the Phase Shift D,and the Vertical