Question

find the amplitude, period, and phase shift for ( y = 7sinleft(x+\frac{pi}{6}
ight) ). write the period and phase shift as an exact value involving ( pi ).
amplitude: 7
period:
phase shift: units to the left.

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\). For \(y = 7\sin(x+\frac{\pi}{6})\), we can rewrite it as \(y = 7\sin(1\times(x-(-\frac{\pi}{6})))+0\). Here, \(A = 7\), \(B = 1\), \(C=-\frac{\pi}{6}\).

Step2: Calculate the period

The formula for the period of \(y = A\sin(B(x - C))+D\) is \(T=\frac{2\pi}{|B|}\). Since \(B = 1\), then \(T=\frac{2\pi}{|1|}=2\pi\).

Step3: Calculate the phase - shift

The formula for the phase - shift of \(y = A\sin(B(x - C))+D\) is \(C\). Here, \(C =-\frac{\pi}{6}\), which means the phase - shift is \(\frac{\pi}{6}\) units to the left.

Answer:

Period: \(2\pi\)
Phase shift: \(\frac{\pi}{6}\)