Question

1. (4 pts) identify the amplitude, horizontal phase shift, vertical displacement, and period for the following equation. ( y = 5 + 2sin(6x + pi) ) amplitude ____ horizontal phase shift __ vertical displacement __ period ____

Explanation:

Step1: Recall the general form of a sine function

The general form of a sine function is \( y = A + B\sin(Cx + D) \), where:

• Amplitude is \( |B| \)
• Horizontal phase shift is \( -\frac{D}{C} \)
• Vertical displacement is \( A \)
• Period is \( \frac{2\pi}{|C|} \)

Step2: Identify the values of A, B, C, D from the given equation

For the equation \( y = 5 + 2\sin(6x + \pi) \), we have:

• \( A = 5 \)
• \( B = 2 \)
• \( C = 6 \)
• \( D = \pi \)

Step3: Calculate the amplitude

Amplitude \( = |B| = |2| = 2 \)

Step4: Calculate the horizontal phase shift

Horizontal phase shift \( = -\frac{D}{C} = -\frac{\pi}{6} \)

Step5: Calculate the vertical displacement

Vertical displacement \( = A = 5 \)

Step6: Calculate the period

Period \( = \frac{2\pi}{|C|} = \frac{2\pi}{6} = \frac{\pi}{3} \)

Answer:

Amplitude: \( 2 \)
Horizontal Phase Shift: \( -\frac{\pi}{6} \)
Vertical Displacement: \( 5 \)
Period: \( \frac{\pi}{3} \)