Questions: Given y=-9 sin (8 x-7)+6, determine: a. the amplitude. b. the period. c. the phase shift.

Solution Steps

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

• \( A \) is the amplitude.
• \( B \) affects the period.
• \( C \) determines the phase shift.
• \( D \) is the vertical shift.

For the given function \( y = -9 \sin(8x - 7) + 6 \), the amplitude \( A \) is the absolute value of the coefficient of the sine function. Thus: \[ \text{Amplitude} = |A| = |-9| = 9. \]

The period of a sine function is given by: \[ \text{Period} = \frac{2\pi}{|B|}. \] For the given function, \( B = 8 \), so: \[ \text{Period} = \frac{2\pi}{8} = \frac{\pi}{4}. \]

The phase shift of a sine function is calculated using: \[ \text{Phase Shift} = \frac{C}{B}. \] For the given function, \( C = 7 \) and \( B = 8 \), so: \[ \text{Phase Shift} = \frac{7}{8}. \] Since the function is in the form \( \sin(Bx - C) \), the phase shift is to the right by \( \frac{7}{8} \) units.

Final Answer