For each function, let's identify the values of $a$, $b$, and $c$ in the standard form of the sine function, $f(x) = a \sin(bx + c)$, and then use the formulas provided.

Function (a): $f(x) = -4 \sin(9x - 5\pi)$

1. Amplitude: The amplitude is given by $|a|$.

   • Here, $a = -4$, so $|a| = 4$.
   • Amplitude = 4

2. Period: The period is given by $\frac{2\pi}{|b|}$.

   • Here, $b = 9$, so $\frac{2\pi}{|b|} = \frac{2\pi}{9}$.
   • Period = $\frac{2\pi}{9}$

3. Phase Shift: The phase shift is given by $-\frac{c}{b}$.

   • Here, $c = -5\pi$ (since the expression is $9x - 5\pi$), so $-\frac{-5\pi}{9} = \frac{5\pi}{9}$.
   • Phase Shift = $\frac{5\pi}{9}$

Summary for (a):

• Amplitude = 4
• Period = $\frac{2\pi}{9}$
• Phase Shift = $\frac{5\pi}{9}$

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Function (b): $f(x) = 8 \sin(6 - 7\pi x)$

1. Amplitude: The amplitude is given by $|a|$.

   • Here, $a = 8$, so $|a| = 8$.
   • Amplitude = 8

2. Period: The period is given by $\frac{2\pi}{|b|}$.

   • Here, $b = -7\pi$, so $|b| = 7\pi$.
   • Period = $\frac{2\pi}{7\pi} = \frac{2}{7}$

3. Phase Shift: The phase shift is given by $-\frac{c}{b}$.

   • Here, $c = 6$ and $b = -7\pi$, so $-\frac{6}{-7\pi} = \frac{6}{7\pi}$.
   • Phase Shift = $\frac{6}{7\pi}$

Summary for (b):

• Amplitude = 8
• Period = $\frac{2}{7}$
• Phase Shift = $\frac{6}{7\pi}$

Let me know if you have questions or need further details.

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Additional Practice Questions:

1. Determine the amplitude, period, and phase shift for $g(x) = 3 \cos(4x + \pi)$.
2. What is the period of $h(x) = 5 \sin\left(\frac{1}{3}x\right)$?
3. Calculate the phase shift of $f(x) = 10 \sin(2x - \frac{\pi}{3})$.
4. If the function is $y = -7 \cos(5x + 2)$, what are the amplitude, period, and phase shift?
5. For $f(x) = -2 \sin(\pi x - \frac{5\pi}{4})$, determine the amplitude, period, and phase shift.

Tip:

Remember that the period of sine and cosine functions is directly influenced by the coefficient of $x$ inside the function. The higher this value, the shorter the period.