We are given the function $y = 9 \sin(4x)$. To find the period and the x-values for the five key points, we need to analyze the function.

Step 1: Find the Period

The general form of a sine function is:

$y = a \sin(bx)$

where:

• $a$ is the amplitude,
• $b$ affects the period,
• The period of a sine function is given by:

$\text{Period} = \frac{2\pi}{|b|}$

In our case, $b = 4$. So, the period of the function is:

$\text{Period} = \frac{2\pi}{4} = \frac{\pi}{2}$

Thus, the period of the function is $\frac{\pi}{2}$.

Step 2: Determine the Five Key Points

For one period of a sine function, the five key points occur at:

• $0$,
• $\frac{1}{4} \times \text{Period}$,
• $\frac{1}{2} \times \text{Period}$,
• $\frac{3}{4} \times \text{Period}$,
• $\text{Period}$.

Substituting the period $\frac{\pi}{2}$ into these expressions:

1. $x_1 = 0$
2. $x_2 = \frac{1}{4} \times \frac{\pi}{2} = \frac{\pi}{8}$
3. $x_3 = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}$
4. $x_4 = \frac{3}{4} \times \frac{\pi}{2} = \frac{3\pi}{8}$
5. $x_5 = \frac{\pi}{2}$

Final Answer:

• The period is $\frac{\pi}{2}$.
• The five key x-values are:

$x_1 = 0, \quad x_2 = \frac{\pi}{8}, \quad x_3 = \frac{\pi}{4}, \quad x_4 = \frac{3\pi}{8}, \quad x_5 = \frac{\pi}{2}.$

Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. What is the amplitude of the function $y = 9 \sin(4x)$?
2. How would the period change if the equation were $y = 9 \sin(2x)$?
3. How do you find the phase shift of a sine function?
4. What are the key points for the cosine function $y = \cos(4x)$?
5. How does the period affect the graph of the sine function?

Tip: The period of a sine or cosine function is always inversely proportional to the value of $b$ in $y = \sin(bx)$ or $y = \cos(bx)$.