he period of yequals9 sine 4 x is​ _______, so the​ x-values for the five key points are x 1equals​_______, x 2equals​_______, x 3equals​_______, x 4equals​_______, and x 5equals​_______.

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 \( y = \cos(bx) \).

The period of y = 9 sin(4x) is _______, so the x-values for the five key points are x1 = _______, x2 = _______, x3 = _______, x4 = _______, and x5 = _______.

Discover how to find the period of the trigonometric function y = 9 sin(4x) and calculate the x-values for the five key points. This problem involves trigonometric concepts and formulas, ideal for students in Grades 9-11.

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