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The problem asks to determine the period of the function \( y = 4 \sin(2x) \) and find the five key points on the x-axis.

### Step 1: Find the period
For a sine function of the form \( y = A \sin(Bx) \), the period is given by the formula:
\[
\text{Period} = \frac{2\pi}{|B|}
\]
Here, \( B = 2 \), so the period is:
\[
\text{Period} = \frac{2\pi}{2} = \pi
\]

### Step 2: Determine the five key points
The key points for a sine function are typically at:
- \( x_1 \): Start of the period
- \( x_2 \): First quarter of the period (maximum point)
- \( x_3 \): Half of the period (zero point)
- \( x_4 \): Three-quarters of the period (minimum point)
- \( x_5 \): Full period

Since the period is \( \pi \), these points occur as follows:
\[
x_1 = 0, \quad x_2 = \frac{\pi}{4}, \quad x_3 = \frac{\pi}{2}, \quad x_4 = \frac{3\pi}{4}, \quad x_5 = \pi
\]

### Final Answer:
- The period of \( y = 4 \sin(2x) \) is \( \pi \).
- The five key x-values are:
  - \( x_1 = 0 \)
  - \( x_2 = \frac{\pi}{4} \)
  - \( x_3 = \frac{\pi}{2} \)
  - \( x_4 = \frac{3\pi}{4} \)
  - \( x_5 = \pi \)

Let me know if you need further details or have questions! Here are some related questions to deepen your understanding:

1. What effect does the amplitude \( A = 4 \) have on the graph of \( y = 4 \sin(2x) \)?
2. How does the value of \( B \) in \( y = A \sin(Bx) \) affect the graph's frequency?
3. What are the corresponding y-values at each of the key points for the function \( y = 4 \sin(2x) \)?
4. How would the key points change if the function was \( y = \sin(2x) \) without the amplitude?
5. How can you calculate the key points if the sine function had a phase shift?

### Tip:
Always remember that the period of \( \sin(Bx) \) or \( \cos(Bx) \) is calculated by \( \frac{2\pi}{|B|} \), while vertical stretching or shrinking is controlled by the amplitude \( A \).

The period of y = 4 sin 2x is ______, so the x-values for the five key points are x₁ = ______, x₂ = ______, x₃ = ______, x₄ = ______, and x₅ = ______.

Solve for the period and key points of the sine function y = 4 sin 2x. Learn how to calculate the period and find the five key x-values, including 0, π/4, π/2, 3π/4, and π, based on the function's periodicity. Suitable for high school students in Grades 10-12.

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