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Let's solve the problem step by step:

### Given Function:
\[ f(x) = 3\sin(2x) \]

### a. Amplitude of \( f(x) \)
The amplitude of a sine function \( y = a\sin(bx) \) is given by the absolute value of the coefficient \( a \).

For \( f(x) = 3\sin(2x) \), the amplitude is:
\[ \text{Amplitude} = |3| = 3 \]

### b. Period of \( f(x) \)
The period of a sine function \( y = a\sin(bx) \) is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function.

For \( f(x) = 3\sin(2x) \), the coefficient \( b \) is 2. Therefore, the period is:
\[ \text{Period} = \frac{2\pi}{2} = \pi \]

### c. Sketch the graph of \( f(x) \) for \( 0 \leq x \leq 2\pi \)
To sketch the graph:

1. The function has an amplitude of 3, so the graph will oscillate between \( -3 \) and \( 3 \).
2. The period is \( \pi \), so one complete cycle of the sine wave will occur over the interval \( [0, \pi] \). Since \( x \) ranges from \( 0 \) to \( 2\pi \), the graph will show two complete cycles.

### Graph Characteristics:
- At \( x = 0 \), \( f(0) = 3\sin(0) = 0 \).
- At \( x = \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 3\sin(\pi) = 0 \).
- At \( x = \frac{\pi}{4} \), \( f\left(\frac{\pi}{4}\right) = 3\sin\left(\frac{\pi}{2}\right) = 3 \).
- At \( x = \frac{3\pi}{4} \), \( f\left(\frac{3\pi}{4}\right) = 3\sin\left(\frac{3\pi}{2}\right) = -3 \).

Given these points, the graph will oscillate between \( 3 \) and \( -3 \) with a period of \( \pi \), completing two cycles from \( 0 \) to \( 2\pi \).

Would you like me to generate a sketch of this graph, or do you have any specific questions?

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Here are 5 follow-up questions based on the above discussion:

1. How would the graph of \( f(x) = 3\sin(2x) \) change if the amplitude were doubled?
2. What effect does changing the period have on the frequency of the sine wave?
3. Can you determine the phase shift of \( f(x) = 3\sin(2x) \) if we add a phase constant?
4. How would the graph of \( f(x) \) change if it were \( 3\cos(2x) \) instead of \( 3\sin(2x) \)?
5. How do you find the maximum and minimum values of the function \( f(x) \)?

**Tip:** When sketching trigonometric functions, always identify the amplitude, period, phase shift, and vertical shift to accurately determine the graph's shape and key points.

Consider the function f(x) = 3 sin(2x). a. State the amplitude of f(x). b. Determine the period of f(x). c. Sketch the graph of f(x), for 0 ≤ x ≤ 2π.

Explore the amplitude, period, and graph of the sine function f(x) = 3sin(2x). Learn how to determine these key features and sketch the graph over the interval 0 ≤ x ≤ 2π. Suitable for high school students in Grades 10-12.

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