A curve has equation y = 2 + 3 sin 1x for 0 ≤ x ≤ 4π.
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(a) State greatest and least values of y.

Given the equation \( y = 2 + 3\sin(x) \) for \( 0 \leq x \leq 4\pi \), let's determine the greatest and least values of \( y \).

### Analyzing the equation:
- The function \( \sin(x) \) oscillates between \(-1\) and \(1\).
- Therefore, the expression \( 3\sin(x) \) will oscillate between \( -3 \) and \( 3 \).

### Finding the greatest and least values of \( y \):
1. **Greatest value:**
   - When \( \sin(x) = 1 \), the maximum value of \( 3\sin(x) \) is \( 3 \).
   - Thus, \( y = 2 + 3 = 5 \).

2. **Least value:**
   - When \( \sin(x) = -1 \), the minimum value of \( 3\sin(x) \) is \( -3 \).
   - Thus, \( y = 2 - 3 = -1 \).

### Conclusion:
- The **greatest value** of \( y \) is \( 5 \).
- The **least value** of \( y \) is \( -1 \).

Would you like a detailed explanation or have any questions?

### Related Questions:
1. How would the range of \( y \) change if the equation were \( y = 3 + 2\sin(x) \)?
2. What is the period of the function \( y = 2 + 3\sin(x) \)?
3. How does the amplitude of the sine function affect the range of \( y \)?
4. What are the x-values corresponding to the maximum and minimum values of \( y \)?
5. How does shifting the sine function vertically affect the graph?

### Tip:
Remember, the amplitude of a sine or cosine function determines the distance between the maximum or minimum points and the horizontal axis (midline).

Explore the greatest and least values of y for the curve y = 2 + 3 sin(x) over the interval 0 ≤ x ≤ 4π. Understand how the sine function affects the amplitude and range of y with a detailed explanation and examples.

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