To graph the function \( y = 3 \sin \left( \frac{1}{3}x \right) + 4 \), let's break it down step by step:

### General form of a sine function:
The general form of a sine function is:

\[
y = A \sin(Bx - C) + D
\]

Where:
- \( A \) is the amplitude (vertical stretch or compression).
- \( B \) affects the period (horizontal stretch/compression).
- \( C \) is the phase shift (horizontal shift).
- \( D \) is the vertical shift.

### Identifying key parameters in \( y = 3 \sin \left( \frac{1}{3}x \right) + 4 \):
1. **Amplitude**: The coefficient of the sine function is \( A = 3 \). This means the amplitude is 3, so the sine wave will oscillate between \( 4 - 3 = 1 \) and \( 4 + 3 = 7 \) because of the vertical shift (see below).
   
2. **Period**: The period of a sine function is given by \( \frac{2\pi}{|B|} \). In this case, \( B = \frac{1}{3} \), so the period is:

\[
\text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi
\]

This means the sine wave repeats every \( 6\pi \) units along the x-axis.

3. **Vertical Shift**: The \( +4 \) means the entire graph is shifted up by 4 units. Instead of oscillating around 0, the sine wave now oscillates around \( y = 4 \).

4. **No Phase Shift**: There is no horizontal shift since \( C = 0 \).

### Steps to graph:
1. Start by plotting the midline, which is the line \( y = 4 \).
2. The wave oscillates between 1 and 7, since the amplitude is 3 and the vertical shift is 4.
3. The period is \( 6\pi \), so one full cycle of the sine wave happens between \( x = 0 \) and \( x = 6\pi \).
4. Key points for the sine wave:
   - At \( x = 0 \), \( y = 4 \) (midline).
   - At \( x = \frac{6\pi}{4} = \frac{3\pi}{2} \), \( y = 7 \) (maximum).
   - At \( x = 3\pi \), \( y = 4 \) (midline).
   - At \( x = \frac{9\pi}{2} \), \( y = 1 \) (minimum).
   - At \( x = 6\pi \), \( y = 4 \) (back to midline).

### Visual representation:
- The wave starts at \( y = 4 \), reaches a maximum of 7, a minimum of 1, and repeats every \( 6\pi \) units.

Would you like to explore more details on graphing trigonometric functions?

### Related questions:
1. How does changing the amplitude affect the graph of a sine function?
2. What happens if the period is reduced by increasing \( B \) in the sine equation?
3. How do you graph cosine functions, and how do they compare to sine functions?
4. What are phase shifts, and how do they affect trigonometric graphs?
5. How do you determine the period of other trigonometric functions like tangent?

**Tip**: When graphing trigonometric functions, always start by identifying key transformations like amplitude, period, and shifts.

Learn how to graph the sine function y = 3sin(1/3x) + 4 by identifying key transformations like amplitude, period, and vertical shift. This step-by-step solution covers important trigonometric concepts, such as sine wave oscillation and graphing techniques, suitable for students in Grades 10-12.

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