Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $$ y=\sin \left(\frac{1}{9} x\right) $$

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To graph the function $$ y = \sin\left(\frac{1}{9}x\right) $$ by hand using transformations of the standard sine function, follow these steps:

### 1. **Start with the Standard Sine Function**
The standard sine function is:
$$ y = \sin(x) $$
Its key features are:
- **Amplitude:** 1
- **Period:** $$ 2\pi $$
- **Key Points per Cycle:**
  - Start at $$ (0, 0) $$
  - Peak at $$ \left(\frac{\pi}{2}, 1\right) $$
  - Back to zero at $$ (\pi, 0) $$
  - Trough at $$ \left(\frac{3\pi}{2}, -1\right) $$
  - Return to zero at $$ (2\pi, 0) $$

### 2. **Identify the Transformation**
The given function modifies the standard sine function by altering its argument:
$$ y = \sin\left(\frac{1}{9}x\right) $$
This represents a **horizontal stretch** of the standard sine curve.

### 3. **Determine the Horizontal Stretch Factor**
The transformation inside the sine function affects the period of the sine wave. For a function $$ y = \sin(bx) $$, the period $$ T $$ is given by:
$$ T = \frac{2\pi}{b} $$

In your function:
$$ b = \frac{1}{9} $$
So the period becomes:
$$ T = \frac{2\pi}{\frac{1}{9}} = 18\pi $$

### 4. **Apply the Transformation**
- **Horizontal Stretch:** Multiply the x-coordinates of the key points of the standard sine function by the stretch factor (which is $$ \frac{1}{b} = 9 $$).

**Transformed Key Points:**
- Start at $$ (0, 0) $$
- Peak at $$ \left(\frac{\pi}{2} \times 9, 1\right) = \left(\frac{9\pi}{2}, 1\right) $$
- Back to zero at $$ (\pi \times 9, 0) = (9\pi, 0) $$
- Trough at $$ \left(\frac{3\pi}{2} \times 9, -1\right) = \left(\frac{27\pi}{2}, -1\right) $$
- Return to zero at $$ (2\pi \times 9, 0) = (18\pi, 0) $$

### 5. **Sketch the Graph**
1. **Draw the Axes:** Label the x-axis with multiples of $$ \pi $$ up to $$ 18\pi $$ to accommodate the new period.
2. **Plot the Transformed Key Points:** Mark the points calculated above.
3. **Draw the Sine Curve:** Connect the key points with a smooth, wave-like curve, maintaining the amplitude of 1.

### 6. **Review the Graph**
- **Amplitude:** Remains 1 (no vertical stretch/compression).
- **Period:** Increased to $$ 18\pi $$, meaning the sine wave completes one full cycle over a longer interval.
- **Frequency:** Decreased due to the longer period.

### **Final Graph Overview**
The graph of $$ y = \sin\left(\frac{1}{9}x\right) $$ is a horizontally stretched version of the standard sine curve, stretching the period from $$ 2\pi $$ to $$ 18\pi $$. This results in a wave that oscillates more slowly along the x-axis but maintains the same amplitude.

### **Visual Representation:**

```
y
|
1 |        *                   *                   *
  |       / \                 / \                 / \
0 |*-----/---\-----*-----*-----\---\-----*-----*-----\-----*-----*----> x
  |    /     \   / \   / \    /     \   / \    /     \   / \
-1|        *                   *                   *
|
```

*Note: The asterisks (*) represent the key points at intervals of $$ \frac{9\pi}{2} $$.

By following these steps, you can accurately graph $$ y = \sin\left(\frac{1}{9}x\right) $$ by hand using transformations of the standard sine function.
To graph $$ y = \sin\left(\frac{1}{9}x\right) $$, start with the standard sine curve and stretch it horizontally by a factor of 9. This increases the period to $$ 18\pi $$, making the wave oscillate more slowly along the x-axis while keeping the amplitude at 1.

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. \[ y=\sin \left(\frac{1}{9} x\right) \]

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