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. $$ y=\sin \left(\frac{1}{9} x\right) $$

Question

UpStudy AI · Accepted Answer

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

Begin with the basic sine function:

$$
y = \sin(x)
$$

This graph has:

- **Amplitude**: 1
- **Period**: $$ 2\pi $$
- **Phase Shift**: 0
- **Vertical Shift**: 0

### 2. Identify the Transformation

The given function is:

$$
y = \sin\left(\frac{1}{9}x\right)
$$

This can be rewritten as:

$$
y = \sin\left(\frac{x}{9}\right)
$$

Comparing this with the general form of the sine function:

$$
y = \sin(bx)
$$

We see that $$ b = \frac{1}{9} $$.

### 3. Determine the Effect of the Transformation

The coefficient $$ b $$ affects the **period** of the sine function. The period $$ T $$ is calculated as:

$$
T = \frac{2\pi}{|b|} = \frac{2\pi}{\frac{1}{9}} = 18\pi
$$

**Interpretation**:
- Since $$ b = \frac{1}{9} < 1 $$, the graph is **stretched horizontally** by a factor of $$ \frac{1}{b} = 9 $$.
- The **frequency** decreases, resulting in a longer period.

### 4. Apply the Transformation to the Graph

**Standard Sine Function Characteristics**:
- **Starts at**: $$ (0, 0) $$
- **First peak**: $$ \left(\frac{\pi}{2}, 1\right) $$
- **First zero crossing**: $$ (\pi, 0) $$
- **First trough**: $$ \left(\frac{3\pi}{2}, -1\right) $$
- **Completes one cycle at**: $$ (2\pi, 0) $$

**Transformed Function Characteristics**:
- **Amplitude**: Remains 1 (no vertical stretch/compression)
- **Period**: $$ 18\pi $$
  
**Key Points**:
- **Starts at**: $$ (0, 0) $$
- **First peak**: $$ \left(9 \times \frac{\pi}{2}, 1\right) = \left(\frac{9\pi}{2}, 1\right) $$
- **First zero crossing**: $$ 9\pi $$
- **First trough**: $$ \left( \frac{27\pi}{2}, -1 \right) $$
- **Completes one cycle at**: $$ 18\pi $$

### 5. Sketch the Graph

1. **Draw the Axes**:
   - $$ x $$-axis and $$ y $$-axis.
   - Label the axes appropriately.

2. **Mark Key Points**:
   - Plot the transformed key points:
     - $$ (0, 0) $$
     - $$ \left(\frac{9\pi}{2}, 1\right) $$
     - $$ (9\pi, 0) $$
     - $$ \left(\frac{27\pi}{2}, -1\right) $$
     - $$ (18\pi, 0) $$

3. **Draw the Sine Curve**:
   - Start at $$ (0, 0) $$.
   - Rise to the first peak at $$ \left(\frac{9\pi}{2}, 1\right) $$.
   - Descend back to zero at $$ (9\pi, 0) $$.
   - Continue to the trough at $$ \left(\frac{27\pi}{2}, -1\right) $$.
   - Return to zero at $$ (18\pi, 0) $$.
   - Continue the pattern as needed for multiple cycles.

4. **Add Smooth Curves**:
   - Ensure the sine wave smoothly oscillates through the key points.
   - The wave should be symmetric around the centerline (the $$ x $$-axis in this case).

### 6. Final Graph

Here's a conceptual sketch of the graph:

```
y
|
1  |                      *                      *
   |                    *   *                  *   *
0  |-------------------*-------*------------*-------*-----------
   |                *           *        *           *
-1 |              *               *  *               *
   |
   |________________________________________________________ x
          0        9π/2      9π      27π/2      18π
```

- Each `*` represents key points on the sine curve.
- The wave completes one full cycle from $$ 0 $$ to $$ 18\pi $$.
- The graph oscillates between $$ y = 1 $$ and $$ y = -1 $$.

### 7. Additional Notes

- **Amplitude**: Remains unchanged at 1; the graph does not stretch or compress vertically.
- **Period**: Extended to $$ 18\pi $$, making each cycle longer than the standard sine function.
- **Frequency**: Decreased since the sine wave cycles less frequently over the same $$ x $$-interval compared to the standard sine function.

By understanding how the coefficient $$ \frac{1}{9} $$ affects the period of the sine function, you can effectively graph $$ y = \sin\left(\frac{1}{9}x\right) $$ using basic transformations of the standard sine curve.
To graph $$ y = \sin\left(\frac{1}{9}x\right) $$, start with the standard sine function and stretch it horizontally by a factor of 9. This increases the period to $$ 18\pi $$. Plot key points at $$ (0, 0) $$, $$ \left(\frac{9\pi}{2}, 1\right) $$, $$ (9\pi, 0) $$, $$ \left(\frac{27\pi}{2}, -1\right) $$, and $$ (18\pi, 0) $$, then draw a smooth sine wave through these points.

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. \[ y=\sin \left(\frac{1}{9} x\right) \]

Real Tutor Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Trigonometry Questions