Emily is using another wavy pattern on a large cake that she wants to be consistent. She decides to use the function $$ f(x)=4 \cot x $$ to create the pattern. Help Emily to determine its stretching factor and phase shift, and then graph it for two periods. Identify the correct pattern below. Enter the exact answers. Stretching factor = Number Phase shift: Click for List

Question

UpStudy AI · Accepted Answer

It seems that I am unable to generate a graph at this moment. However, I can describe how to graph the function $$ f(x) = 4 \cot x $$ for two periods.

### Steps to Graph $$ f(x) = 4 \cot x $$:

1. **Identify Key Points**:
   - The cotangent function has vertical asymptotes where $$ \cot x $$ is undefined, which occurs at $$ x = n\pi $$ for $$ n \in \mathbb{Z} $$. For the interval $$ [0, 2\pi] $$, the asymptotes are at $$ x = 0 $$, $$ x = \pi $$, and $$ x = 2\pi $$.
   - The function crosses the x-axis at $$ x = \frac{\pi}{2} $$ and $$ x = \frac{3\pi}{2} $$.

2. **Plot the Asymptotes**:
   - Draw vertical dashed lines at $$ x = 0 $$, $$ x = \pi $$, and $$ x = 2\pi $$.

3. **Plot the Key Points**:
   - At $$ x = \frac{\pi}{2} $$, $$ f\left(\frac{\pi}{2}\right) = 4 \cot\left(\frac{\pi}{2}\right) = 0 $$.
   - At $$ x = \frac{3\pi}{2} $$, $$ f\left(\frac{3\pi}{2}\right) = 4 \cot\left(\frac{3\pi}{2}\right) = 0 $$.

4. **Behavior Near Asymptotes**:
   - As $$ x $$ approaches $$ 0 $$ from the right, $$ f(x) $$ approaches $$ +\infty $$.
   - As $$ x $$ approaches $$ 0 $$ from the left, $$ f(x) $$ approaches $$ -\infty $$.
   - As $$ x $$ approaches $$ \pi $$ from the left, $$ f(x) $$ approaches $$ +\infty $$.
   - As $$ x $$ approaches $$ \pi $$ from the right, $$ f(x) $$ approaches $$ -\infty $$.
   - The same behavior occurs at $$ x = 2\pi $$.

5. **Sketch the Curve**:
   - Connect the points smoothly, ensuring the curve approaches the asymptotes and crosses the x-axis at the identified points.

### Summary of Results:
- **Stretching Factor**: $$ 4 $$
- **Phase Shift**: $$ 0 $$

If you have graphing software or a graphing calculator, you can input the function $$ f(x) = 4 \cot x $$ to visualize it accurately.
Stretching factor: 4 Phase shift: 0

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Trigonometry Questions