Create a table and graph the following: 10. $$ f(x)=4 \csc \left(\frac{x}{4}-\frac{\pi}{2}\right)+2 $$

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Function by following steps:
- step0: Determine the period:
  $$f\left(xight)=-4\sec\left(\frac{x}{4}ight)+2$$
- step1: Calculate:
  $$f\left(xight)=-4\sec\left(\frac{1}{4}xight)+2$$
- step2: The period of the function is $$\frac{2\pi }{\left|\frac{1}{4}ight|}:$$
  $$\frac{2\pi }{\left|\frac{1}{4}ight|}$$
- step3: Calculate:
  $$\frac{2\pi }{\frac{1}{4}}$$
- step4: Divide the terms:
  $$8\pi $$
Analyze the y intercept of the function $$ f(x)=4 \csc \left(\frac{x}{4}-\frac{\pi}{2}ight)+2 $$
Function by following steps:
- step0: Find the y-intercept:
  $$f\left(xight)=-4\sec\left(\frac{x}{4}ight)+2$$
- step1: Set $$x$$=0$$:$$
  $$f\left(0ight)=-4\sec\left(\frac{0}{4}ight)+2$$
- step2: Simplify:
  $$f\left(0ight)=-2$$
Analyze the x intercept of the function $$ f(x)=4 \csc \left(\frac{x}{4}-\frac{\pi}{2}ight)+2 $$
Function by following steps:
- step0: Find the $$x$$-intercept/zero:
  $$f\left(xight)=-4\sec\left(\frac{x}{4}ight)+2$$
- step1: Set $$f\left(xight)$$=0$$:$$
  $$0=-4\sec\left(\frac{x}{4}ight)+2$$
- step2: Swap the sides:
  $$-4\sec\left(\frac{x}{4}ight)+2=0$$
- step3: Move the constant to the right side:
  $$-4\sec\left(\frac{x}{4}ight)=0-2$$
- step4: Remove 0:
  $$-4\sec\left(\frac{x}{4}ight)=-2$$
- step5: Change the signs:
  $$4\sec\left(\frac{x}{4}ight)=2$$
- step6: Divide both sides:
  $$\frac{4\sec\left(\frac{x}{4}ight)}{4}=\frac{2}{4}$$
- step7: Divide the numbers:
  $$\sec\left(\frac{x}{4}ight)=\frac{1}{2}$$
- step8: Rearrange the terms:
  $$x 
otin \mathbb{R}$$
- step9: There is no $$x$$-intercept$$:$$
  $$	extrm{No x-intercept}$$
It seems that I am unable to generate a graph at this moment. However, I can guide you on how to plot the function using graphing software or tools like Desmos or GeoGebra.

### Steps to Plot the Function
1. **Open a Graphing Tool**: Use a graphing calculator or an online graphing tool like Desmos.
2. **Input the Function**: Enter the function $$ f(x) = 4 \csc \left( \frac{x}{4} - \frac{\pi}{2} ight) + 2 $$.
3. **Set the Range**: Adjust the x-axis to cover the interval from $$ 0 $$ to $$ 8\pi $$ (approximately $$ 25.13 $$).
4. **Observe the Graph**: Look for the vertical asymptotes where the function is undefined, and note the behavior at the y-intercepts and critical points.

### Summary of Key Points
- **Domain**: Excludes points where $$ x = 4n\pi + 2\pi $$.
- **Range**: $$ (-\infty, -2] \cup [6, \infty) $$.
- **Y-Intercept**: $$ f(0) = -2 $$.
- **X-Intercept**: None.
- **Periodicity**: The function has a period of $$ 8\pi $$.

If you have any specific questions or need further assistance, feel free to ask!
To graph the function $$ f(x) = 4 \csc \left( \frac{x}{4} - \frac{\pi}{2} ight) + 2 $$, follow these steps: 1. **Create a Table of Values**: - $$ x = 0 $$: $$ f(0) = -2 $$ - $$ x = 2\pi $$: $$ f(2\pi) = 6 $$ - $$ x = 4\pi $$: $$ f(4\pi) = -2 $$ - $$ x = 6\pi $$: $$ f(6\pi) = 6 $$ - $$ x = 8\pi $$: $$ f(8\pi) = -2 $$ 2. **Plot the Points** on a graph with the x-axis ranging from $$ 0 $$ to $$ 8\pi $$. 3. **Draw the Graph**: - The function has vertical asymptotes at $$ x = 4n\pi + 2\pi $$ for integer values of $$ n $$. - It reaches its maximum value of 6 and minimum value of -2. - The graph will have a period of $$ 8\pi $$. This will give you a clear visual representation of the function's behavior over one period.

Create a table and graph the following: 10. \( f(x)=4 \csc \left(\frac{x}{4}-\frac{\pi}{2}\right)+2 \)

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