Graph the following function: 
y
=
tan
(
1
4
x
+
π
24
)
+
1
 
Drag the movable black point to shift the function, the red points to set the vertical asymptotes, and the blue point at the correct set of coordinates. You may click on a point to verify its coordinates. Note that only one period of the function is shown.

To graph the function \( y = \tan\left(\frac{1}{4}x + \frac{\pi}{24}\right) + 1 \), we need to understand its key characteristics, including the period, phase shift, vertical shift, and vertical asymptotes. Let's go through these steps:

### Step 1: Analyze the Function

**Standard Tangent Function:**  
The standard form of the tangent function is:
\[
y = \tan(bx + c)
\]
where \(b\) affects the period and \(c\) affects the phase shift.

**Given Function:**  
\[
y = \tan\left(\frac{1}{4}x + \frac{\pi}{24}\right) + 1
\]

1. **Period:**

   The period of the standard tangent function \(\tan(x)\) is \(\pi\). For a modified tangent function, the period \(P\) is given by:
   \[
   P = \frac{\pi}{b}
   \]
   where \(b = \frac{1}{4}\).

   So, the period is:
   \[
   P = \frac{\pi}{\frac{1}{4}} = 4\pi
   \]

2. **Phase Shift:**

   The phase shift \(\phi\) is calculated by solving:
   \[
   bx + c = 0 \quad \Rightarrow \quad \frac{1}{4}x + \frac{\pi}{24} = 0
   \]
   Solving for \(x\):
   \[
   \frac{1}{4}x = -\frac{\pi}{24} \quad \Rightarrow \quad x = -\frac{\pi}{6}
   \]

   Therefore, the phase shift is \(x = -\frac{\pi}{6}\).

3. **Vertical Shift:**

   The entire function is shifted upward by 1 unit due to the "+1" at the end.

4. **Vertical Asymptotes:**

   The vertical asymptotes of the tangent function occur where the argument is an odd multiple of \(\frac{\pi}{2}\):
   \[
   \frac{1}{4}x + \frac{\pi}{24} = \frac{\pi}{2} + k\pi \quad \text{for integer } k
   \]
   Solving for \(x\):
   \[
   \frac{1}{4}x = \frac{\pi}{2} - \frac{\pi}{24} + k\pi
   \]
   \[
   \frac{1}{4}x = \frac{12\pi}{24} - \frac{\pi}{24} + k\pi
   \]
   \[
   \frac{1}{4}x = \frac{11\pi}{24} + k\pi
   \]
   \[
   x = 4\left(\frac{11\pi}{24} + k\pi\right)
   \]
   \[
   x = \frac{11\pi}{6} + 4k\pi
   \]

   The vertical asymptotes occur at \(x = \frac{11\pi}{6} + 4k\pi\).

### Step 2: Graph the Function

Let's graph the function \(y = \tan\left(\frac{1}{4}x + \frac{\pi}{24}\right) + 1\):

- **Vertical Asymptotes:** Set the red points at \(x = \frac{11\pi}{6}\) and \(x = \frac{11\pi}{6} + 4\pi\).
- **Shift:** The function is shifted up by 1 unit, so adjust the movable black point accordingly.
- **Plotting the Points:** The tangent function will cross the midline at the phase shift and repeat every period (\(4\pi\)).

To visualize this graph, the key points are:

- **Period**: \(4\pi\)
- **Phase Shift**: \(-\frac{\pi}{6}\)
- **Vertical Shift**: 1 unit up

### Example Graph

Here's a rough sketch of one period of the function:

```plaintext
|
|            /
|          /
|        /
|      /
|    /
|  /
|/  
|   \        
|     \      
|       \    
|         \  
|           \
|             \
|
```

### Step 3: Drag the Points

1. **Movable Black Point:** Adjust the function vertically by 1 unit.
2. **Red Points (Vertical Asymptotes):** Position them at \(x = \frac{11\pi}{6}\) and \(x = \frac{11\pi}{6} + 4\pi\).
3. **Blue Point:** Place it at a point on the curve, for example at the x-intercept or another significant point like the maximum or minimum within the period.

### Final Coordinates

- **Phase Shift Coordinate:** \((- \frac{\pi}{6}, 1)\)
- **Midline Crossing:**  Adjust this coordinate accordingly.
- **Vertical Asymptotes:** At \(x = \frac{11\pi}{6}\) and \(x = \frac{11\pi}{6} + 4\pi\).

The graph should reflect the periodic and asymptotic nature of the tangent function with these adjustments. 

### Interactive Graph

For an interactive graph where you can adjust the points directly, consider using a graphing tool like [Desmos](https://www.desmos.com/calculator) or GeoGebra to visualize and manipulate the function.

---

## Follow-Up Questions

1. **How does the period of \(4\pi\) affect the function's behavior compared to the standard \(\pi\) period?**
2. **What impact does the vertical shift of +1 have on the range of the function?**
3. **How can the phase shift \(-\frac{\pi}{6}\) be represented on a unit circle?**
4. **What happens to the tangent function as \(x\) approaches each vertical asymptote?**
5. **How would changing \(\frac{1}{4}\) to another value affect the period?**
6. **What are the coordinates of the points where the function intersects its new midline?**
7. **Can we find a general formula for the vertical asymptotes of this function?**
8. **How does the transformation of the tangent function help in graphing other trigonometric functions?**

## Tip

Remember, when graphing trigonometric functions, always consider transformations like scaling, shifting, and periodic adjustments. Understanding how these affect the graph will make visualization much easier!

Learn how to graph the function y = tan(1/4x + π/24) + 1. Understand its period, phase shift, vertical shift, and vertical asymptotes with detailed step-by-step guidance. Suitable for high school students studying trigonometry.

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