For a trigonometric equation 𝑦=15𝑡𝑎𝑛(𝜋𝑥3+2), find its stretching factor, period, phase shift, vertical asymptotes, and domain.

To analyze the trigonometric equation \( y = 15 \tan \left( \frac{\pi x}{3} + 2 \right) \), we will determine its stretching factor, period, phase shift, vertical asymptotes, and domain. 

### Stretching Factor
The stretching factor of the function is the coefficient that multiplies the tangent function. Here, the coefficient is 15. Therefore, the stretching factor is:
\[ \text{Stretching factor} = 15 \]

### Period
The period of the tangent function \(\tan(kx)\) is \(\frac{\pi}{|k|}\). In this equation, \(k = \frac{\pi}{3}\). Thus, the period \(P\) is:
\[ P = \frac{\pi}{\left| \frac{\pi}{3} \right|} = \frac{\pi}{\frac{\pi}{3}} = 3 \]

### Phase Shift
The phase shift of the tangent function \(\tan(kx + \phi)\) is given by \(-\frac{\phi}{k}\). Here, \( \phi = 2 \) and \( k = \frac{\pi}{3} \). Thus, the phase shift is:
\[ \text{Phase shift} = -\frac{2}{\frac{\pi}{3}} = -\frac{2 \cdot 3}{\pi} = -\frac{6}{\pi} \]

### Vertical Asymptotes
For the function \( y = a \tan(kx + \phi) \), vertical asymptotes occur where the argument \( kx + \phi \) is an odd multiple of \(\frac{\pi}{2}\), i.e.,
\[ kx + \phi = \left( n + \frac{1}{2} \right) \pi, \quad n \in \mathbb{Z} \]

Substituting \( k = \frac{\pi}{3} \) and \( \phi = 2 \):
\[ \frac{\pi x}{3} + 2 = \left( n + \frac{1}{2} \right) \pi \]
\[ \frac{\pi x}{3} = \left( n + \frac{1}{2} \right) \pi - 2 \]
\[ x = 3 \left( n + \frac{1}{2} \right) - \frac{6}{\pi} \]
\[ x = 3n + \frac{3}{2} - \frac{6}{\pi} \]

So, the vertical asymptotes occur at:
\[ x = 3n + \frac{3}{2} - \frac{6}{\pi}, \quad n \in \mathbb{Z} \]

### Domain
The domain of \( y = 15 \tan \left( \frac{\pi x}{3} + 2 \right) \) is all real numbers except where the vertical asymptotes occur. Therefore, the domain is:
\[ x \in \mathbb{R} \setminus \left\{ 3n + \frac{3}{2} - \frac{6}{\pi} \mid n \in \mathbb{Z} \right\} \]

### Summary
- **Stretching Factor:** \( 15 \)
- **Period:** \( 3 \)
- **Phase Shift:** \( -\frac{6}{\pi} \)
- **Vertical Asymptotes:** \( x = 3n + \frac{3}{2} - \frac{6}{\pi} \)
- **Domain:** \( x \in \mathbb{R} \setminus \left\{ 3n + \frac{3}{2} - \frac{6}{\pi} \mid n \in \mathbb{Z} \right\} \)

Would you like any further details or explanations?

Here are 8 related questions you might be interested in:
1. How do you derive the period of a trigonometric function in general?
2. What are the steps to determine the phase shift of a trigonometric function?
3. Can you explain how to find vertical asymptotes for other trigonometric functions like sine and cosine?
4. How does the stretching factor affect the graph of a trigonometric function?
5. What is the difference between horizontal and vertical shifts in trigonometric functions?
6. How do you find the domain and range of a trigonometric function?
7. Can you give an example of a trigonometric function with a different period and phase shift?
8. How would the graph of this function look like with its key features labeled?

**Tip:** When analyzing trigonometric functions, always start by identifying the key parameters: amplitude, period, phase shift, and vertical shift. This helps in sketching the graph accurately and understanding the function's behavior.

Explore the stretching factor, period, phase shift, vertical asymptotes, and domain of the trigonometric equation y = 15 tan(πx/3 + 2). Understand the mathematical properties and graphical representation of this function. Suitable for advanced high school students and beyond.

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