Consider the graph of y equals 6 minus 5 tan open parentheses 4 x minus straight pi close parentheses.

Determine the vertical stretch or compression, and if there is a reflection, period, phase shift, and vertical shift.

To analyze the function \( y = 6 - 5 \tan(4x - \pi) \), we need to identify its key characteristics, including vertical stretch or compression, reflection, period, phase shift, and vertical shift.

1. **Vertical Stretch or Compression:**
   The coefficient of the tangent function is \(-5\). This means there is a vertical stretch by a factor of 5, and because it is negative, there is also a reflection over the x-axis.

2. **Reflection:**
   The negative sign in front of the 5 indicates a reflection over the x-axis.

3. **Period:**
   The period of the tangent function is normally \(\pi\). The period is affected by the coefficient of \(x\) inside the tangent function. Here, the coefficient is 4. The period of the tangent function is given by:
   \[
   \text{Period} = \frac{\pi}{|b|} = \frac{\pi}{4}
   \]
   where \(b\) is the coefficient of \(x\) inside the tangent function.

4. **Phase Shift:**
   The phase shift is determined by the horizontal translation inside the argument of the tangent function. In \( \tan(4x - \pi) \), the phase shift is found by setting \(4x - \pi = 0\) and solving for \(x\):
   \[
   4x - \pi = 0 \implies x = \frac{\pi}{4}
   \]
   This indicates a phase shift to the right by \(\frac{\pi}{4}\).

5. **Vertical Shift:**
   The vertical shift is determined by the constant term added or subtracted from the function. Here, it is \(6\). This means the entire graph is shifted up by 6 units.

Summarizing all these transformations:
- **Vertical Stretch:** By a factor of 5
- **Reflection:** Over the x-axis
- **Period:** \(\frac{\pi}{4}\)
- **Phase Shift:** Right by \(\frac{\pi}{4}\)
- **Vertical Shift:** Up by 6 units

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How to graph \( y = 6 - 5 \tan(4x - \pi) \)?
2. What is the effect of changing the coefficient in front of the tangent function?
3. How to determine the asymptotes of the tangent function after transformations?
4. How to find the x-intercepts of \( y = 6 - 5 \tan(4x - \pi) \)?
5. How do transformations affect the domain and range of the tangent function?

**Tip:** When analyzing trigonometric functions, always start by identifying the basic function's properties before applying transformations.

Explore the transformations of the function y = 6 - 5 tan(4x - π), including vertical stretch, reflection, period, phase shift, and vertical shift. This problem involves trigonometric concepts and graphing functions in high school mathematics.

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