To graph the function $y = 3\cot(3x + \pi) - 2$, let's analyze its key properties.

Step 1: Identify Transformations

The parent function is $y = \cot x$, which has:

• Vertical asymptotes at $x = k\pi$ (where $k$ is an integer).
• Period of $\pi$.
• Zeroes at $x = \frac{\pi}{2} + k\pi$.

Now, let's analyze the transformations:

1. Horizontal Compression by a factor of 3

   • The period of $y = \cot(3x)$ is given by:
     $\frac{\pi}{|B|} = \frac{\pi}{3}$
   • This means that the cotangent function repeats every $\frac{\pi}{3}$.

2. Phase Shift

   • The phase shift is determined by setting $3x + \pi = 0$, solving for $x$: $x = -\frac{\pi}{3}$
   • This means the graph shifts left by $\frac{\pi}{3}$.

3. Vertical Stretch by 3

   • The amplitude is scaled by 3, making the function steeper.

4. Vertical Shift Down by 2

   • The entire graph moves down by 2 units.

Step 2: Locate Asymptotes

The vertical asymptotes occur where $3x + \pi = k\pi$, solving for $x$:

$x = \frac{k\pi - \pi}{3} = \frac{k\pi}{3} - \frac{\pi}{3}$

Thus, asymptotes occur at:

$x = \frac{k\pi}{3} - \frac{\pi}{3}, \quad k \in \mathbb{Z}$

Step 3: Plot Key Points

1. Find where $y = -2$ (midline).
2. Find x-values where $y = 0$ (cotangent zeroes).
3. Sketch the asymptotes and the decreasing behavior of cotangent.

Graph Description

• The function has vertical asymptotes at $x = \frac{k\pi}{3} - \frac{\pi}{3}$.
• The period is $\frac{\pi}{3}$, meaning it repeats every $\frac{\pi}{3}$.
• The graph is vertically stretched and shifted downward.
• The cotangent function is decreasing between asymptotes.

Would you like me to generate an actual graph for you? Let me know if you have any questions!

Further Exploration Questions

1. How would the graph change if we replaced $\cot$ with $\tan$?
2. What happens if we change the coefficient 3 in front of $x$ to another number?
3. How does a vertical shift affect the asymptotes of a cotangent function?
4. Can you find the x-intercepts of the transformed function?
5. How would the period change if the equation was $y = \cot(5x)$?

Tip:

Remember that cotangent is undefined where sine is zero, so always check for vertical asymptotes first when graphing!