To create a complete graph of the function $f(x) = 3e^{-x} \sin(x)$ over the interval $[-\pi, \pi]$, let's follow these graphing guidelines:

Step 1: Identify Key Characteristics

1. Domain: The function $f(x) = 3e^{-x} \sin(x)$ is defined on the entire real line, but we are only interested in $x \in [-\pi, \pi]$.
2. Range: Since $e^{-x}$ is always positive and $\sin(x)$ oscillates between -1 and 1, the range will depend on this oscillation and the exponential decay/growth.
3. Symmetry: The function is neither even nor odd, so it lacks symmetry around the origin or the y-axis.
4. Asymptotes: There are no vertical or horizontal asymptotes within this interval, though $e^{-x}$ generally trends towards zero as $x$ increases for large values beyond this interval.
5. Behavior at Critical Points: We'll identify critical points by finding the derivative of $f(x)$ and setting it to zero.

Step 2: First Derivative and Critical Points

To find the critical points, calculate $f'(x)$ using the product rule.

1. $f(x) = 3e^{-x} \sin(x)$
2. Using the product rule, $f'(x) = 3 \left( -e^{-x} \sin(x) + e^{-x} \cos(x) \right) = 3e^{-x}(-\sin(x) + \cos(x))$.

Set $f'(x) = 0$: $3e^{-x}(-\sin(x) + \cos(x)) = 0.$ Since $e^{-x} \neq 0$ for real $x$, we get $-\sin(x) + \cos(x) = 0$ or $\tan(x) = 1$.

Step 3: Solving for $x$

Within $x \in [-\pi, \pi]$, the solutions to $\tan(x) = 1$ are:

1. $x = \frac{\pi}{4}$
2. $x = -\frac{3\pi}{4}$.

Step 4: Second Derivative and Concavity

For concavity analysis, calculate the second derivative $f''(x)$ and determine where it changes sign. This will indicate intervals of concave up or down, aiding in understanding the curve’s behavior.

Step 5: Plotting the Graph

• Plot points at key locations: Evaluate $f(x)$ at $x = -\pi, -\frac{3\pi}{4}, 0, \frac{\pi}{4}, \pi$.
• Determine concavity: Use the second derivative to establish concave up/down intervals.
• Sketch the function: The graph should oscillate due to $\sin(x)$, with amplitudes modulated by $e^{-x}$.

Would you like a plotted graph for this function, or further analysis on any of these steps?

Here are five related questions for deeper understanding:

1. How does the behavior of $e^{-x}$ affect the oscillation of $\sin(x)$ in this graph?
2. What would happen to the graph if the interval were larger, say $[-2\pi, 2\pi]$?
3. How do critical points help in determining the shape of the graph?
4. Can we identify local maximums or minimums on this interval for this function?
5. How would the graph change if the coefficient 3 were different, say 1 or 5?

Tip: When graphing, analyzing the first and second derivatives can significantly clarify where a function increases, decreases, or changes concavity, helping in sketching the graph accurately.