To graph the function \( y = 3 \sin(x + \pi) \), we start by identifying the general form of the sine function, which is expressed as \( y = a \sin(b(x - h)) + k \). In this case, we can see that the amplitude \( a \) is 3, indicating that the graph will oscillate between 3 and -3 on the y-axis. The absence of a vertical shift means \( k = 0 \).

Next, we analyze the term \( x + \pi \). To fit the general form, we rewrite this as \( x - (-\pi) \). This reveals that \( h = -\pi \). The phase shift can be calculated using the formula \( \text{Phase Shift} = \frac{h}{b} \). Here, \( b = 1 \) (since there is no coefficient in front of \( x \)), leading to a phase shift of \( \frac{-\pi}{1} = -\pi \). This indicates a shift of \( \pi \) units to the left.

Now, we can graph the basic sine function \( y = 3 \sin(x) \). The standard sine graph starts at the origin, reaches a peak of 3 at \( \frac{\pi}{2} \), crosses the x-axis at \( \pi \), and reaches a valley of -3 at \( \frac{3\pi}{2} \). However, due to the phase shift of \( \pi \) units to the left, we adjust the key points accordingly:

• The peak at \( \frac{\pi}{2} \) shifts to \( -\frac{\pi}{2} \).
• The x-intercept at \( \pi \) shifts to \( 0 \).
• The valley at \( \frac{3\pi}{2} \) shifts to \( \frac{\pi}{2} \).

Thus, the graph of \( y = 3 \sin(x + \pi) \) will start at the center (0), peak at \( -\frac{\pi}{2} \) with a value of 3, cross through 0 at \( -\pi \), and reach a valley at \( -\frac{3\pi}{2} \) with a value of -3. The graph will continue to oscillate in this manner, maintaining the amplitude of 3 and the periodic nature of the sine function.

In summary, the graph of \( y = 3 \sin(x + \pi) \) exhibits a phase shift of \( \pi \) units to the left, with an amplitude of 3, resulting in a wave pattern that oscillates between 3 and -3.