图中显示了 $y = a\tan(bx + c)$ 函数图像的一部分。该图像在 $x = -4\pi$ 和 $x = 4\pi$ 处有垂直渐近线，并穿过点 P 和 Q。

1. Determine the period of the tangent graph from the given asymptotes

The distance between consecutive vertical asymptotes gives the period of the tangent graph.
$\text{Period} = 4\pi - (-4\pi) = 8\pi$

2. Relate the period to the parameter $b$ and solve for $b$

The period of $y = a\tan(bx + c)$ is given by $\frac{\pi}{b}$.
$\frac{\pi}{b} = 8\pi$
$b = \frac{\pi}{8\pi} = \frac{1}{8}$

1. Determine the amplitude, period, and vertical shift of the function

For $y = 4 + 3\sin 2x$:

• Amplitude $= 3$
• Vertical shift $= 4$ (principal axis is $y = 4$)
• Period $= \frac{360^\circ}{2} = 180^\circ$

2. Identify the key coordinates and sketch the graph for $0^\circ \le x \le 360^\circ$

Since the period is $180^\circ$, there are two full cycles in the domain $0^\circ \le x \le 360^\circ$.
Maximum value $= 4 + 3 = 7$
Minimum value $= 4 - 3 = 1$
Key points for the first cycle:

• Start: $(0^\circ, 4)$
• Max: $(45^\circ, 7)$
• Axis crossing: $(90^\circ, 4)$
• Min: $(135^\circ, 1)$
• End of cycle 1: $(180^\circ, 4)$
  Key points for the second cycle:
• Max: $(225^\circ, 7)$
• Axis crossing: $(270^\circ, 4)$
• Min: $(315^\circ, 1)$
• End of cycle 2: $(360^\circ, 4)$
  Draw a smooth continuous curve through these points.