下图显示了当 $0 \le x \le \dfrac{2\pi}{3}$ 弧度时的曲线 $y = a\cos bx + c$，其中 $a$、$b$ 和 $c$ 是整数。该曲线在 $\left(\dfrac{\pi}{3}, 1\right)$ 处有一个极大值点，且在 $\left(\dfrac{2\pi}{3}, -3\right)$ 处有一个极小值点。

1. Determine the period of the curve from the given maximum and minimum points

The horizontal distance between a maximum point and the adjacent minimum point represents half of the period.
$\frac{1}{2} \text{ Period} = \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3}$
$\text{Period} = 2 \times \frac{\pi}{3} = \frac{2\pi}{3}$

2. Use the formula for the period of a cosine function to find $b$

The period of $y = a\cos bx + c$ is given by $\frac{2\pi}{b}$.
$\frac{2\pi}{b} = \frac{2\pi}{3}$
$b = 3$

1. Explain the geometric effect of the parameter $c$ on the graph.

The parameter $c$ represents a vertical translation of the curve by $c$ units.

2. Calculate the value of $c$ using the maximum and minimum values of the curve.

The value of $c$ is the average of the maximum and minimum $y$-values (the principal axis).
$c = \frac{\text{Maximum} + \text{Minimum}}{2}$
$c = \frac{1 + (-3)}{2}$
$c = \frac{-2}{2} = -1$

1. Calculate the amplitude of the curve

$\text{Amplitude} = \frac{\text{Maximum value} - \text{Minimum value}}{2}$
$\text{Amplitude} = \frac{1 - (-3)}{2} = \frac{4}{2} = 2$

2. Determine the value of $a$ and write the equation of the curve

From parts (a) and (b), $b = 3$ and $c = -1$.
The equation is of the form $y = a\cos 3x - 1$.
Substitute the maximum point $(\frac{\pi}{3}, 1)$:
$1 = a\cos\left(3 \times \frac{\pi}{3}\right) - 1$
$1 = a\cos(\pi) - 1$
$1 = -a - 1$
$a = -2$

The equation of the curve is $y = -2\cos 3x - 1$.