The function $f$ is defined by $f(x) = 4 \cos ax + b$ for $0 \le x \le \pi$, where $a$ and $b$ are constants. The period of $f$ is $\frac{2\pi}{3}$ and the function has a maximum value of 2.

1. State the amplitude from the given function

The amplitude of $f(x) = 4 \cos ax + b$ is given by the coefficient of the cosine term.
$\text{Amplitude} = 4$

1. Find the value of $a$ using the given period.

The period of $f(x) = 4 \cos ax + b$ is given by $\frac{2\pi}{a}$.
$\frac{2\pi}{a} = \frac{2\pi}{3}$
$a = 3$

2. Find the value of $b$ using the given maximum value.

The maximum value of $\cos ax$ is $1$.
The maximum value of $f(x)$ is $4(1) + b = 4 + b$.
Given that the maximum value is $2$:
$4 + b = 2$
$b = -2$

1. Determine the full equation of the function using results from earlier parts

From the given period $\frac{2\pi}{3}$, $a = 3$.
From the maximum value of 2 and amplitude 4, $4 + b = 2 \implies b = -2$.
Thus, $f(x) = 4 \cos 3x - 2$.

2. Find the $x$-coordinates where the graph crosses the $x$-axis

Set $f(x) = 0$:
$4 \cos 3x - 2 = 0$
$\cos 3x = \frac{1}{2}$
For $0 \le x \le \pi$, the domain for $3x$ is $0 \le 3x \le 3\pi$.
$3x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3}$
$x = \frac{\pi}{9}, \frac{5\pi}{9}, \frac{7\pi}{9}$

3. Identify key points (maximum, minimum, and endpoints) for the sketch

At $x = 0$, $y = 2$ (maximum).
At $x = \frac{\pi}{3}$, $y = 4(-1) - 2 = -6$ (minimum).
At $x = \frac{2\pi}{3}$, $y = 4(1) - 2 = 2$ (maximum).
At $x = \pi$, $y = 4(-1) - 2 = -6$ (minimum).
The graph completes 1.5 cycles in the interval $0 \le x \le \pi$.