Example 10 When $a$ takes all real values from 0 to 5, the number of integer $b$ that satisfies $3 b=a(3 a-8)$ is $\qquad$
(1997, National Junior High School Competition)
Analysis: From $3 b=a(3 a-8)$, we have $b=a^{2}-$ $\frac{8}{3} a$. This is a quadratic function, and its graph is a parabola. When $a$ takes all real numbers from 0 to 5, finding the number of integer $b$ is equivalent to finding the number of integers between the maximum and minimum values of $b$.

Solution: First, draw the graph of $b=a^{2}-\frac{8}{3} a$ (note $0 \leqslant a \leqslant 5$). From the graph, we know that when $0 \leqslant a \leqslant 5$, the minimum value of $b$ is $\frac{-\left(-\frac{8}{3}\right)^{2}}{4}$ $=-\frac{16}{9}$, and the maximum value of $b$ is $f(5)=\frac{35}{3}$. There are 13 integers between $-\frac{16}{9}$ and $\frac{35}{3}$. Therefore, the number of integer $b$ is 13.