Example 8 In the Cartesian coordinate system, a point whose both the x-coordinate and y-coordinate are integers is called an integer point. The image of the quadratic function $y=-x^{2}+6 x-\frac{27}{4}$ and the x-axis enclose a closed figure which is colored red. The number of integer points inside this red region and on its boundary is ( ).
(A) 5
(B) 6
(C) 7
(D) 8
(2008, National Junior High School Mathematics Competition, Zhejiang Province Preliminary)
Explanation: As shown in Figure 5, the graph of the quadratic function $y=-x^{2}+6 x-\frac{27}{4}$ intersects the $x$-axis at two points $\left(\frac{3}{2}, 0\right)$ and $\left(\frac{9}{2}, 0\right)$.

Between $x=\frac{3}{2}$ and $x=\frac{9}{2}$, there are three integers: $2, 3, 4$.

When $x=2, 4$, $y=\frac{5}{4}$, the integers that satisfy $0 \leqslant y \leqslant \frac{5}{4}$ are 0, 1, so there are 4 integer points;

When $x=3$, $y=\frac{9}{4}$, the integers that satisfy $0 \leqslant y \leqslant \frac{9}{4}$ are $0, 1, 2$, so there are 3 integer points.
In total, there are 7 integer points. Therefore, the answer is $(\mathrm{C})$.