Example 2 The vertex of the parabola $y=a x^{2}+b x+c$ is $(4,-11)$, and the x-coordinates of its two intersections with the $x$-axis are one positive and one negative. Then the positive number(s) among $a 、 b 、 c$ is/are ( ).
(A) only $a$
(B) only $b$
(C) only $c$
(D) $a 、 b$
Explanation: Given the vertex is $(4,-11)$, the parabola intersects the $x$-axis at two points. It is easy to see that the graph opens upwards, i.e., $a>0$. Let the $x$-coordinates of the two intersection points of the parabola with the $x$-axis be $x_{1}$ and $x_{2}$, i.e., $x_{1}$ and $x_{2}$ are the two roots of the corresponding equation $a x^{2}+b x+c=0$. Given that $x_{1} x_{2}0$, it follows that $b<0$. Therefore, the answer is (A).