3. From the eight numbers $-3,-2,-1,0,1,2,3,4$, any three different numbers are taken as the coefficients of the quadratic function $f(x)=a x^{2}+b x+c(a \neq 0)$. If the graph of the quadratic function passes through the origin, and its vertex is in the first quadrant or the third quadrant, how many such quadratic functions are there? $\qquad$
Hui Shi 24.
Analysis: Quadratic functions can be divided into two major categories: one with the vertex in the first quadrant; the other with the vertex in the third quadrant, and then examine the signs of the vertex coordinates separately.
Since the graph passes through the origin, we have $c=0$, so the quadratic function can be written in the form $f(x)=a x^{2}+b x$.
Also, since $f(x)=a\left(x+\frac{b}{2 a}\right)^{2}-\frac{b^{2}}{4 a}$, its vertex coordinates are $\left(-\frac{b}{2 a}, \frac{b^{2}}{4 a}\right)$.
If the vertex is in the first quadrant, then $-\frac{b}{2 a}>0, -\frac{b^{2}}{4 a}>0$, hence $a<0, b<0$, and such quadratic functions have $A_{3}^{1} \cdot A_{4}^{1}=12$; if the vertex is in the third quadrant, then $-\frac{b}{2 a}<0, -\frac{b^{2}}{4 a}<0$, hence $a>0, b>0$, and such quadratic functions have $A_{4}^{2}=12$. By the principle of addition, the number of quadratic functions that satisfy the conditions is $A_{3}^{1} \cdot A_{4}^{1}+A_{4}^{2}=24$.