8. If the graph of a quadratic function $f(x)=a x^{2}+b x+c(a \neq 0)$ passes through two distinct points $(r, k)$ and $(s, k)$, what is $f(r+s)$ ?
(A) $2 k$
(B) $c$
(C) $k-c$
(D) $2 k-c$
(E) None of the above
8. Answer: (B)
Let $g(x)=f(x)-k$. Then $g(r)=f(r)-k=k-k=0$. Similarly, $g(s)=0$. Therefore $r$ and $s$ are roots of the quadratic equation $g(x)=a x^{2}+b x+c-k=0$, from which we deduce that $r+s=-\frac{b}{a}$. Hence
$$
f(r+s)=f\left(-\frac{b}{a}\right)=a\left(-\frac{b}{a}\right)^{2}+b\left(-\frac{b}{a}\right)+c=c .
$$