By Vieta's formulas, the sum of the zeros is $ -\frac{b}{a}, $ and the sum of the coefficients is $ \frac{c}{a}, $ so $ b = -c $. Then the sum of the coefficients is $ a + b + c = a, $ which is the coefficient of $ x^2 $. Thus, the answer is $ \boxed{\text{(A)}} $. To see that none of the other choices can work, consider $ f(x) = -2x^2 - 4x + 4 $. The sum of the zeros, the product of the zero, and the sum of the coefficients are all $ -2 $. The coefficient of $ x $ is $ -4, $ the $ y $-intercept of the graph of $ y = f(x) $ is 4, the $ x $-intercepts are $ -1 \pm \sqrt{3}, $ and the mean of the $ x $-intercepts is $ -1, $ so none of the other choices work.