14.
由条件知 ax1^2+c = ax2^2+c
=> x1^2 = x^2
= x1 = -x2 (因x1≠x2)
=>x1+x2 = 0
=>函数值为 a*0^2+c = c
19.
(1)因抛物线经过远点O,所以其抛物线方程为 y = ax^2+bx
入水点坐标为 (4+1-3,-10),即(2,-10)
又,顶点坐标为(-b/2a,2/3)
代入抛物线方程得
-10 = 4a + 2b =>b = -2a - 5
2/3 = a*b^2/4a^2 + b*(-b/2a) = -b^2/4a
联立解二元二次方程组,得
a = -25/6,b = 10/3
a = -3/2 ,b = -2
由图可知,抛物线对称轴的x坐标大于0,因此a,b必定异号,
所以a = -25/6,b = 10/3
得抛物线方程：y = -25/6*x^2 + 10/3*x
(2) 运动员在空中调整好入水姿势时的x坐标为 (18/5+1-3) = 8/5
代入抛物线方程得
y = -25/6 * 64/25 + 10/3* 8/5 = -16/3 < -5
此时,显然跳水运动员距水面高度小于5m了,因此跳水会失误.