如图,抛物线y=ax2+bx﹣4与x轴交于A(4,0)、B(﹣2,0)两点,与y轴交于点C,点P是线段AB上一动点(端点
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(1) y = a(x - 4)(x + 2)

x = 0,y = -8a = -4,a = 1/2

y = (x - 4)(x + 2)/2 = x²/2 - x - 4

(2)C(0,-4)

BC = 2√5

P(p,0),p > -2

BP² = (p + 2)²

AC和PD的斜率均为1,PD的方程:y = x - p

BC的方程:x/(-2) + y/(-4) = 1

联立,D((p-4)/3,(2p + 4)/3)

BD² = [(p - 4)/3 + 2]² + [(2p + 4)/3]² = 5(p+2)²/9

BD = (√5)(p+2)/3

BP² = BD•BC

(p + 2)² = 2√5* (√5)(p+2)/3

p = 4/3

P(4/3,0)

(3)PD² = [(p - 4)/3 -p]² + [(2p + 4)/3]² = 8(p+2)²/9

PD = 2(√2)(p+2)/3

AC = 4√2

PD的方程:y = x - p,x - y - p = 0

A与PD的距离h = |4 - 0 - p|/√2 = (4 - p)/√2

△PCD的面积S = 梯形ACDP的面积 - △APC的面积

= (1/2)(PD + AC)*h - (1/2)AP*|C的纵坐标|

= (1/2)[2(√2)(p+2)/3 + 4√2]*(4 - p)/√2 - (1/2)(4 - p)*4

= -p²/3 + 2p/3 + 8/3

= -(p - 1)²/3 + 3

p = 1时,S最大

P(1,0)

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