如图,在平面直角坐标系中,抛物线y=ax方+c与x轴正半轴,y轴正半轴分别交于点A（4,0）B（0,4）,点C为抛物线y

如图,在平面直角坐标系中,抛物线y=ax方+c与x轴正半轴,y轴正半轴分别交于点A（4,0）B（0,4）,点C为抛物线y=ax方+c上的点,且OC=AC 边长为4的正方形PQMN的顶点P在直线AC上,顶点Q,M,N都在直线AC的左下方,边PN,PQ分别与x轴,y轴平行,设点M的横坐标为m,
（1）求a,c的值
（2）求点C坐标和直线AC所对应的函数关系式
（3）设正方形PQMN与△OAC重叠部分面积为S,求S与m的函数关系式
（4）直接写出正方形PQMN的各边与抛物线y=ax方+c共有两个交点时m的取值范围
主要是第3.4问

(1)
B(0,4),c = 4
过A(4,0):16a + 4 = 0,a = -1/4
(2)
AC = OC,C在OA的中垂线x = 2上,x = 2,y = (-1/4)*4 + 4 = 3
C(2,3)
AC:(y - 0)/(3 - 0) = (x - 4)/(2 - 4),y = -3x/2 + 6
(3)
令P(p,6 - 3p/2),p = m + 4 (a)
OC的方程:y = 3x/2
(i) p ≥ 4即m ≥ 0
正方形PQMN全部在x轴上或下方,S = 0
(ii) 2 < p < 4即-2 < m < 0
正方形PQMN与△OAC重叠部分为梯形OLPT
y = 3x/2中取y = 6 - 3p/2,x = 4 -p
T(4 - p,6 - 3p/2)
S = (1/2)(TP + OL)LP
= (1/2)(p - 4 + p + p)(6 - 3p/2)
= (1/2)(3p - 4)(6 - 3p/2)
= (3/4)m(3m + 8) (p = m + 4)
(iii) 4/3 < p ≤ 2即 -8/3 < m ≤ -2
(p = 4/3时,P(4/3,4),MQ在x轴上)
正方形PQMN与△OAC重叠部分为△OLK
y = 3x/2中取x = p,y = 3p/2,K(p,3p/2)
S = (1/2)OL*LK = (1/2)*p*(3p/2) = (3/4)p² = (3/4)(m + 4)²
(iv) 2/3 < p ≤ 4/3即-10/3 < m ≤ -8/3
(p = 2/3时,Q(2,4/3)在OC上)
正方形PQMN与△OAC重叠部分为△HQK
Q的纵坐标 = P的纵坐标 - 4 = 6 - 3p/2 - 4 = 2 - 3p/2
Q(p,2 - 3p/2)
y= 3x/2中取y = 2 - 3p/2,x = 4/3 - p
H(4/3 - p)
y= 3x/2中取x = p,y = 3p/2
K(p,3p/2)
S = (1/2)HQ*QK
= (1/2)(p - 4/3 + p)(3p/2 - 2 + 3p/2)
= (1/3)(3p - 2)²
= (1/3)(3m + 10)²
(v) p ≤ -2/3即 m ≤ -10/3
正方形PQMN全部在OC上或左上方,S = 0
(4)
P(p,6 - 3p/2)
p > 4时,无公共点
p = 4时,只有一个公共点
2 < p < 4时,无公共点
p = 2 (m = -2)时,有2个公共点(P,N在抛物线上) (i)
3 < 6 - 3p/2 < 4即4/3 < p < 2时,有4个公共点
p = 4/3时,有3个公共点
-4/3 < p < 4/3 (-16/3 < m < -8/3)时,有2个公共点 (ii)
p = -4/3时,有1个公共点
p < -4/3时,无公共点
结合(i)(ii),m =-2或-16/3 < m < -8/3