已知二次函数图像的顶点坐标为m(1,0)直线y=x+m与该二次函数的图像交于A,B两点,其中A点的坐标为（3,4）

(1)顶点M(1,0),二次函数可表达为y = a(x - 1)²过A(3,4):a(3 - 1)² = 4,a = 1二次函数:y = (x - 1)² = x² - 2x + 1A(3,4)在直线y = x + m上:3 + m = 4,m = 1直线:y = x + 1(2)y = x + 1 = x² - 2x + 1x² - 3x = x(x - 3) = 0,x = 0 (x = 3,点A)B(0,1)B关于x轴的对称点为B'(0,-1)AB'的方程:(y + 1)/(4 + 1) = (x - 0)/(3 - 0)取y = 0,x = 3/5Q(3/5,0)(∆OBQ与∆OB'Q全等,BQ = B'Q,B',Q,A在一条直线上)(3) y = x + 1,x = a,y = a + 1,D(a,a + 1)y = (x - 1)²,x = a,y = (a - 1)²,E(a,(a - 1)²)DE = a + 1 - (a - 1)² = -a² + 3a = -(a - 3/2)² + 9/4a = 3/2时,DE最大,为9/4y = x + 1,x = 1,y = 2,N(1,2)MN = 2MN与DE平行,只需MN = DE即可-a² + 3a = 2a² - 3a + 2 = (a - 1)(a - 2) = 0a = 1,MN与DE重合,舍去a = 2,P(2,0)