抛物线y=4x^2+(b-1)x+3的顶点在直线y=-x上,求抛物线的解析式

y = 4x² + (b - 1)x + 3
= [2x + (b - 1)/4]² + [-(b - 1)² + 48]/16 (经配方而得)
所以顶点坐标( -(b - 1)/8,[-(b - 1)² + 48]/16 )
且顶点在直线y=-x上,
代入x = -(b - 1)/8,y = [-(b - 1)² + 48]/16
得:[-(b - 1)² + 48]/16 = (b - 1)/8
化简:(b - 1)² + 2(b - 1) - 48 = 0
[(b - 1) - 6][(b - 1) + 8] = 0
(b - 1) - 6 = 0 或(b - 1) + 8 = 0
b = 7 或 b = -7
所以抛物线的解析式:y = 4x² + 6x + 3 或 y = 4x² - 8x + 3 (其中,x:实数)