已知二次函数f(x)的最大值是4分之1,且他的图像过两点A（0,0）,B(1,-6）,求此二次函数的解析式.
1个回答

y = a(x+b)² + c

∵最大值是1/4,∴c=1/4

将(0,0)代入,

0 = a(0+b)² + 1/4

0 = ab²+1/4

a = -1/(4b²)...(*)

将(1,-6)代入,

-6 = a(1+b)² + 1/4

0 = a(1+b)² + 25/4

将(*)代入,0 = -1/(4b²) * (1+b)² + 25/4

0 = -1*(1+b)² + 25b²

0 = -(1+2b+b²)+25b²

0 = -1-2b+24b²

b = -1/6 or 1/4

当b = -1/6,a = -9

当b = 1/4,a = -4

∴二次函数方程是

y = -4(x+1/4)² + 1/4,i.e.y = -4x² - 2x

或y = -9(x-1/6)² + 1/4,i.e.y = -9x² + 3x

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