Question

设二次函数fx满足f(x+2)=f(2-x),其图像与y轴交与(0,1),在x轴上截得线段长为2根号2,求fx

Answer 1

由f(2+x)=f(2-x),得f(x)图像的对称轴为x=2

又∵在x轴上截得线段长为2√2

∴两零点分别为2+√2,2-√2

设f(x)=a[x-(2+√2)][x-(2-√2)]

把点(0,1)代入得

a=1/2

∴y=(1/2)[x-(2+√2)][x-(2-√2)]

=(1/2)[(x-2)-√2][(x-2)+√2]

=(1/2)(x-2)²-1

=(1/2)x²-2x+1