设二次函数f(x)满足f(x-2)=f(-x-2)且经过(0,1),又被X轴截得的线段长度为2√2求f(x
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![设二次函数f(x)满足f(x-2)=f(-x-2)且经过(0,1),又被X轴截得的线段长度为2√2求f(x](/uploads/image/z/3524943-39-3.jpg?t=%E8%AE%BE%E4%BA%8C%E6%AC%A1%E5%87%BD%E6%95%B0f%28x%29%E6%BB%A1%E8%B6%B3f%28x-2%29%3Df%28-x-2%29%E4%B8%94%E7%BB%8F%E8%BF%87%280%2C1%29%2C%E5%8F%88%E8%A2%ABX%E8%BD%B4%E6%88%AA%E5%BE%97%E7%9A%84%E7%BA%BF%E6%AE%B5%E9%95%BF%E5%BA%A6%E4%B8%BA2%E2%88%9A2%E6%B1%82f%28x)
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设二次函数f(x)满足f(x-2)=f(-x-2)且经过(0,1),又被X轴截得的线段长度为2√2求f(x
设二次函数f(x)满足f(x-2)=f(-x-2)且经过(0,1),又被X轴截得的线段长度为2√2求f(x
设二次函数f(x)满足f(x-2)=f(-x-2)且经过(0,1),又被X轴截得的线段长度为2√2求f(x
f(x-2)=f(-x-2)
则:对称轴为x=-2
所以,可设f(x)=a(x+2)^2+c
将(0,1)代入,得:
4a+c=1
c=1-4a
f(x)=a(x+2)^2+1-4a=ax^2+4ax+1
当ax^2+4ax+1=0
x1+x2=-4
x1x2=1/a
(x1-x2)^2=(x1+x2)^2-4x1x2=16-(4/a)=(2√2)^2=8
4/a=8
a=1/2
所以:
f(x)=(1/2)x^2+2x+1
f(x-2)=f(-x-2)
则:对称轴为x=-2
所以,可设f(x)=a(x+2)^2+c
将(0,1)代入,得:
4a+c=1
c=1-4a
f(x)=a(x+2)^2+1-4a=ax^2+4ax+1
当ax^2+4ax+1=0
x1+x2=-4
x1x2=1/a
(x1-x2)^2=(x1+x2)^2-4x1x2=16-(4/a)=(2√2)^2=8
4/a=8
a=1/2
所以:
f(x)=(1/2)x^2+2x+1