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Let f(x)=3x^2 - 4x - x^2 + 7x + 8. Find the constant k such that f(x)=f(k-x) for all real numbers x.

 May 22, 2022
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Simplify this a little:

 

f(x) = 3x^2 - 4x - x^2 + 7x + 8 = 2x^2 + 3x + 8

 

This is a quadratic function, which means its graph is a parabola. The axis of symmetry of the graph is \(x = - \dfrac{3}{2(2)} = -\dfrac34\). (It is always x = -b/(2a) for the parabola y = ax^2 + bx + c.)

 

That means \(f(x) = f\left(2\left(-\dfrac34\right) - x\right) = f\left(-\dfrac32 - x\right)\). The required constant is -3/2.

 May 22, 2022

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