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Find the real number $k$ such that $f(x) = f(k - x)$ for all real numbers $x$, where
f(x) = 3x^2 - 4x - x^2 + 11x - x^2 + 7x

 Feb 17, 2024
 #1
avatar+394 
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Here is my method:

We simplify, 

\(f(x)={x}^{2}+14x\).

We might be tempted to substitute \(f(k-x)\), into the equation, but let's see if there's any other method.

We know \(f(k-x)\) represents a transformation.

First \(f(x+k)\) represents a translation left of k

Secondly, \(f(-x+k)\), represents a reflection across the y axis after the transformation.

First we know that the shape and size of the function after the transformation didn't change, because these transformations don't change the shape and size of an object.

But what we can do if calculate the shift of the quadratic through the roots. The original roots of the quadratic are 0 and -14.

Lets trace the path of the root 0, through this transformation.

We shift right left k, 0 becomes (-k, 0). Reflecting across the y axis, this becomes (k, 0). We know that (k, 0) is not the original root, but we see that for f(x) to map to f(k-x), then (k, 0) must represent the root (-14, 0)!!!

So, therefore k = -14.

 Feb 17, 2024

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