Don't understand use of auxiliary function in proving theorems

How was the auxiliary function to be used in proving the theorem below determined? I don't understand why this function was chosen, and how the author arrived at the decision of choosing this function? This is the first time I came across the use of auxiliary functions in proving a theorem, so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.
http://www.mathhelpforum.com/math-he...3&d=1290060131
Thanks a lot for reading.
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Originally Posted by ssadi

... so I am a bit confused whether the function is chosen by pure intuition or is there a method behind this.

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The important thing is that works and you understand why works. All the same, in this case, analyze:

1.- translates vertically the graph of in such a way that .

2.- The equation of the straight line containing the endpoints of the graph is:



Regards.

Fernando Revilla