Reading Parameters off an Absolute-Value Graph

Problem

The figure shows the graph of the function

f(x)=|x-p|+|kx-q|-|2x-r|,\qquad k>0.

As the graph indicates, $f$ is constant at a positive value on the far left, then increases to a maximum, then decreases (crossing the $x$ -axis), and is constant at a negative value on the far right.

Determine, with proof, which of the following triples $(p,q,r)$ could produce this graph:

1. $(2,-1,1)$

2. $(1,-1,3)$

3. $(1,-1,2)$

4. $(3,-1,1)$

Answer

Solution

Difficulty	5/10
Topics	functions, Absolute Value, Casework

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