Problem: Draw the graph (on plain paper) of f(x)= min { |x| -1, |x-1| – 1, |x-2|-1}

Discussion: The easiest way to solve this problem is to draw the graph of all these three pieces of functions mentioned, and pick ones which are minimum.

Graph of y = |x| is same as, y =  x, where x is non negative. For negative values of x, it is the graph of y = x reflected about x axis.

Screen Shot 2015-11-19 at 9.29.26 PM

Now we apply transformations to find the remaining graphs.

  • Graph of y = |x|-1 can be found by lowering the graph of y = |x| by 1 unit along y axis.
    Screen Shot 2015-11-19 at 9.30.31 PM
  • Graph of y = |x-1| – 1 can be found by first shift the graph of y = |x| along positive direction of x axis by 1 unit, and then lowering it by 1 unit along y axis.
    Screen Shot 2015-11-19 at 9.30.43 PM
  • Graph of y = |x-2| – 1 can be found by first shift the graph of y = |x| along positive direction of x axis by 2 unit, and then lowering it by 1 unit along y axis.

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