Every absolute value function has certain characteristics that relate it to the basic absolute value function (window #1).  This is true for other functionsas well.  In fact, we describe all of these related functions as a family of functions. 

Consider the graphs of  f(x) = |x| , g(x) = |x| + 2, and p(x) = |x| - 5 (window #2).  What in general can you say about the graph of  r(x) = |x| + k?  How will its graph be different from the original function? 

Consider the graphs of g(x) = |x - 4| and p(x) = |x + 1| (window #3).  What in general can you say about the graph of  r(x) = |x - h|?  How will its graph be different from the original function? 
 

Graphing Calculator Explorations

Using your findings from above, determine how the graph of  r(x) = |x - 3| - 4  will be different from the original.  Determine the h and k value so that r(x) = |x - h| + k  is the function with the following graph:  What in general can you say about the graph of  r(x) = a|x|?  (Hint:  Try different a-values, such as fractions, whole numbers, and negatives.) Write an absolute value function that is inverted, moved to the left three, and moved up six.  Verify your result graphically.