Investigation: Graphing Absolute Value Functions

Graphing Absolute Value Functions

1. The graphs of the functions \(y = 2x + 1\) and \(y = \frac{1}{2}\left(x - 3 \right)^2 - 4\) are shown.
   
   
   

   
   
   
1. Predict what the graphs of \(y = \left| {2x + 1} \right|\) and \(y = \left| {\frac{1}{2}\left( {x - 3} \right)^2 - 4} \right|\) will look like.
   
   
2. Use tables of values to check your predictions from part a.
   
   
3. Predict how the graph of \(y = \left| f(x) \right|\) relates to the graph of \(y = f(x)\), for any function.
   
   
2. Open Absolute Value of a Function applet.
   
   
1. Make sure the “linear” box is turned on. The \(m\)- and \(b\)-values can be adjusted to represent different linear functions. Predict what \(y = \left| f(x) \right|\) will look like based on \(y = f(x)\). Turn on \(y = \left| f(x) \right|\) to check your prediction. Repeat the process for other \(m\)- and \(b\)-values.
   
   
2. Turn on the “quadratic” box. The \(a\)-, \(p\)-, and \(q\)-values can be adjusted to represent different quadratic functions. Predict what \(y = \left| g(x) \right|\) will look like based on \(y = g(x)\). Turn on \(y = \left| g(x) \right|\) to check your prediction. Repeat the process for other \(a\)-, \(p\)-, and \(q\)-values.