Parent Functions and Transformations

A family of graphs is a group of graphs that display one or more similar characteristics. The parent graph, which is the graph of the parent function, is the simplest of the graphs in a family. This is the graph that is transformed to create other members in a family of graphs.

Example 1: Identify the type of function represented by the graph.

b) Y2 = |x| + 2, what do you notice? Example 2: Using your calculator, graph the following, in this order: a) Y1 = |x|  b) Y2 = |x| + 2, what do you notice? c) Y2 = |x| – 2, what do you notice? d) Y2 = |x + 2|, what do you notice?   e) Y2 = |x – 2|, what do you notice?

b) Y2 = –|x|, what do you notice? c) Y2 = |3x|, what do you notice? Example 3: Using your calculator, graph the following, in this order: a) Y1 = |x| b) Y2 = –|x|, what do you notice? c) Y2 = |3x|, what do you notice?   d) Y2 = , what do you notice?

Transformations of a parent graph may appear in a different location, flip over an axis, or appear to have been stretched or compressed. The transformed graph may resemble the parent graph, or it may not.

A translation moves a figure up, down, left, or right.  

*When a constant k is added to or subtracted from a parent function, the result f(x) ± k is a translation of the graph up or down.

*When a constant h is added to or subtracted from x before evaluating a parent function, the result f(x ± h), is a translation left or right.

A reflection flips a figure over a line called the line of reflection.

*When a parent function is multiplied by –1, the result –f(x) is a reflection of the graph in the x-axis.

*When a parent function is multiplied by –1, the result f(–x) is a reflection of the graph in the y-axis.

A dilation shrinks or enlarges a figure proportionally A dilation shrinks or enlarges a figure proportionally. When the variable in a linear parent function is multiplied by a nonzero number, the slope of the graph changes.  

*When a nonlinear parent function is multiplied by a nonzero number, the function is stretched or compressed vertically.

*Coefficients greater than 1 cause the graph to be stretched vertically, and coefficients between 0 and 1 cause the graph to be compressed vertically.

Example 4: Describe the transformation in y = (x + 1)2 Example 4: Describe the transformation in y = (x + 1)2. Then graph the function

Example 5: Describe the transformation in y = |x| – 4 Example 5: Describe the transformation in y = |x| – 4. Then graph the function.

Example 6: Describe the transformation in y = –|x| Example 6: Describe the transformation in y = –|x|. Then graph the function.

Example 7: Describe the transformation on y = 1 2 |x| Example 7: Describe the transformation on y = 1 2 |x|. Then graph the function.

Example 8: Describe all transformations on y = 2|x – 3|+ 1 Example 8: Describe all transformations on y = 2|x – 3|+ 1. Then graph the function.

a) +4 translates f(x) = |x| right 4 units. Example 9: Which of the following is not an accurate description of the transformations in the function   a) +4 translates f(x) = |x| right 4 units. b) –2 translates f(x) = |x| down 2 units. c) translates f(x) = |x| across the x-axis d) +4 translates f(x) = |x| left 4 units.