Section 2 7 Parent Functions and Transformations A

Section 2. 7 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.

• Example 2: Using your calculator, graph the following, in this order: • a) Y 1 = |x| • b) Y 2 = |x| + 2, what do you notice? • c) Y 2 = |x| – 2, what do you notice? • d) Y 2 = |x + 2|, what do you notice? • e) Y 2 = |x – 2|, what do you notice?

• Example 3: Using your calculator, graph the following, in this order: • a) Y 1 = |x| • b) Y 2 = –|x|, what do you notice? • c) Y 2 = |3 x|, what do you notice? • d) Y 2 = , 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. 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. Then graph the function

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

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

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

• 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.