Pre – Calculus Unit 1

Section 1.5 Notes: Parent Functions and Transformations

Objectives • Identify, graph, and describe parent functions. • Identify and graph transformations of parent functions

Family of Functions

A group of functions with graphs that display similar characteristics Parent function: simplest of the functions in a family

Family: 𝑓(𝑥) = 2√𝑥 − 3 𝑓(𝑥) = √𝑥 + 3 − 4 𝑓(𝑥) = −√𝑥 − 5 Parent: 𝑓(𝑥) = √𝑥

Step Function

A piece-wise function in which the graph resembles a set of stairs Most well known step function is greatest integer function:

Example 1: Describe the following characteristics of the graph of the parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. a) f(x) = 1

𝑥 b) f(x) = x 2

Transformations

A change in the location, size, or shape of a graph Rigid transformations: change only position Nonrigid: distort graph shape

Translation: rigid transformation that shifts the graph (up or down, left or right)

Example 2: Use the graph of f (x) = x3 to graph the given function. a) g (x) = x3 – 2. b) g (x) = (x – 1)3. c) g (x) = (x – 1)3 – 2

Reflection: rigid transformation that produced a mirror image with respect to a line Example 3: Describe how the graphs of f(x) = √𝑥 and g(x) are related. Then write an equation for g(x). a) b)

Dilation: nonrigid transformation that compresses or expands a graph Example 4: a) Identify the parent function f (x) of g(x) = 3

𝑥 , and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on

the same axes. b) Identify the parent function f (x) of g (x) = –|4x|, and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes.

Example 5: a) Graph b) Graph

Example 7: a) Use the graph of f (x) = x 2 – 4x + 3 to graph the function g(x) = |f (x)|. b) Use the graph of f (x) = x 2 – 4x + 3 to graph the function h (x) = f (|x|).