Develop a familiarity with the graphs of basic functions (Toolbox Functions).

Apply the transformations (shifts, reflections, stretches, and compressions) to basic graphs to obtain more general graphs.

Determine the transformations that result in a given graph.

Summary of the Toolbox Functions

Before we study the graphs of general functions, we must build up a library, or toolbox, of basic functions and their graphs. The "toolbox functions" are the constant functions, the linear functions, the squaring and cubing functions (eventually general power functions), the square and cube root functions (eventually -th roots), and the absolute value function. We must become familiar with the following characteristics of these functions and their graphs:

Domain

Range

Interesting features (vertices, flat spots, opens up/down, asymptotes, intercepts, etc.)

Symmetry

End (long-term) behavior

Increasing/Decreasing

Extreme values

Examples

Discuss the characteristics of each of the toolbox functions.

Transformations

This table summarizes typical transformations of a given function/graph.

Transformed Function

Effect on Graph

Graph of shifted units up

Graph of shifted units left

Graph of reflected about -axis

Graph of reflected about -axis

Graph of stretched vertically

Graph of compressed vertically

Graph of compressed horizontally

Graph of stretched horizontally

Examples

Describe, then sketch, the graph of .

Describe the graph of .

Describe the graph of . How about ?

Important idea: When a function is transformed, many of its special features, such as domain, range, and intervals on which it is increasing/decreasing, change in corresponding ways.

Discuss the features of the function . Domain? Range? Vertex?