Section 3.1 - Basic Graphs and Transformations

Section Objectives

  1. Develop a familiarity with the graphs of basic functions (Toolbox Functions).
  2. Apply the transformations (shifts, reflections, stretches, and compressions) to basic graphs to obtain more general graphs.
  3. 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:



Examples




Transformations

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

Transformed FunctionEffect 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













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.