College Algebra Lecture Notes Section 2.6 Page 7 of 7

Section 2.6: Toolbox Functions and Transformations

Big Idea: Knowing the graphs of a few basic functions and how those graphs change when the functions are transformed allows you to sketch any algebraic function quickly.

Big Skill: .You should be able to graph these “toolbox functions” and their linear transformations.

A. The Toolbox Functions

·  The toolbox functions include:

o  The identity function

o  The absolute value function

o  The even power functions (squaring, fourth power, sixth power, etc.)

o  The odd power functions (cubing, fifth power, etc.)

o  The even root functions (square root, fourth root, etc.)

o  The odd root functions (cube root, fifth root, etc.)

Practice:

  1. Graph the identity function .
  1. Graph the absolute value function .
  1. Graph the even power functions . Notice the pattern for and .
  1. Graph the odd power functions . Notice the pattern for and .
  1. Graph the even root functions .
  1. Graph the odd root functions .

B. Vertical and Horizontal Shifts

·  The so-called linear transformations of a function produce simple and predictable changes to the appearance of the graph of the transformed function.

·  These linear transformations include:

o  Vertical shifts: adding a constant to a function; f(x) vs. f(x) + k

o  Horizontal shifts: adding a constant to the independent variable; f(x) vs. f(x + k)

o  Vertical reflections: negative one times the function ; f(x) vs.- f(x)

o  Horizontal reflections: negative one times the independent varible; f(x) vs. f(-x)

o  Vertical stretching: a constant times the function ; f(x) vs. kf(x)

o  Horizontal stretching: a constant times the independent variable; f(x) vs. f(kx)

Vertical Translations of a Graph

Given k > 0 and any function whose graph is determined by y = f(x):

·  The graph of y = f(x) + k is the graph of f(x) shifted up by k units.

·  The graph of y = f(x) - k is the graph of f(x) shifted down by k units.

Horizontal Translations of a Graph

Given h > 0 and any function whose graph is determined by y = f(x):

·  The graph of y = f(x + h) is the graph of f(x) shifted left by h units.

·  The graph of y = f(x - h) is the graph of f(x) shifted right by h units.

Practice:

  1. Graph the functionsand .
  1. Graph the functionsand .


C. Vertical and Horizontal Reflections

Vertical Reflections of a Graph

For the graph of any function y = f(x), the graph of y = -f(x) is the graph of f(x) reflected across the x-axis.

Horizontal Reflections of a Graph

For the graph of any function y = f(x), the graph of y = f(-x) is the graph of f(x) reflected across the y-axis.

Practice:

  1. Graph .
  1. Graph .


D. Vertically Stretching / Compressing a Basic Graph

Stretches and Compressions of a Graph

For the graph of any function y = f(x), the graph of y = af(x) is:

·  The graph of f(x) stretched vertically if |a| > 1.

·  The graph of f(x) compressed vertically if |a| < 1.

For the graph of any function y = f(x), the graph of y = f(ax) is:

·  The graph of f(x) compressed horizontally if |a| > 1.

·  The graph of f(x) stretched horizontally if |a| < 1.

Practice:

  1. Graph …
  1. Graph …


E. Transformations of a General Function

General Transformations of a Graph

From the graph of a function y = f(x), the graph of y = af(x ± h) ± k can be obtained by applying the following transformations in the following order:

·  Horizontal shifts.

·  Reflections.

·  Stretches or compressions

·  Vertical shifts

Practice:

  1. Graph …
  1. Graph …