College Algebra 9th txtbk

188 CHAPTER 3 FUNCTIONS
3-3 Transformations of Functions
Z A Library of Elementary Graphs
Z Shifting Graphs Horizontally and Vertically
Z Reflecting Graphs
Z Stretching and Shrinking Graphs
Z Even and Odd Functions
We have seen that the graph of a function can provide valuable insight into the information
provided by that function. But there is a seemingly endless variety of functions out there,
and it seems like an insurmountable task to learn about so many different graphs. In this
section, we will see that relationships between the formulas for certain functions lead to
relationships between their graphs as well. For example, the functions
g(x) x 2 2
can be expressed in terms of the function
h(x) (x 2) 2
k(x) 2x 2
f(x) x 2 as follows:
g(x) f (x) 2
h(x) f (x 2)
k(x) 2f(x)
We will see that the graphs of functions g, h, and k are closely related to the graph of
function f.
Once we understand these relationships, knowing the graph of a very simple function
like f (x) x 2 will enable us to learn about the graphs of many related functions.
Z A Library of Elementary Graphs
As you progress through this book, you will encounter a number of basic functions that you
will want to add to your library of elementary functions. Figure 1 shows six basic functions
that you will encounter frequently. You should know the definition, domain, and range of
each of these functions, and be able to draw their graphs.
f(x)
5
g(x)
h(x)
5
5
5 5
x
5 5
x
5 5
x
5
(a) Identity function
f(x) x
Domain: R
Range: R
(b) Absolute value function
g(x) |x|
Domain: R
Range: [0, )
(c) Square function
h(x) x 2
Domain: R
Range: [0, )