LESSON 9 GRAPHS
You should learn to:
1.
2.
3.
4.
5.
Use the Vertical Line Test to determine whether a graph represents a function.
Determine intervals for which functions are increasing, decreasing, or constant.
Determine relative maximum and minimum values of functions.
Identify and graph piecewise functions.
Identify even and odd functions.
Terms to know: interval notation, vertical line test, increasing, decreasing, constant, relative maximum, relative
minimum, step function, greatest integer function, even function, odd function.
Example 1: Find the domain and range for the graph of f ( x ) shown below.
Use interval notation to express your answers.
( , ) open
[ , ] closed
does not include endpoints
includes endpoints
[ , ) or ( , ] half open (or half closed)
includes one endpoint
y


Domain :
x






Range :

1, 2  ,  2, 4 
 2,1



Example 2: Find the domain and range of f ( x)  x  3
a. Algebraically
b. Graphically
f ( x)  x  3
x3
x 3  0
3
3
x3
Vertical Line Test: A set of points in a coordinate plane represents the graph of y as a function of x , if
and only if no vertical line intersects the graph more than once.
Example 3: Use the Vertical Line Test to determine whether each graph represents a function of x .
a.
b.
y
NO


YES






x






NO








y



c.
y


x






















x





A function is increasing when its graph has positive slope, decreasing when its graph has negative slope,
constant when its graph has zero slope.
Example 4: List the open intervals on which the graph of f ( x ) below is:
y

a. Increasing:

 3, 6


b. Decreasing:
 3, 0 

x











c. Constant:
 0,3

(Always list in terms of x)
A relative maximum point for a function is a high point (peak) in an interval.
A relative minimum point is a low point (valley) in an interval.
Example 5: Find the relative maximum and minimum points of g ( x)  x 4  4 x 2  2 .
Relative Maximum:
 0, 2 
Minimum:  1.414, 2  , 1.414, 2 
Window: 10,10   10,10
 1
 x  6 , x  4
Example 6: Sketch the graph of f ( x)   2
 x  5 ,
x >4
without using a calculator.
A function is even if its graph is symmetric to the y-axis and odd if its graph is symmetric to the origin.
y
y
  x, y 
 x, y 
 x, y 
x
x
  x,  y 
Symmetric to the y-axis (even function)
Symmetric to the origin (odd function)
y
( x, y )
x
( x,  y )
Symmetry to the x-axis (not a function)
For even functions, f ( x)  f ( x)
If ( x, y ) is on the graph then ( x, y ) is also on the graph.
For odd functions, f ( x)   f ( x)
If  x, y  is on the graph then   x,  y  is also on the graph.
Informally: find a few ordered pairs of solutions with opposite x-values and look for symmetry
Example 7: Determine whether each function is even, odd, or neither without graphing.
a. f ( x)  x 2  1
b.
(2,3) and ( 2,3)
(4,15) and (4,15)
g ( x)  x 3  1
(1, 2) and (1, 0)
so, it ' s NEITHER
so it ' s EVEN
c.
h( x)  x3  x
(2, 6) and ( 2, 6)
(3, 24) and ( 3, 24)
so, it ' s ODD
Example 8: Use your calculator to determine whether each function is even, odd, or neither by graphing.
a. f ( x)  4 x3  6 x
b. g ( x)  x 3  1
ODD
ASSIGNMENT 9
Pages 38-41 (1, 4, 7, 9, 16, 18, 20, 28, 34, 47, 61, 70, 77, 112, 115, 116)
Page 26: (50, 54)
EVEN