Math 95
8.5 "Graphs of Nonlinear Functions"
Objectives:
*
Graph nonlinear functions.
*
Translate graphs of functions.
*
Re‡ect graphs of functions.
*
Find the domain and range of a rational function.
Graphs of Nonlinear Functions
We have seen that the graph of a linear function is a line. We will now consider several examples of nonlinear
equations whose graphs are not lines.
Square Function or Parabola:
Example 1: (Square Function)
Graph f (x) = x2 and …nd its domain and range.
x
y
y 10
(x; y)
8
6
4
2
-6
-4
-2
2
4
6
-2
Domain:
x
Range:
Cube Function:
Example 2: (Cube Function)
Graph f (x) = x3 and …nd its domain and range.
x
y
y
(x; y)
8
6
4
2
-4
-2
-2
2
4
x
-4
-6
-8
Domain:
Range:
Page: 1
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
8.5
Absolute Value Function:
Example 3: (Absolute Value Function)
Graph f (x) = j x j and …nd its domain and range.
x
y
y
(x; y)
6
4
2
-6
-4
-2
2
4
6
-2
x
-4
-6
Domain:
Range:
Translations of Graphs
Translations:
"Vertical Translations"
If f is a function and k represents a positive number, then:
The graph of
is identical to the graph of y = f (x)
except that it is translated k units upward.
The graph of
is identical to the graph of y = f (x)
except that it is translated k units downward.
Example 4: (Square Function/Vertical Translation)
Graph f (x) = x2 + 3 and …nd its domain and range.
y 12
x
y
x
y
(x; y)
10
8
6
4
2
-6
-4
-2
2
-2
Domain:
4
6
x
Range:
Page: 2
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
8.5
Translations:
"Horizontal Translation"
If f is a function and h is a positive number, then:
The graph of
is identical to the graph of y = f (x)
except that it is translated h units to the right.
The graph of
is identical to the graph of y = f (x)
except that it is translated h units to the left.
Example 5: (Cube Function/Horizontal Translation)
Graph f (x) = (x
2)
3
and …nd its domain and range.
y
x
y
x
y
8
(x; y)
6
4
2
-4
-2
2
-2
4
x
-4
-6
-8
Domain:
Range:
Re‡ections of Graphs
Re‡ection:
"Re‡ection of a Graph"
The graph of
is the graph of y = f (x) re‡ected about the x
axis
Example 6: (Re‡ection & Vertical Translation)
Graph f (x) =
x2 + 5 and …nd its domain and range.
y
x
y
x
y
x
y
(x; y)
6
4
2
-6
-4
-2
2
-2
4
6
x
-4
-6
Page: 3
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
8.5
Example 7: (Horizontal & Vertical Translation )
Graph f (x) = jx + 2j
4 and …nd its domain and range.
y
x
y
x
y
x
y
6
4
(x; y)
2
-6
-4
-2
2
-2
4
6
x
-4
-6
Finding the Domain and Range of a Rational Function
Since division by 0 is unde…ned any value that makes the denominator 0 in a rational function must be excluded from
the domain of the function.
Example 8: (Finding the domain and range of a rational function)
Find the domain of the following rational functions.
3x + 2
a) f (x) = 2
x +x 6
b)
Page: 4
x2 + 1
x 2
Notes by Bibiana Lopez