Section 8.1: Graphs of Rational Functions and Reducing Rational Expressions.
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Section 8.1: Graphs of Rational Functions and Reducing
Rational Expressions
8.1 Lecture Guide: Graphs of Rational Functions and Reducing Rational Expressions
Objective: Determine the domain of a rational function.Objective: Identify the vertical asymptotes of the graph of a rational function.
Rational Function
Algebraically
P x
f xQ x
is a rational function if P x and Q x
are polynomials and 0Q x
Verbally
A rational function is defined as the ratio of two polynomials.
Algebraic Example
1f x
x
Domain of a Rational Function:
The domain of a rational function must exclude values that would cause division by zero.
Algebraic Example The domain of 1
f xx
is ( ,0) (0, )
Numerical Example Verbal Example
Only zero is excluded from the domain to prevent division by zero.Note that zero causes an error message to appear in the table of values for this function.
Graphical Example
4.7,4.7,1 3.1,3.1,1 by
Verbal Example Also note that there is a break in the graph at 0x
Characteristics of the graph of 1f x
x
• The domain of this function does not include __________.
• The function is _________________________, there is a break in the graph at 0.x The graph consists of two
• The graph approaches the _________________________0x asymptotically.
• The graph approaches the _________________________0y asymptotically.
line
line
unconnected branches.
Determine the domain of the each rational function.
2
1
6f x
x x
1.
2.
Determine the domain of the each rational function.
2 9
9
xf x
3.
Determine the domain of the each rational function.
2
4
4 4
xf x
x x
4. 2
7
2 9 35
xf x
x x
Determine the domain of the each rational function.
Use the given table to determine the domain and vertical asymptotes of each rational function
2
7
5 6
xf x
x x
5.
Domain:
Vertical Asymptotes:
Use the given table to determine the domain and vertical asymptotes of each rational function
6.
Domain:
Vertical Asymptotes:
2
2
2
4 5
xf x
x x
Objective : Reduce a rational expression to lowest terms.
Reducing a Rational Expression to Lowest Terms
Algebraically
If A,B and C are polynomials and 0B and 0C thenAC A
BC B
Verbally 1. Factor both the numerator and the denominator of the rational expression.2. Divide the numerator and the denominator by any common nonzero factors.
Algebraic Example
1
2
3 23 6
2
xx
x x
2x x 1
3 for 2xx
7.
Reduce each rational expression.
4 6
7 4
3
12
x y
x y
8.
Reduce each rational expression.
2 49
3 21
x
x
9.
Reduce each rational expression.
3 5
15 9
x y
y x
10.
Reduce each rational expression.
2 2
7 7a b
a b
11.
Reduce each rational expression.
2
2
4
12
x x
x x
12.
Reduce each rational expression.
2 2
2 2
4 9
4 12 9
x y
x xy y
13.
Reduce each rational expression.
2
11
121
x
x
14.
Reduce each rational expression.
2 3 7
2 3 7
x y
x y
Fill in the missing numerator or denominator:
15.
23 4 21
5 ???
x x x
x
Fill in the missing numerator or denominator:
16. 2
2 ?
3 2 15
x
x x x
Section 8.1 Using the Language and Symbolism of Mathematics
1. A rational expression is defined as the _________________________ of two polynomials.
2. Division by zero is _________________________.
3. A function defined by P x
f xQ x
where P x and
Q x are polynomials and 0Q x is called a is called a
_________________________ function.
4. To determine the values excluded from the domain of a rational function, we look at the values for which the _________________________ is equal to __________.
5. A vertical line that a graph approaches but does not touch is called a vertical _________________________.
6. A horizontal line that a graph approaches is called a horizontal _________________________.
7. The graph of the rational function 5
( )4
f xx
will have the line 4x as a vertical ___________________.
8. A rational expression is in _________________________ _________________________ when the numerator and denominator have no common factor other than 1 or 1.
9. If A,B and C are polynomials and 0B and 0C then
AC
BC __________.
10. If two polynomials are opposites, their ratio is __________.
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