3.5 - 1 Rational function A function of the form where p(x) and q(x) are polynomials, with q(x) ≠...
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Transcript of 3.5 - 1 Rational function A function of the form where p(x) and q(x) are polynomials, with q(x) ≠...
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Rational function
A function of the form
( )( ) ,
( )p x
xq x
f
where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.
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Rational Function
Some examples of rational functions are
2
2 2
1 1 3 3 6( ) , ( ) , and ( )
2 5 3 8 16x x x
x x xx x x x x
f f f
Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.
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Domain: (– , 0) (0, ) Range: (– , 0) (0, )
RECIPROCAL FUNCTION 1
( )xx
f
x y
– 2 – ½ – 1 – 1
– ½ – 2
0 undefined
½ 2
1 1
2 ½
decreases on the intervals (–,0) and (0, ).
1( )x
xf
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Asymptotes
Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b:
1. If (x) as x a, then the line is a vertical asymptote.2. If (x) b as x , then the line y = b is a horizontal asymptote.
( )( ) ,
( )p x
xq x
f
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Determining Asymptotes
To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures.1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.
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Determining Asymptotes
2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:(a) If the numerator has lower degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).
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Determining Asymptotes
2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:
(b) If the numerator and denominator have the same degree, and the function is of the form
where an, bn ≠ 0,
then the horizontal asymptote has equation.n
n
ay
b
0
0
( ) ,n
nn
n
a x ax
b x b
f
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Determining Asymptotes
2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:
(c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.
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Steps for Graphing Functions
A comprehensive graph of a rational function exhibits these features:1. all x- and y-intercepts;2. all asymptotes: vertical, horizontal, and/or oblique;3. the point at which the graph intersects its nonvertical asymptote (if there is any such point);4. the behavior of the function on each domain interval determined by the verticalasymptotes and x-intercepts.
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Graphing a Rational Function
Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps.Step 1 Find any vertical asymptotes.Step 2 Find any horizontal or oblique asymptotes.Step 3 Find the y-intercept by evaluating (0).
( )( )
( )p x
xq x
f
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Graphing a Rational Function
Step 4 Find the x-intercepts, if any, by solving (x) = 0 . (These will be the zeros of the numerator, p(x).)
Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving (x) = b or(x) = mx + b.
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Graphing a Rational Function
Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts.
Step 7 Complete the sketch.
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Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Interval Test Point
Value of (x)
Sign of (x) Graph Above or Below
x-Axis
(– , – 3) – 4 Negative Below
(– 3, – 1) – 2 Positive Above
(– 1, ½ ) 0 Negative Below
(½ , ) 2 Positive Above
15
15
13
13
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 1 To find the vertical asymptote(s), solve x2 + 8x + 16 = 0.
2 8 16 0x x Set the denominator equal to 0.
2( 4) 0x Factor.
4x Zero-factor property.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Since the numerator is not 0 when x = – 4, the vertical asymptote has the equation x = – 4.
4x Zero-factor property.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 2 We divide all terms by x2 to get the equation of the horizontal asymptote.
Leading coefficient of numerator
Leading coefficient of denominator
31
y
or 3.y
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 3 The y-intercept is (0) = – 3/8.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 4 To find the x-intercept(s), if any, we solve (x) = 0.
2
2
3 3 68 16
0x x
x x
23 3 6 0x x Set the numerator equal
to 0.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 4 23 3 6 0x x Set the numerator equal to 0.
2 2 0x x Divide by 3.
( 2)( 1) 0x x Factor.
2 or 1x x Zero-factor property
The x-intercepts are – 1 and 2.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 5 We set (x) = 3 and solve to locate the point where the graph intersects the horizontal asymptote.
Multiply by x2 + 8x + 16.
2
2
3 3 68 16
3x x
x x
2 23 3 6 3 24 48x x x x
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 5
Multiply by x2 + 8x + 16.2 23 3 6 3 24 48x x x x
3 6 24 48x x Subtract 3x2.
27 54x Subtract 24x; add 6.
2x Divide by – 27.
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 52x Divide by – 27.
The graph intersects its horizontal asymptote at (– 2, 3).
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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Solution
Graph2
2
3 3 6( ) .
8 16x x
xx x
f
Step 6 and 7 Some of the other points that lie on the graph are
These are used to complete the graph.
1 210,9 , 8,13 , and 5, .
8 3
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Behavior of Graphs of Rational Functions Near Vertical Asymptotes
Suppose that (x) is defined by a rational expression in lowest terms. If n is the largest positive integer such that (x – a)n is a factor of the denominator of (x), the graph will behave in the manner illustrated.