2.6: Graphs of Rational Functions Rational Functions...
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14 2.6: Graphs of Rational Functions Example 1: Sketch the graph then find the domain of f and use limits to describe its behavior at values of x not in its domain using limits. a) 3 1 ) ( x x f b) 2 1 ) ( x x f x y x y Rational Functions Let f and g be polynomial functions with g(x) ≠ 0. Then the function given by () () () fx rx gx is a rational function Horizontal and Vertical asymptotes: In the graph of a function y = f(x), the line y = b is a horizontal asymptote of the graph of f if lim () x fx b of or lim () x fx b of The line x = a is a vertical asymptote of the graph of f if lim () x a fx o rf or lim () x a fx o rf
Transcript of 2.6: Graphs of Rational Functions Rational Functions...
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2.6: Graphs of Rational Functions Example 1: Sketch the graph then find the domain of f and use limits to describe its behavior at values of x not in its domain using limits.
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Rational Functions Let f and g be polynomial functions with g(x) ≠ 0. Then the function given by
( )( )( )
f xr xg x
is a rational function
Horizontal and Vertical asymptotes: In the graph of a function y = f(x), the line y = b is a horizontal asymptote of the graph of f if
lim ( )x
f x bo�f
or lim ( )x
f x bof
The line x = a is a vertical asymptote of the graph of f if
lim ( )x a
f x�o
rf or lim ( )x a
f xo
rf
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Example 2: Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the rational function.
a) 6
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Steps to Graph a Rational Function
The graph of 0
0
......
)()(
bxbaxa
xgxfy m
m
nn
����
has the following characteristics:
1. End behavior asymptote:
If mn � , the end behavior asymptote is the horizontal asymptote y = 0.
If mn , the end behavior asymptote is the horizontal asymptote m
n
ba
y
If mn ! , the end behavior asymptote is the slant asymptote which is the quotient polynomial function )(xqy where )()()()( xrxqxgxf � . There is no horizontal asymptote.
2. Holes: This occurs when you can factor the numerator and denominator and “cancel” something out.
3. Vertical asymptotes: These occur at the zeros of the denominator, provided that the zeros are not also zeros of the numerator of equal or greater multiplicity.
4. X-intercepts: These occur at the zeros of the numerator, which are not also zeros of the denominator.
5. Y-intercept: This is the value of f(0), if defined.
6. Plot Points: Evaluate the function at x-values on each side of the vertical asymptotes to find points on the graph.
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c) 9
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