October 23, 2014
Find all of the zeros.
f(x) = x3 - x2 - x - 2
October 23, 2014
(2.6) Graphs of Rational Functions
Objective: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
Why: Rational functions are used in calculus and scientific
appications such as inverse proportions.
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
Rational Function:
r(x) =
f(x)
g(x)
,
where f(x) and g(x) are polynomials
The Reciprocal Function
f(x) =
1
x
Analyze the function and use limits to describe the
behavior at the value(s) of x not in its domain.
Domain: ____________________________________
Range: _____________________________________
Continuity: __________________________________
Increasing/Decreasing behavior: _________________
Symmetry: __________________________________
Boundedness: _______________________________
Local Extrema: _______________________________
Horizontal Asymptotes: ________________________
Vertical Asymptotes: __________________________
End behavior: ________________________________
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
Find the domain and use limits to describe its behavior at
values of x not in its domain.
f(x) =
1
x-3
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
1
x
and use limits to describe the corresponding behavior.
Describe the transformations made to f(x) =
a. f(x) = 1 + 1
x
c. f(x) =
1
x+1
b. f(x) = 1 - 2
x
d. f(x) =
1
x-2
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
1
x
and use limits to describe the corresponding behavior.
Describe the transformations made to f(x) =
1. g(x) = 2
x+3
2. h(x) = 3x - 7
x-2
October 23, 2014
Graph of a Rational Function
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
1. End behavior asymptote:
N < D then horizontal asymptote is y = 0
N=D
N>D
then horizontal asymptote is
y=
an
bm
then no horizontal asymptote, but a slant
asymptote: y = q(x)
2. Vertical asymptotes: occur at zeros of
denominator, provided they are not also zeros of
the numerator.
3. x-intercepts: zeros of numerator, provided they are not
also zeros of the denominator.
4. y-intercept: the value of f(0), if defined.
October 23, 2014
October 23, 2014
Graph f(x) =
x3
x2- 9
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
October 23, 2014
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
Graph f(x) = 2x2 - 2
x2 - 4
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
Graph f(x) = x3 - 3x2 + 3x + 1
x-1
October 23, 2014
Obj: To describe the graphs of rational functions, identify
horizontal and vertical asymptotes, and predict the end behavior.
HW:
HR: (2.6) Pg. 225: 1-7odd, 11-18, 37-43odd