October 25, 2012
2.6
Graphs of Rational Functions
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 25, 2012
Find the domain and use limits to describe its behavior at
values of x not in its domain.
f(x) =
1
x-3
October 25, 2012
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 25, 2012
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 25, 2012
Graph of a Rational Function
1. End Behavior Asymptote:
N<D
then horizontal asymptote: y = 0
N=D
then horizontal asymptote: y = lead. coef. num
lead coef. den
N>D
then no horizontal asymptote, but a
quotient asymptote. (Divide to find equation)
2. Vertical asymptotes: occur at zeros of den., provided
they are not also zeros of the num.
3. x-intercepts: zeros of the num, provided they are not
also zeros of the den.
4. y-intercept: the value of f(0), if defined
October 25, 2012
Graph f(x) = 2x2 - 2
x2 - 4
October 25, 2012
Graph f(x) =
x3
x -9
2
October 25, 2012
f(x) = 3x - 6
x+2
October 25, 2012
Graph f(x) = x3 - 3x2 + 3x + 1
x-1