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Precalculus
Lesson 2.6 – Lesson 2.7
Rational Functions, Asymptotes
& Graphs of Rational Functions
A rational function is a function of the
form
N ( x)
f ( x) 
D( x)
where N and D are polynomial functions
and N is not the zero polynomial. The
domain consists of all real numbers
except those for which the denominator
D is 0.
Find the domain of the rational functions
x 1
(a) R( x)  2
x  8 x  12
x4
(b) R( x )  2
x  16
5
(c) R( x )  2
x 9
Vertical Asymptotes
N ( x)
f ( x) 
D( x)
If the rational function is in lowest terms (no
common factors of N and D), then:
The graph of f has vertical asymptotes at the
zeros of D(x).
x=c
y
x
y
x=c
x
Find the vertical asymptotes, if any, of the graph
of the rational functions:
3
(a) R( x )  2
x 1
x 5
(b) R( x )  2
x 1
x3
(c) R( x )  2
x  x  12
N ( x) an x n  an 1 x n 1    a1 x  a0
f ( x) 

D( x) bm x m  bm1 x m1    b1 x  b0
in which the degree of the numerator is n and the
degree of the denominator is m.
1. If n < m, then the line y = 0 (x-axis) is a horizontal
asymptote of the graph of f.
2. If n = m, then the line y = an / bm is a horizontal
asymptote of the graph of f.
3. If n = m + 1, then y = ax + b is a slant (oblique)
asymptote of the graph of f. Found using long division.
4. If n > m + 1, the graph of f has neither a horizontal
nor a slant (oblique) asymptote.
y
y = f(x)
y=L
x
y
y=L
x
y = f(x)
If an asymptote is neither horizontal nor
vertical it is called oblique.
y
x
Find the horizontal asymptotes of the graph of the
rational function, if any:
3x  4 x  15
(a) R( x )  3
2
x  4x  7x  1
2
2 x2  4 x  1
(b) R( x ) 
2
3x  x  5
Find the slant asymptote of the graph of the rational
function:
x  4x  1
(c) R( x ) 
x2
2
Point discontinuity is like a hole in a graph. If the
original related expression is undefined for x a but
the simplified expression is defined for x a, then there
is a hole in the graph at x a.
Analytical and Numerical Explanations
•Determine the domains of f and g.
•Simplify f and find any vertical asymptotes.
•Complete the table.
•Explain how the two functions differ.
2x  8
f ( x)  2
x  9 x  20
x
f(x)
g(x)
0
1
2
2
g ( x) 
x5
3
4
5
6
Plotting data on your graphing
calculator:
To Enter Data in a List:
• Press the STAT button.
• Press 1 to edit.
• L1 will be your
independent variable (x).
• L2 will be your
dependent variable (y).
Year
Number, N
1993
813
1994
941
1995
962
1996
1053
1997
1132
1998
1194
1999
1205
2000
1244
2001
1254
2002
1262
The information that you entered into your lists
represents the number N of threatened and
endangered species in the United States from 19932002. The data can be approximated using the
equation:
42.58t 2  690
N
0.03t 2  1
t represents the year,
with t = 3 corresponding to 1993
•Use your graphing calculator to plot the data and graph the function in the same
viewing window. How closely does the function represent the data?
•Use the function to estimate the number of threatened and endangered species in
2006.
•Would this model be useful for estimating the number of threatened and endangered
species in future years? Explain?
Graphing Rational Functions
• Simplify the function if possible.
• Find and plot the y-intercept (if any) by evaluating f(0).
• Find the zeros of the numerator by solving the equation
N(x) = 0. Then plot the corresponding x-intercepts.
• Find the zeros of the denominator by solving the equation
D(x) = 0. Then sketch the corresponding vertical asymptotes
using dashed vertical lines.
• Find and sketch the horizontal asymptotes (if any) using
dashed horizontal lines.
• Find and sketch any slant asymptotes (if any) using a
dashed line.
• Find and plot any holes.
• Plot at least one point between and one point beyond each
x-intercept and vertical asymptote.
• Use smooth curves to draw your graph.
Sketch the Graph by Hand:
3
f ( x) 
x2
Sketch the Graph by Hand:
2x 1
f ( x) 
x
Sketch the Graph by Hand:
x
f ( x)  2
x x2
Sketch the Graph by Hand:
x2  9
f ( x)  2
x  2x  3
Sketch the Graph by Hand:
x2  x  2
f ( x) 
x 1
Application: Finding a Minimum Area
(Calculus Preview)
A rectangular page is designed to contain 48 square
inches of print. The margins on each side of the page
are 1.5” wide. The margins at the top and bottom are
each 1” deep. What should the dimensions of the
page be so that the minimum amount of paper is
used?