2.3 Polynomial and
Rational Functions
• Identify a Polynomial Function
• Identify a Rational Function
• Find Vertical and Horizontal Asymptotes for
Rational Functions
• Review finding x and y intercepts of graphs
Polynomial and rational functions are often
used to express relationships in application
problems.
Scary Math Definition for Polynomial
Be sure to know the end behavior properties
(2 and 3 below).
Scary Math Definition for Rational Function
Go forward a few slides to see the
easy-to-understand explanation.
DEFINITION:
The line x = a is a vertical asymptote
if any of the following limit statements
are true:
lim f x   
lim f x   
xa
xa
lim f x   
lim f x   .
xa
xa
We will learn about limits in section 3.1
•If c makes the denominator
zero, but doesn’t make the
numerator zero, then x = c is a
vertical asymptote.
•If c makes both the
denominator and the numerator
zero, then there is a hole at x=c
Example of hole
Example 2: Determine the vertical
asymptotes of the function given by
x( x  2)
f ( x) 
x( x  1)( x  1)
Example 2
There are Vertical Asymptotes at
x = 1 and x = -1.
There isn‘t a vertical asymptote at x = 0.
Since 0 makes both the numerator and
denominator equal zero, there is a hole
where x = 0.
• Since x = 1 and x = –1 make
the denominator 0, but don’t
make the numerator 0, x = 1
and x = –1 are vertical
asymptotes.
• x=0 is not a vertical asymptote
since it makes both the
numerator and denominator 0.
The line y = b is a horizontal
asymptote if either or both of the
following limit statements are true:
lim f x   b
x
or
lim f x   b.
x
We will learn about limits in section 3.1.
The graph of a rational function may or may not
cross a horizontal asymptote. Horizontal
asymptotes are found by comparing the degree
of the numerator to the degree of the
denominator.
3 cases
Same: y = leading coefficient/leading
coefficient
BOB: y = 0 (bottom degree bigger)
TUB: undefined-no H.A. (top degree bigger
Bob and tub are not in the textbook.
Determine the horizontal asymptote
of the function given by
3x  2x  4
f (x) 
.
2
2x  x  1
2
Example of vertical and
horizontal asymptotes
8
Find the intercepts of y 
5 - 3x
Intercepts
• The x-intercepts occur at values for
which y = 0. For a fraction to = 0,
the numerator must equal 0. Since
8 ≠ 0, there are no x-intercepts.
• To find the y-intercept, let x = 0.
8
y
5
y-intercept
(0, 8/5)
Suppose the average cost per unit C
in dollars, to produce x units of a product is
given by
500
C x  
x  30
(a) find C (10), C (50), C (100)
(b) How much would 10 units cost?
(c) Identify any intercepts & asymptotes.
Graph the function to verify your answers.
(a) $12.50, $6.25, $3.85
(b) $12.50 x 10 = $125.00
(c) V.A. x = -30
H.A. y = 0
no x-intercepts
y-intercept (0, 50/3)