P.o.D.
1.) Find the real, rational zeros of 𝑓(𝑥 ) =
𝑥 4 − 𝑥 3 + 𝑥 2 − 3𝑥 − 6
2.) Find a fourth degree polynomial
function with real coefficients that has
-1,-1, and 3i as zeros.
1.) X= -1, 2
2.) 𝑓(𝑥 ) = 𝑥 4 + 2𝑥 3 + 10𝑥 2 + 18𝑥 + 9
2.6 – Rational Functions
Learning Target: be able to find the
domains of rational functions; find
horizontal and vertical asymptotes; use
rational functions to model real-world
problems.
Rational Function:
- A fractional equation with a variable
expression in both the numerator and
denominator.
Recall: Domain is all the possible inputs
(or x-values) of a function.
EX: Find the domain of 𝑓(𝑥 ) =
3𝑥
𝑥−1
and
discuss the behavior of f near any
excluded x-values.
𝑥 − 1 ≠ 0 → 𝑥 ≠ 1. The domain is all
real numbers except x=1.
As x approaches 1 from the
left, the graph tends towards negative
infinity. As x approaches 1 from the
right, the graphs tends towards positive
infinity.
*This is an example of an asymptote.
This is also known as the MC Hammer
Theorem.
https://www.youtube.com/watch?v=z5X5zh00rdg
Definitions of Vertical and Horizontal
Asymptotes:
1. The line x=a is a vertical asymptote
of the graph f if
𝑓(𝑥 ) → ∞ 𝑜𝑟 𝑓 (𝑥 ) → −∞ 𝑎𝑠
𝑥 → 𝑎, either from the right or from
the left.
2. The line y=b is a horizontal
asymptote of the graph f if f(x)b
as 𝑥 → ∞ 𝑜𝑟 𝑥 → −∞.
EX: Find the horizontal and vertical
asymptotes of 𝑓(𝑥 ) =
5𝑥 2
.
𝑥 2 −1
To find a horizontal asymptote, we need
to look at the degree of the numerator
and denominator. If the degrees are
different, no horizontal asymptote exists.
If the degree is the same, the horizontal
asymptote can be found by dividing the
leading coefficients.
5
𝑦= =5
1
Vertical asymptotes will occur where the
denominator is undefined.
𝑥 2 − 1 = 0 → 𝑥 2 = 1 → 𝑥 = ±√1 →
𝑥 = ±1
Therefore, we have a horizontal
asymptote at y=5, and we have two
vertical asymptotes at x=1 and x= -1.
We could also have determined these
asymptotes graphically.
(show how to use the
calculator to determine asymptotes)
http://www.youtube.com/watch?v=OmwhelahNBY
*Study the “Guidelines for Analyzing
Graphs of Rational Functions” on page
187.
EX: Sketch the graph of 𝑓(𝑥 ) =
1
𝑥+3
and
state its domain. Be sure to label at least
three points and all asymptotes.
The domain is all real
numbers except x= -3. It has a horizontal
asymptote at y=0. It has a vertical
asymptote at x= -3. Some points on the
graph are (-2,1), (-4, -1), and (-7, -1/4).
EX: Sketch the graph of 𝐶 (𝑥 ) =
3+2𝑥
1+𝑥
Vertical Asymptote at
x= -1. Horizontal asymptote at y=2.
.
Points on the curve include (-2,1), (0,3),
and (1, 2.5)
EX: Sketch the graph of 𝑓(𝑥 ) =
3𝑥
.
𝑥 2 +𝑥−2
Vert. Asymp. At x= -2, 1.
Horiz. Asymp. at y=0. Points include
(-4,-1.2), (0,0), and (2,1.5).
Slant (Oblique) Asymptotes:
If the degree of the numerator is exactly
one degree higher than that of the
denominator, a SLANT asymptote will
exist.
- We will find slant asymptotes by
dividing.
EX: Sketch the graph of 𝑓(𝑥 ) =
3𝑥 2 +1
𝑥
and find its slant asymptote.
We can find the slant asymptote by
dividing.
3𝑥 2
𝑥
1
1
𝑥
𝑥
+ = 3𝑥 + . The slant asymptote
will be the linear term after dividing.
Y=3x.
We can graph this with the original
function to confirm that it is the slant
asymptote.
EX: Find the slant asymptote for 𝑓 (𝑥 ) =
𝑥3
.
2𝑥 2 −8
Simply divide the leading terms.
𝑥3
𝑥 1
𝑦= 2= = 𝑥
2𝑥
2 2
Confirm this graphically.
http://www.youtube.com/watch?v=6mIOXC_W7_o
Cost Benefit Model:
EX: The cost C (in millions of dollars)
for removing p% of the industrial and
municipal pollutants discharged into a
river is given by 𝐶 =
255𝑝
100−𝑝
, where
0<p<100. A proposed new law would
require companies to remove 80% of
their pollutants. The current law requires
45%. How much additional cost would
the companies incur as a result of this
law?
255(45) 11475 2295
𝐶 (45%) =
=
=
100 − 45
55
11
= 208.63
255(80) 20400
𝐶 (80%) =
=
= 1020
100 − 80
20
𝐶 = 1020 −
2295
11
=
8925
11
≈
$811.36Million
Upon completion of this lesson, you
should be able to:
1. Find asymptotes
a. Horizontal
b. Vertical
c. Oblique (Slant)
For more information, visit
https://www.youtube.com/watch?v=HeqfhnKncjc
HW Pg. 193 6-72 6ths, 74, 85-88