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f ( x ) = x 2 x + 1

Local behavior: x 1 2 + , f ( x ) , x 1 2 , f ( x )

End behavior: x ± , f ( x ) 1 2

f ( x ) = 2 x x 6

f ( x ) = 2 x x 6

Local behavior: x 6 + , f ( x ) , x 6 , f ( x ) , End behavior: x ± , f ( x ) 2

f ( x ) = x 2 4 x + 3 x 2 4 x 5

f ( x ) = 2 x 2 32 6 x 2 + 13 x 5

Local behavior: x 1 3 + , f ( x ) , x 1 3 , f ( x ) , x 5 2 , f ( x ) , x 5 2 + , f ( x )


End behavior: x ± , f ( x ) 1 3

For the following exercises, find the slant asymptote of the functions.

f ( x ) = 24 x 2 + 6 x 2 x + 1

f ( x ) = 4 x 2 10 2 x 4

y = 2 x + 4

f ( x ) = 81 x 2 18 3 x 2

f ( x ) = 6 x 3 5 x 3 x 2 + 4

y = 2 x

f ( x ) = x 2 + 5 x + 4 x 1

Graphical

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

The reciprocal function shifted up two units.

V . A .   x = 0 , H . A .   y = 2

Graph of a rational function.

The reciprocal function shifted down one unit and left three units.

The reciprocal squared function shifted to the right 2 units.

V . A .   x = 2 ,   H . A .   y = 0

Graph of a rational function.

The reciprocal squared function shifted down 2 units and right 1 unit.

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.

p ( x ) = 2 x 3 x + 4

V . A .   x = 4 ,   H . A .   y = 2 ; ( 3 2 , 0 ) ; ( 0 , 3 4 )

Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.

q ( x ) = x 5 3 x 1

s ( x ) = 4 ( x 2 ) 2

V . A .   x = 2 ,   H . A .   y = 0 ,   ( 0 , 1 )

Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.

r ( x ) = 5 ( x + 1 ) 2

f ( x ) = 3 x 2 14 x 5 3 x 2 + 8 x 16

V . A .   x = 4 ,   x = 4 3 ,   H . A .   y = 1 ; ( 5 , 0 ) ; ( 1 3 , 0 ) ; ( 0 , 5 16 )

Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.

g ( x ) = 2 x 2 + 7 x 15 3 x 2 14 + 15

a ( x ) = x 2 + 2 x 3 x 2 1

V . A .   x = 1 ,   H . A .   y = 1 ; ( 3 , 0 ) ; ( 0 , 3 )

Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.

b ( x ) = x 2 x 6 x 2 4

h ( x ) = 2 x 2 +   x 1 x 4

V . A .   x = 4 ,   S . A .   y = 2 x + 9 ; ( 1 , 0 ) ; ( 1 2 , 0 ) ; ( 0 , 1 4 )

Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.

k ( x ) = 2 x 2 3 x 20 x 5

w ( x ) = ( x 1 ) ( x + 3 ) ( x 5 ) ( x + 2 ) 2 ( x 4 )

V . A .   x = 2 ,   x = 4 ,   H . A .   y = 1 , ( 1 , 0 ) ; ( 5 , 0 ) ; ( 3 , 0 ) ; ( 0 , 15 16 )

Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.

z ( x ) = ( x + 2 ) 2 ( x 5 ) ( x 3 ) ( x + 1 ) ( x + 4 )

For the following exercises, write an equation for a rational function with the given characteristics.

Vertical asymptotes at x = 5 and x = 5 , x -intercepts at ( 2 , 0 ) and ( 1 , 0 ) , y -intercept at ( 0 , 4 )

y = 50 x 2 x 2 x 2 25

Vertical asymptotes at x = 4 and x = 1 , x- intercepts at ( 1 , 0 ) and ( 5 , 0 ) , y- intercept at ( 0 , 7 )

Vertical asymptotes at x = 4 and x = 5 , x -intercepts at ( 4 , 0 ) and ( 6 , 0 ) , Horizontal asymptote at y = 7

y = 7 x 2 + 2 x 24 x 2 + 9 x + 20

Vertical asymptotes at x = 3 and x = 6 , x -intercepts at ( 2 , 0 ) and ( 1 , 0 ) , Horizontal asymptote at y = 2

Vertical asymptote at x = 1 , Double zero at x = 2 , y -intercept at ( 0 , 2 )

y = 1 2 x 2 4 x + 4 x + 1

Vertical asymptote at x = 3 , Double zero at x = 1 , y -intercept at ( 0 , 4 )

For the following exercises, use the graphs to write an equation for the function.

Graph of a rational function with vertical asymptotes at x=-3 and x=4.

y = 4 x 3 x 2 x 12

Graph of a rational function with vertical asymptotes at x=-3 and x=4.
Graph of a rational function with vertical asymptotes at x=-3 and x=3.

y = 9 x 2 x 2 9

Graph of a rational function with vertical asymptotes at x=-3 and x=4.
Graph of a rational function with vertical asymptote at x=1.

y = 1 3 x 2 + x 6 x 1

Graph of a rational function with vertical asymptote at x=-2.
Graph of a rational function with vertical asymptotes at x=-3 and x=2.

y = 6 ( x 1 ) 2 ( x + 3 ) ( x 2 ) 2

Graph of a rational function with vertical asymptotes at x=-2 and x=4.

Numeric

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

f ( x ) = 1 x 2

x 2.01 2.001 2.0001 1.99 1.999
y 100 1,000 10,000 –100 –1,000
x 10 100 1,000 10,000 100,000
y .125 .0102 .001 .0001 .00001

Vertical asymptote x = 2 , Horizontal asymptote y = 0

f ( x ) = x x 3

f ( x ) = 2 x x + 4

x –4.1 –4.01 –4.001 –3.99 –3.999
y 82 802 8,002 –798 –7998
x 10 100 1,000 10,000 100,000
y 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote x = 4 , Horizontal asymptote y = 2

f ( x ) = 2 x ( x 3 ) 2

f ( x ) = x 2 x 2 + 2 x + 1

x –.9 –.99 –.999 –1.1 –1.01
y 81 9,801 998,001 121 10,201
x 10 100 1,000 10,000 100,000
y .82645 .9803 .998 .9998

Vertical asymptote x = 1 , Horizontal asymptote y = 1

Technology

For the following exercises, use a calculator to graph f ( x ) . Use the graph to solve f ( x ) > 0.

f ( x ) = 2 x + 1

f ( x ) = 4 2 x 3

( 3 2 , )

Graph of f(x)=4/(2x-3).

f ( x ) = 2 ( x 1 ) ( x + 2 )

f ( x ) = x + 2 ( x 1 ) ( x 4 )

( 2 , 1 ) ( 4 , )

Graph of f(x)=(x+2)/(x-1)(x-4).

f ( x ) = ( x + 3 ) 2 ( x 1 ) 2 ( x + 1 )

Extensions

For the following exercises, identify the removable discontinuity.

f ( x ) = x 2 4 x 2

( 2 , 4 )

f ( x ) = x 3 + 1 x + 1

f ( x ) = x 2 + x 6 x 2

( 2 , 5 )

f ( x ) = 2 x 2 + 5 x 3 x + 3

f ( x ) = x 3 + x 2 x + 1

( 1 , 1 )

Real-world applications

For the following exercises, express a rational function that describes the situation.

A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.

A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.

C ( t ) = 8 + 2 t 300 + 20 t

For the following exercises, use the given rational function to answer the question.

The concentration C of a drug in a patient’s bloodstream t hours after injection in given by C ( t ) = 2 t 3 + t 2 . What happens to the concentration of the drug as t increases?

The concentration C of a drug in a patient’s bloodstream t hours after injection is given by C ( t ) = 100 t 2 t 2 + 75 . Use a calculator to approximate the time when the concentration is highest.

After about 6.12 hours.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let x = length of the side of the base.

A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/ square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let x = length of the side of the base.

A ( x ) = 50 x 2 + 800 x . 2 by 2 by 5 feet.

A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let x = radius.

A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let x = radius.

A ( x ) = π x 2 + 100 x . Radius = 2.52 meters.

A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let x = radius.

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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