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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

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f ( x ) = 2 x x 6

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

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

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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

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

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

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

y = 2 x + 4

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

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

y = 2 x

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

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

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The reciprocal squared function shifted to the right 2 units.

V . A .   x = 2 ,   H . A .   y = 0

Graph of a rational function.
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The reciprocal squared function shifted down 2 units and right 1 unit.

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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.
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q ( x ) = x 5 3 x 1

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

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

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

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

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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

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

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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

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

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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

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

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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

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

y = 9 x 2 x 2 9

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Graph of a rational function with vertical asymptote at x=1.

y = 1 3 x 2 + x 6 x 1

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

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

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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

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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

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

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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

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Technology

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

f ( x ) = 4 2 x 3

( 3 2 , )

Graph of f(x)=4/(2x-3).
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f ( x ) = 2 ( x 1 ) ( x + 2 )

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

( 2 , 1 ) ( 4 , )

Graph of f(x)=(x+2)/(x-1)(x-4).
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f ( x ) = ( x + 3 ) 2 ( x 1 ) 2 ( x + 1 )

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Extensions

For the following exercises, identify the removable discontinuity.

f ( x ) = x 2 4 x 2

( 2 , 4 )

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

( 2 , 5 )

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

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

( 1 , 1 )

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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.

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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

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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?

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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.

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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.

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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.

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

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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.

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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.

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Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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