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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

what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
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Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
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Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
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ARC
A child is a member of community not society elucidate ?
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Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
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nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
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Jonathan
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generally amnesi how long yrs memory loss
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interpersonal relationships
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What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
What are the treatment for autism?
Magret Reply
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
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Jharna
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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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