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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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