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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

(1+cosA)(1-cosA)=sin^2A
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questions
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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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