<< Chapter < Page Chapter >> Page >
Graph of f(x)=1/x with its vertical asymptote at x=0.

Vertical asymptote

A vertical asymptote    of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a . We write

As  x a , f ( x ) ,   or as  x a , f ( x ) .

End behavior of f ( x ) = 1 x

As the values of x approach infinity, the function values approach 0. As the values of x approach negative infinity, the function values approach 0. See [link] . Symbolically, using arrow notation

As  x , f ( x ) 0 , and as  x , f ( x ) 0.

Graph of f(x)=1/x which highlights the segments of the turning points to denote their end behavior.

Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote , a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line y = 0. See [link] .

Graph of f(x)=1/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.

Horizontal asymptote

A horizontal asymptote    of a graph is a horizontal line y = b where the graph approaches the line as the inputs increase or decrease without bound. We write

As  x  or  x ,   f ( x ) b .

Using arrow notation

Use arrow notation to describe the end behavior and local behavior of the function graphed in [link] .

Graph of f(x)=1/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.

Notice that the graph is showing a vertical asymptote at x = 2 , which tells us that the function is undefined at x = 2.

As  x 2 , f ( x ) ,  and as  x 2 + ,   f ( x ) .

And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at y = 4. As the inputs increase without bound, the graph levels off at 4.

As  x ,   f ( x ) 4  and as  x ,   f ( x ) 4.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

End behavior: as x ± ,   f ( x ) 0 ; Local behavior: as x 0 ,   f ( x ) (there are no x - or y -intercepts)

Got questions? Get instant answers now!

Using transformations to graph a rational function

Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.

Shifting the graph left 2 and up 3 would result in the function

f ( x ) = 1 x + 2 + 3

or equivalently, by giving the terms a common denominator,

f ( x ) = 3 x + 7 x + 2

The graph of the shifted function is displayed in [link] .

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

Notice that this function is undefined at x = 2 , and the graph also is showing a vertical asymptote at x = 2.

As  x 2 ,   f ( x ) , and as   x 2 + ,   f ( x ) .

As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at y = 3.

As  x ± ,   f ( x ) 3.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.

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

The function and the asymptotes are shifted 3 units right and 4 units down. As x 3 , f ( x ) , and as x ± , f ( x ) 4.

The function is f ( x ) = 1 ( x 3 ) 2 4.

Got questions? Get instant answers now!

Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function f ( x ) = 3 x + 7 x + 2 . This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

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
what is titration
John Reply
what is physics
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
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
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?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 5

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask