# 3.7 Rational functions  (Page 2/16)

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## Vertical asymptote

A vertical asymptote    of a graph is a vertical line $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ where the graph tends toward positive or negative infinity as the inputs approach $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ We write

## End behavior of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}$

As the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approach infinity, the function values approach 0. As the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approach negative infinity, the function values approach 0. See [link] . Symbolically, using arrow notation

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 $\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See [link] .

## Horizontal asymptote

A horizontal asymptote    of a graph is a horizontal line $\text{\hspace{0.17em}}y=b\text{\hspace{0.17em}}$ where the graph approaches the line as the inputs increase or decrease without bound. We write

## Using arrow notation

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

Notice that the graph is showing a vertical asymptote at $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ which tells us that the function is undefined at $\text{\hspace{0.17em}}x=2.$

And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at $\text{\hspace{0.17em}}y=4.\text{\hspace{0.17em}}$ As the inputs increase without bound, the graph levels off at 4.

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

End behavior: as Local behavior: as (there are no x - or y -intercepts)

## 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

$f\left(x\right)=\frac{1}{x+2}+3$

or equivalently, by giving the terms a common denominator,

$f\left(x\right)=\frac{3x+7}{x+2}$

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

Notice that this function is undefined at $\text{\hspace{0.17em}}x=-2,\text{\hspace{0.17em}}$ and the graph also is showing a vertical asymptote at $\text{\hspace{0.17em}}x=-2.$

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 $\text{\hspace{0.17em}}y=3.$

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.

The function and the asymptotes are shifted 3 units right and 4 units down. As $\text{\hspace{0.17em}}x\to 3,f\left(x\right)\to \infty ,\text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to ±\infty ,f\left(x\right)\to -4.$

The function is $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{\left(x-3\right)}^{2}}-4.$

## Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{3x+7}{x+2}.\text{\hspace{0.17em}}$ 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.

what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?