Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an
x -intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of
x -intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.
Writing rational functions from intercepts and asymptotes
If a
rational function has
x -intercepts at
$\text{\hspace{0.17em}}x={x}_{1},{x}_{2},\mathrm{...},{x}_{n},\text{\hspace{0.17em}}$ vertical asymptotes at
$\text{\hspace{0.17em}}x={v}_{1},{v}_{2},\dots ,{v}_{m},\text{\hspace{0.17em}}$ and no
$\text{\hspace{0.17em}}{x}_{i}=\text{any}{v}_{j},\text{\hspace{0.17em}}$ then the function can be written in the form:
where the powers
$\text{\hspace{0.17em}}{p}_{i}\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}{q}_{i}\text{\hspace{0.17em}}$ on each factor can be determined by the behavior of the graph at the corresponding intercept or asymptote, and the stretch factor
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ can be determined given a value of the function other than the
x -intercept or by the horizontal asymptote if it is nonzero.
Given a graph of a rational function, write the function.
Determine the factors of the numerator. Examine the behavior of the graph at the
x -intercepts to determine the zeroes and their multiplicities. (This is easy to do when finding the “simplest” function with small multiplicities—such as 1 or 3—but may be difficult for larger multiplicities—such as 5 or 7, for example.)
Determine the factors of the denominator. Examine the behavior on both sides of each vertical asymptote to determine the factors and their powers.
Use any clear point on the graph to find the stretch factor.
Writing a rational function from intercepts and asymptotes
Write an equation for the rational function shown in
[link] .
The graph appears to have
x -intercepts at
$\text{\hspace{0.17em}}x=\u20132\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asymptotes. The one at
$\text{\hspace{0.17em}}x=\u20131\text{\hspace{0.17em}}$ seems to exhibit the basic behavior similar to
$\text{\hspace{0.17em}}\frac{1}{x},\text{\hspace{0.17em}}$ with the graph heading toward positive infinity on one side and heading toward negative infinity on the other. The asymptote at
$\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ is exhibiting a behavior similar to
$\text{\hspace{0.17em}}\frac{1}{{x}^{2}},\text{\hspace{0.17em}}$ with the graph heading toward negative infinity on both sides of the asymptote. See
[link] .
We can use this information to write a function of the form
$$f(x)=a\frac{(x+2)(x-3)}{(x+1){(x-2)}^{2}}.$$
To find the stretch factor, we can use another clear point on the graph, such as the
y -intercept
$\text{\hspace{0.17em}}(0,\mathrm{\u20132}).$
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
GREAT ANSWER THOUGH!!!
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
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.
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
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
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.
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
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?
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...