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Writing rational functions

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 x = x 1 , x 2 , ... , x n , vertical asymptotes at x = v 1 , v 2 , , v m , and no x i = any  v j , then the function can be written in the form:

f ( x ) = a ( x x 1 ) p 1 ( x x 2 ) p 2 ( x x n ) p n ( x v 1 ) q 1 ( x v 2 ) q 2 ( x v m ) q n

where the powers p i or q i on each factor can be determined by the behavior of the graph at the corresponding intercept or asymptote, and the stretch factor a 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.

  1. 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.)
  2. Determine the factors of the denominator. Examine the behavior on both sides of each vertical asymptote to determine the factors and their powers.
  3. 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] .

Graph of a rational function.

The graph appears to have x -intercepts at x = –2 and x = 3. At both, the graph passes through the intercept, suggesting linear factors. The graph has two vertical asymptotes. The one at x = –1 seems to exhibit the basic behavior similar to 1 x , with the graph heading toward positive infinity on one side and heading toward negative infinity on the other. The asymptote at x = 2 is exhibiting a behavior similar to 1 x 2 , with the graph heading toward negative infinity on both sides of the asymptote. See [link] .

Graph of a rational function denoting its vertical asymptotes and x-intercepts.

We can use this information to write a function of the form

f ( x ) = a ( 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 ( 0 , –2 ) .

−2 = a ( 0 + 2 ) ( 0 3 ) ( 0 + 1 ) ( 0 2 ) 2 −2 = a 6 4 a = 8 6 = 4 3

This gives us a final function of f ( x ) = 4 ( x + 2 ) ( x 3 ) 3 ( x + 1 ) ( x 2 ) 2 .

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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
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Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Practice Key Terms 5

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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