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It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior    of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function

f ( x ) = 3 x 5 x 2 x + 3

with end behavior

f ( x ) 3 x 5 x = 3 x 4 ,

the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

x ± ,   f ( x )

Horizontal asymptotes of rational functions

The horizontal asymptote    of a rational function can be determined by looking at the degrees of the numerator and denominator.

  • Degree of numerator is less than degree of denominator: horizontal asymptote at y = 0.
  • Degree of numerator is greater than degree of denominator by one : no horizontal asymptote; slant asymptote.
  • Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients.

Identifying horizontal and slant asymptotes

For the functions below, identify the horizontal or slant asymptote.

  1. g ( x ) = 6 x 3 10 x 2 x 3 + 5 x 2
  2. h ( x ) = x 2 4 x + 1 x + 2
  3. k ( x ) = x 2 + 4 x x 3 8

For these solutions, we will use f ( x ) = p ( x ) q ( x ) ,   q ( x ) 0.

  1. g ( x ) = 6 x 3 10 x 2 x 3 + 5 x 2 : The degree of p = degree of   q = 3 , so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at y = 6 2 or y = 3.
  2. h ( x ) = x 2 4 x + 1 x + 2 : The degree of p = 2 and degree of q = 1. Since p > q by 1, there is a slant asymptote found at x 2 4 x + 1 x + 2 .
    -2 1 4 1 2 12    1 6 13

    The quotient is x 6 and the remainder is 13. There is a slant asymptote at y = x 6.

  3. k ( x ) = x 2 + 4 x x 3 8 : The degree of p = 2   < degree of q = 3 , so there is a horizontal asymptote y = 0.
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Identifying horizontal asymptotes

In the sugar concentration problem earlier, we created the equation C ( t ) = 5 + t 100 + 10 t .

Find the horizontal asymptote and interpret it in context of the problem.

Both the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is t , with coefficient 1. In the denominator, the leading term is 10 t , with coefficient 10. The horizontal asymptote will be at the ratio of these values:

t ,   C ( t ) 1 10

This function will have a horizontal asymptote at y = 1 10 .

This tells us that as the values of t increase, the values of C will approach 1 10 . In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or 1 10 pounds per gallon.

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Identifying horizontal and vertical asymptotes

Find the horizontal and vertical asymptotes of the function

f ( x ) = ( x 2 ) ( x + 3 ) ( x 1 ) ( x + 2 ) ( x 5 )

First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at x = 1 , 2 , and  5 , indicating vertical asymptotes at these values.

The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as x ± ,   f ( x ) 0. This function will have a horizontal asymptote at y = 0. See [link] .

Graph of f(x)=(x-2)(x+3)/(x-1)(x+2)(x-5) with its vertical asymptotes at x=-2, x=1, and x=5 and its horizontal asymptote at y=0.
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Questions & Answers

can you not take the square root of a negative number
Sharon Reply
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
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
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  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
Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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