End –Behavior Asymptotes

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End – Behavior Asymptotes

Going beyond horizontal Asymptotes

We will..1. Learn how to find horizontal asymptotes

without simplifying.2. Learn how to find an oblique asymptote.3. Learn how to find x-intercepts.4. Utilize our knowledge to graph rational

functions.

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• An end-behavior asymptote is an asymptote used to describe how the ends of a function behave.

• It is possible to determine these asymptotes without much work.

• Rational functions behave differently when the numerator isn’t a constant.

• There are two types of end-behavior asymptotes a rational function can have:• (1) horizontal• (2) oblique

Graph the following functions in Desmos.• Estimate their end-behavior asymptote.

• What do you notice about the highest degree terms in the numerator and denominator for every function?

• Look at g(x) and h(x). What do you notice about the graph and the numerator?

• These ALL have horizontal asymptotes of 0.• The numerator will give you the x-intercept after the

rational function is simplified.

! " = −3"& + " + 12"* − 4, " = " − 2

"& − 1 ℎ " = " + 6"& − 2" − 8

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So far we have learned…

1. If n < m, then the end behavior is a horizontal asymptote y = 0.

2. After you simplify the rational function, set the numerator equal to 0 and solve. The solutions are the x-intercepts.

! " = $%"% + $%'("%'( + ⋯+ $("( + $* "*+,", + +,'(",'( +⋯+ +("( + +* "*

Look at the degree of the leading term for the numerator and the denominator.



• What do you notice about the coefficients of the highest degree term in every function?

• These ALL are horizontal asymptotes using the quotient of the leading coefficients.

! " = −3"& + " + 12"& − 4

+ " = 2"& + " − 2"& − 1, " = " + 3

" − 5

ℎ " = "& − " − 2"& − 2" − 8

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1. If n < m, then the end behavior is a horizontal asymptote y = 0.

2. If n = m, then the end behavior is a horizontal asymptote ! = #$

%&.


' ( = )*(* + )*,-(*,- + ⋯+ )-(- + )/ (/01(1 + 01,-(1,- +⋯+ 0-(- + 0/ (/

Look at the leading coefficient for the numerator and the denominator.



• What do you notice about the graphs of these functions?• These ALL are oblique asymptotes NOT horizontal. • We use long or synthetic division to find them.

! " = −3"& + " + 12" − 4

+ " = 2", + " − 2" − 1

ℎ " = ". − 2" + 1/ " = 4"& − 3" − 7

2" + 3

1 " = "& − " − 22" − 8

−20, 20 x −30, 30

−20, 20 x −30, 30

−30, 30 x −300, 500−5, 5 x −30, 30

−30, 30 x −250, 150

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Let’s find the oblique asymptote using long division.

! " = 4"% − 3" − 72" + 3

−20, 20 x −30, 30

. " = 2"/ + " − 2" − 1

−5, 5 x −30, 30

ℎ " = "3 − 2" + 1

−30, 30 x −300, 500

4 " = "% − " − 22" − 8

−20, 20 x −30, 30

6 " = −3"% + " + 12" − 4

−30, 30 x −250, 150


1. If n < m, then the end behavior is a horizontal asymptote y = 0.2. If n = m, then the end behavior is a horizontal asymptote ! = #$

%&.

3. If n > m, then the end behavior is an oblique asymptote and is found using long/synthetic division.


' ( = )*(* + )*,-(*,- + ⋯+ )-(- + )/ (/01(1 + 01,-(1,- +⋯+ 0-(- + 0/ (/

Look at the degree for the numerator and the denominator.

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Here’s a synopsis of rational functions:

Practice1. Domain2. Range3. Vertical asymptote(s)4. Holes5. Horizontal or oblique asymptote6. X-intercept(s)7. Y-intercept(s)8. Does the function cross the

horizontal or oblique asymptote?

! " = " + 6"& + 7" + 6

( " = −""& − 4"

ℎ " = 4"" + 1

- " = " + 4" − 4

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Let’s review thus far..1. How do you know if a rational function will have a

horizontal asymptote or oblique asymptote? 2. How do you find horizontal asymptotes without

simplifying?3. How do you to find an oblique asymptote of a

rational function?4. How do you find x-intercepts of a rational function

algebraically?5. How do you find y-intercepts of a rational function

algebraically?

Mathia time

Keep your notes out.

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End –Behavior Asymptotes