12/11/18
1
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.
12/11/18
2
• 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
12/11/18
3
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.
! " = $%"% + $%'("%'( + ⋯+ $("( + $* "*+,", + +,'(",'( +⋯+ +("( + +* "*
Look at the degree of the leading term for the numerator and the denominator.
Graph the following functions in Desmos.• Estimate their end-behavior asymptote.
• What do you notice about the highest degree terms in the numerator and denominator for every function?
• What do you notice about the coefficients of the highest degree term in every function?
• These ALL are horizontal asymptotes using the quotient of the leading coefficients.
! " = −3"& + " + 12"& − 4
+ " = 2"& + " − 2"& − 1, " = " + 3
" − 5
ℎ " = "& − " − 2"& − 2" − 8
12/11/18
4
So far we have learned…
1. If n < m, then the end behavior is a horizontal asymptote y = 0.
2. If n = m, then the end behavior is a horizontal asymptote ! = #$
%&.
3. After you simplify the rational function, set the numerator equal to 0 and solve. The solutions are the x-intercepts.
' ( = )*(* + )*,-(*,- + ⋯+ )-(- + )/ (/01(1 + 01,-(1,- +⋯+ 0-(- + 0/ (/
Look at the leading coefficient for the numerator and the denominator.
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?
• 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
12/11/18
5
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
So far we have learned…
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.
4. After you simplify the rational function, set the numerator equal to 0 and solve. The solutions are the x-intercepts.
' ( = )*(* + )*,-(*,- + ⋯+ )-(- + )/ (/01(1 + 01,-(1,- +⋯+ 0-(- + 0/ (/
Look at the degree for the numerator and the denominator.
12/11/18
6
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
12/11/18
7
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.
Top Related