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Home > Algebra calculators > Asymptotes of a function example
11. Asymptotes of a function example ( Enter your problem )
( Enter your problem )
  1. `y=1/(x+4)` Example-1
  2. `y=(x^2+2x-3)/(x^2-5x-6)` Example-2
  3. `y=(x+2)/(x^2+2x-8)` Example-3
  4. `y=(x^3-8)/(x^2+9)` Example-4
 		Other related methods
  1. Domain of a function
  2. Range of a function
  3. Inverse of a function
  4. Properties of a function
  5. Parabola Vertex of a function
  6. Parabola focus
  7. axis symmetry of a parabola
  8. Parabola Directrix
  9. Intercept of a function
  10. Parity of a function
  11. Asymptotes of a function
3. `y=(x+2)/(x^2+2x-8)` Example-3
(Previous example)

4. `y=(x^3-8)/(x^2+9)` Example-4


4. `y=(x^3-8)/(x^2+9)`, find Asymptotes of a function

Solution:
Asymptotes :
Now, find domain of `y=(x^3-8)/(x^2+9)`

`y=(x^3-8)/(x^2+9)`

`x^2+9=0`

`=>x^2+9=0`

`=>x^2=0-9`

`=>x^2=-9`

Domain : all x
Vertical asymptote : none
The highest power in the numerator is 3
The highest power in the denominator is 2
Slant(Oblique) asymptote :
Final Solution
 ```x``+``0`  
`color{blue}{x^2+9}``` `x^3``+` `0x^2``+` `0x``-` `8`  
 ``−`x^3``+`−`9x` `x xx (color{blue}{x^2+9})`
 `` `0x^2``-` `9x``-` `8`  
 ``−`0x^2``+`−`0` `color{green}{0} xx (color{blue}{x^2+9})`
 `-` `9x``-` `8`  

Final answer `= "Quotient" + (color{Magenta}{"Remainder"})/(color{blue}{"Divisor"})`.
`:.` Final answer = `x+0 + (color{Magenta}{-9x-8})/(color{blue}{x^2+9})`
 
Here, Divisor = `x^2+9`
Dividend = `x^3-8`
Quotient = `x+0`
Remainder = `-9x-8`


Step by step division solution

Step - 1 :
1. Divide the first term of the dividend by the first term of the divisor : `(x^3)/(x^2)=color{green}{x}`

2. Write down the calculated result `color{green}{x}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{x} xx (color{blue}{x^2+9})=color{red}{x^3+9x}`

4. Subtract this result from the dividend
`(x^3+0x^2+0x-8)-(color{red}{x^3+9x})=color{Magenta}{0x^2-9x-8}`

 ```x`  
`color{blue}{x^2+9}``` `x^3``+` `0x^2``+` `0x``-` `8`  
 ``−`x^3``+`−`9x` `color{green}{x} xx (color{blue}{x^2+9})`
 `` `0x^2``-` `9x``-` `8`  


Step - 2 :
1. Divide the first term of the dividend by the first term of the divisor : `(0x^2)/(x^2)=color{green}{0}`

2. Write down the calculated result `color{green}{0}` in the upper part of the table.

3. Multiply it by the divisor `color{green}{0} xx (color{blue}{x^2+9})=color{red}{0x^2+0}`

4. Subtract this result from the remainder
`(0x^2-9x-8)-(color{red}{0x^2+0})=color{Magenta}{-9x-8}`

 ```x``+``0`  
`color{blue}{x^2+9}``` `x^3``+` `0x^2``+` `0x``-` `8`  
 ``−`x^3``+`−`9x` `x xx (color{blue}{x^2+9})`
 `` `0x^2``-` `9x``-` `8`  
 ``−`0x^2``+`−`0` `color{green}{0} xx (color{blue}{x^2+9})`
 `-` `9x``-` `8`  


Slant(Oblique) asymptote : `y=x`

Horizontal asymptote : none

This material is intended as a summary. Use your textbook for detail explanation.
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3. `y=(x+2)/(x^2+2x-8)` Example-3
(Previous example)


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