Feel free to modify and personalize this study guide by clicking “Customize.”
Complete the table.
|___________||A horizontal line marking a specific value toward which the graph of a function may approach, but will generally not cross|
|___________||function that is a ratio of two polynomials|
|___________||A single point that is not possible on the graph of an otherwise continuous function|
Remember: Rational functions are defined as where and are polynomials
In your own words, what is an asymtote? ________________________________________
How do you find asymtotes?
To find a vertical asymtote:
Set the denominator to 0
Solve for x
This graph has two vertical asymtotes and one horizontal asymtote:
For more on vertical asymtotes, click here.
To find a horizontal asymtote:
Put in standard form
Remove all terms but the one where x has the largest exponent in numerator and denominator
If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote crosses the axis at
If the degree of the denominator and the numerator are the same, the horizontal asymptote equals to the ratio of the leading coefficients
If the degree of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote
If the degree of the numerator is one greater than the degree of the denominator, then there is an oblique asymtote. To find this asymtote:
- Divide the numerator and the denominator
- Re-write that function in the form where is the quotient and is the remainder
- is the asymtote
Oblique asymtotes are diagonal lines:
For more on oblique asymtotes, click here.
Holes occur when one term in the numerator cancels with one term in the denominator. It is not an asymtote, it is merely a missing point.
For more on holes, click here.