How do you find slant asymptotes?
Understand the Problem
The question is asking how to determine the slant (or oblique) asymptotes of a rational function. This involves analyzing the degrees of the polynomial in the numerator and the denominator and applying polynomial long division if the degree of the numerator is exactly one more than that of the denominator.
Answer
y = x + 2
Answer for screen readers
The final answer is y = x + 2
Steps to Solve
- Identify the degrees of the polynomials
Check the degree of the numerator and the degree of the denominator of your rational function. Slant asymptotes occur if the degree of the numerator is exactly one more than the degree of the denominator.
- Perform polynomial long division
Divide the numerator by the denominator using polynomial long division or synthetic division if applicable. The result will be a linear expression plus a remainder.
- Extract the linear term
The quotient from the polynomial division gives the equation of the slant asymptote. Ignore the remainder term as it becomes insignificant as x approaches infinity.
Example
For the rational function, let’s say ( \frac{x^2 + 3x + 5}{x + 1} ):
- The degree of the numerator ((x^2)) is 2.
- The degree of the denominator ((x)) is 1.
- Since 2 = 1 + 1, we can proceed with the division.
- Setup the division
Divide (x^2 + 3x + 5) by (x + 1):
$$ x^2 \div x = x $$
- Multiply & subtract
Multiply (x) by (x + 1):
$$ x \cdot (x + 1) = x^2 + x $$
Subtract (x^2 + x) from (x^2 + 3x + 5):
$$ (x^2 + 3x + 5) - (x^2 + x) = 2x + 5 $$
- Divide again
Divide (2x) by (x):
$$ 2x \div x = 2 $$
Multiply & subtract again:
$$ 2 \cdot (x + 1) = 2x + 2 $$
Subtract (2x + 2) from (2x + 5):
$$ (2x + 5) - (2x + 2) = 3 $$
- Form the slant asymptote equation
The quotient of the division is (x + 2). Therefore, the slant asymptote is:
$$ y = x + 2 $$
The final answer is y = x + 2
More Information
Slant asymptotes are found in rational functions where the degree of the numerator is one more than the degree of the denominator. They show the behavior of the function at infinity.
Tips
A common mistake is failing to correctly perform the polynomial long division, particularly in keeping track of the signs and coefficients. Double-check your work to ensure accuracy.