Slant Asymptote Formula
A rational function is a polynomial ratio in which the denominator polynomial should not be equal to zero. It is a function that is the polynomial ratio. A rational function is any function of one variable, x, that can be expressed as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials such that q(x) β 0. There are three sorts of asymptotes for a rational function, that is, horizontal, vertical, and slant asymptotes.
Slant Asymptote
A slant asymptote is a hypothetical slant line that seems to touch a portion of the graph. A rational function has a slant asymptote only when the degree of the numerator (a) is exactly one more than the degree of the denominator (b). In other words, the deciding condition is, a + 1 = b. For example, a slant asymptote exists for the function f(x) = x + 1 as the degree of the numerator is 1, which is one greater than that of the denominator. The general equation of slant asymptote of a rational function is of the form Q = mx + c, which is called quotient function produced by long dividing the numerator by the denominator.
Slant Asymptote Formula
For a rational function f(x) of the form g(x)/h(x), the slant asymptote, S(x) is of the form:
S(x) =
The value of quotient S(x) is calculated using long division method for the dividend g(x) and divisor h(x).
Example: Obtain the slant asymptote for the function: y = (x2 β 3x β 10)/(x β 5).
Solution:
We have, f(x) = (x2 β 3x β 10)/(x β 5).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x + 2, the slant asymptote for the given function f(x) is,
S(x) = x + 2
Sample Problems on Slant Asymptote Formula
Problem 1. Obtain the slant asymptote for the function: y = (x2 β 2x β 24)/(x + 4).
Solution:
We have, f(x) = (x2 β 2x β 24)/(x + 4).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x β 6, the slant asymptote for the given function f(x) is,
S(x) = x β 6
Problem 2. Obtain the slant asymptote for the function: y = (x2 β 2x β 8)/(x + 2).
Solution:
We have, f(x) = (x2 β 2x β 8)/(x + 2).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x β 4, the slant asymptote for the given function f(x) is,
S(x) = x β 4
Problem 3. Obtain the slant asymptote for the function: y = (x2 β 7x + 10)/(x β 2).
Solution:
We have, f(x) = (x2 β 7x + 10)/(x β 2).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x β 5, the slant asymptote for the given function f(x) is,
S(x) = x β 5
Problem 4. Obtain the slant asymptote for the function: y = (x2 β 3x β 28)/(x β 7).
Solution:
We have, f(x) = (x2 β 3x β 28)/(x β 7).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x + 4, the slant asymptote for the given function f(x) is,
S(x) = x + 4
Problem 5. Obtain the slant asymptote for the function: y = (x2 β 3x β 18)/(x + 3).
Solution:
We have, f(x) = (x2 β 3x β 18)/(x + 3).
Here f(x) has a slant asymptote as the degree of numerator is one more than that of denominator.
Using the slant asymptote formula, we have
As the quotient obtained is x β 6, the slant asymptote for the given function f(x) is,
S(x) = x β 6
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