Slant asymptote - Rational Functions

Slant asymptote

When the polynomial in the numerator is exactly one degree higher than the polynomial in the denominator, there is a slant asymptote in the rational function. To determine the slant asymptote, we need to perform long division.

Related concepts:
Curve sketching

Lessons

Notes:

For a simplified rational function, when the numerator is exactly one degree higher than the denominator, the rational function has a slant asymptote. To determine the equation of a slant asymptote, we perform long division.

1.
Algebraically Determining the Existence of Slant Asymptotes
Without sketching the graph of the function, determine whether or not each function has a slant asymptote:
2.
Determining the Equation of a Slant Asymptote Using Long Division
Determine the equations of the slant asymptotes for the following functions using long division.
3.
Determining the Equation of a Slant Asymptote Using Synthetic Division
Determine the equations of the slant asymptotes for the following functions using long division.

Slant asymptote

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Notes:

For a simplified rational function, when the numerator is exactly one degree higher than the denominator, the rational function has a slant asymptote. To determine the equation of a slant asymptote, we perform long division.

Intro Lesson

Introduction to slant asymptote i) What is a slant asymptote? ii) When does a slant asymptote occur? iii) Overview: Slant asymptote

1.

Algebraically Determining the Existence of Slant Asymptotes Without sketching the graph of the function, determine whether or not each function has a slant asymptote:

a)

a(x)=x2−3x−10x−5a(x) = \frac{x^{2} - 3x - 10}{x - 5}a(x)=​x−5​​x​2​​−3x−10​​

b)

b(x)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5} b(x)=​x−5​​x​2​​−x−6​​

c)

c(x)=5x3−7x2+10x+1c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}c(x)=​x+1​​5x​3​​−7x​2​​+10​​

2.

Determining the Equation of a Slant Asymptote Using Long Division Determine the equations of the slant asymptotes for the following functions using long division.

a)

b(x)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5}b(x)=​x−5​​x​2​​−x−6​​

b)

p(x)=−9x2+x4x3−8p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}p(x)=​x​3​​−8​​−9x​2​​+x​4​​​​

3.

Determining the Equation of a Slant Asymptote Using Synthetic Division Determine the equations of the slant asymptotes for the following functions using long division.

a)

b(x)=x2−x−6x−5b(x) = \frac{x^{2} - x - 6}{x - 5}b(x)=​x−5​​x​2​​−x−6​​

b)

f(x)=x2+1x−3f(x) = \frac{x^{2} + 1}{x - 3}f(x)=​x−3​​x​2​​+1​​

4.

Slant asymptote

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Algebra
Polynomial long division

Algebra
Polynomial synthetic division

or

Introduction to slant asymptote
i) What is a slant asymptote?

ii) When does a slant asymptote occur?

iii) Overview: Slant asymptote

Algebraically Determining the Existence of Slant Asymptotes
Without sketching the graph of the function, determine whether or not each function has a slant asymptote:

$a(x) = \frac{x^{2} - 3x - 10}{x - 5}$

$b(x) = \frac{x^{2} - x - 6}{x - 5}$

$c(x) = \frac{5x^{3} - 7x^{2} + 10}{x + 1}$

Determining the Equation of a Slant Asymptote Using Long Division
Determine the equations of the slant asymptotes for the following functions using long division.

$b(x) = \frac{x^{2} - x - 6}{x - 5}$

$p(x) = \frac{-9x^{2} + x^{4}}{x^{3} - 8}$

Determining the Equation of a Slant Asymptote Using Synthetic Division
Determine the equations of the slant asymptotes for the following functions using long division.

$b(x) = \frac{x^{2} - x - 6}{x - 5}$

$f(x) = \frac{x^{2} + 1}{x - 3}$