rational functions and their asymptotes

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Mathematics
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1.
FLASHCARD QUESTION
Front
What are Vertical Asymptotes?
Back
Vertical asymptotes are lines x = a where a rational function approaches infinity or negative infinity as the input approaches a. They occur at values of x that make the denominator zero, provided the numerator is not also zero at that point.
2.
FLASHCARD QUESTION
Front
How do you find Vertical Asymptotes?
Back
To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.
3.
FLASHCARD QUESTION
Front
What is an Oblique Asymptote?
Back
An oblique (or slant) asymptote is a diagonal line that a function approaches as x approaches infinity or negative infinity. It occurs when the degree of the numerator is one greater than the degree of the denominator.
4.
FLASHCARD QUESTION
Front
How do you find an Oblique Asymptote?
Back
To find an oblique asymptote, perform polynomial long division on the rational function. The quotient (ignoring the remainder) will give the equation of the oblique asymptote.
5.
FLASHCARD QUESTION
Front
What are Horizontal Asymptotes?
Back
Horizontal asymptotes are lines y = b where a rational function approaches a constant value b as x approaches infinity or negative infinity.
6.
FLASHCARD QUESTION
Front
How do you determine Horizontal Asymptotes?
Back
To determine horizontal asymptotes, compare the degrees of the numerator and denominator: If the degree of the numerator is less than the denominator, y = 0; if they are equal, y = leading coefficient of numerator / leading coefficient of denominator; if the numerator's degree is greater, there is no horizontal asymptote.
7.
FLASHCARD QUESTION
Front
What is the significance of the degree of the numerator and denominator?
Back
The degree of the numerator and denominator helps determine the behavior of the rational function at infinity and the existence of horizontal or oblique asymptotes.
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