Rational Functions and Asymptotes

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•
Mathematics
•
9th - 12th Grade
•
Hard
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1.
FLASHCARD QUESTION
Front
What is a rational function?
Back
A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.
2.
FLASHCARD QUESTION
Front
What are vertical asymptotes?
Back
Vertical asymptotes are lines x = a where a rational function approaches infinity or negative infinity as x approaches a. They occur where the denominator Q(x) = 0 and P(a) ≠ 0.
3.
FLASHCARD QUESTION
Front
What are horizontal asymptotes?
Back
Horizontal asymptotes are lines y = b that a rational function approaches as x approaches infinity or negative infinity. They indicate the end behavior of the function.
4.
FLASHCARD QUESTION
Front
How do you find vertical asymptotes of a rational function?
Back
To find vertical asymptotes, set the denominator Q(x) equal to zero and solve for x. The values of x that make Q(x) = 0 are the vertical asymptotes.
5.
FLASHCARD QUESTION
Front
How do you find horizontal asymptotes of a rational function?
Back
To find horizontal asymptotes, compare the degrees of the numerator and denominator: If degree of P < degree of Q, y = 0; if degree of P = degree of Q, y = leading coefficient of P / leading coefficient of Q; if degree of P > degree of Q, there is no horizontal asymptote.
6.
FLASHCARD QUESTION
Front
What is the significance of asymptotes in graphing rational functions?
Back
Asymptotes help identify the behavior of the graph near certain values and guide the overall shape of the graph, indicating where the function does not exist or approaches certain values.
7.
FLASHCARD QUESTION
Front
Back
The horizontal asymptote is y = 2, since the degrees of the numerator and denominator are equal and the leading coefficients are 2 and 1.
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