rational functions and their asymptotes

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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.

Tags

CCSS.HSF-IF.C.7D

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find the horizontal asymptote of a rational function?

Back

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the asymptote is y=0. If they are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q.

Tags

CCSS.HSF-IF.C.7D

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find vertical asymptotes of a rational function?

Back

Vertical asymptotes are found by setting the denominator Q(x) = 0 and solving for x.

Tags

CCSS.HSF-IF.C.7D

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find the slant asymptote of a rational function?

Back

To find the slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

Tags

CCSS.HSF-IF.C.7D

Front

What are the characteristics of the graph of a rational function near vertical asymptotes?

Back

Near vertical asymptotes, the graph of a rational function will approach infinity or negative infinity, indicating a discontinuity.

Tags

CCSS.HSF-IF.C.7D

Front

What happens to the graph of a rational function at horizontal asymptotes?

Back

At horizontal asymptotes, the graph of the rational function approaches the line but may not touch or cross it.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the leading coefficients in determining horizontal asymptotes?

Back

The leading coefficients of the numerator and denominator determine the value of the horizontal asymptote when the degrees of the polynomials are equal.

Tags

CCSS.HSF-IF.C.7D