Asymptotes of Rational Functions

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What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where the function approaches infinity or negative infinity as x approaches a. It occurs when the denominator of a rational function equals zero and the numerator does not equal zero at that point.

Tags

CCSS.HSF-IF.C.7D

Front

How do you find vertical asymptotes of a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes.

Tags

CCSS.HSF-IF.C.7D

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a line y = b that the graph of a function approaches as x approaches infinity or negative infinity. It indicates the end behavior of the function.

Tags

CCSS.HSF-IF.C.7D

Front

How do you determine horizontal asymptotes for rational functions?

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1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. 2. If the degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). 3. If the degree of the numerator is greater, there is no horizontal asymptote.

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. It can be found using polynomial long division.

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 is a removable discontinuity?

Back

A removable discontinuity occurs at a point where both the numerator and denominator of a rational function equal zero. It can be 'removed' by factoring and simplifying the function.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the y-intercept in a rational function?

Back

The y-intercept is the value of the function when x = 0. It can be found by substituting x = 0 into the function.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the x-intercept in a rational function?

Back

The x-intercept is the value of x where the function equals zero. It can be found by setting the numerator equal to zero and solving for x.

Tags

CCSS.HSF-IF.C.7D

Front

What is end behavior in the context of rational functions?

Back

End behavior describes how the values of a function behave as x approaches positive or negative infinity. It is often represented by horizontal asymptotes.

Tags

CCSS.HSF-IF.C.7D