Asymptotes

Front

What is an asymptote?

Back

An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or oblique.

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What are vertical asymptotes?

Back

Vertical asymptotes occur where a function approaches infinity or negative infinity as the input approaches a certain value. They are found by setting the denominator of a rational function to zero.

Front

What are horizontal asymptotes?

Back

Horizontal asymptotes describe the behavior of a function as the input approaches positive or negative infinity. They are determined by the degrees of the numerator and denominator in a rational function.

Front

What is an oblique asymptote?

Back

An oblique (or slant) asymptote occurs when the degree of the numerator is one greater than the degree of the denominator in a rational function. It can be found using polynomial long division.

Front

How do you find vertical asymptotes for the function f(x) = 1/(x^2 - 4)?

Back

Set the denominator equal to zero: x^2 - 4 = 0. This gives x = 2 and x = -2 as vertical asymptotes.

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How do you find horizontal asymptotes for the function f(x) = (3x^2 + 2)/(2x^2 + 5)?

Back

Since the degrees of the numerator and denominator are the same, the horizontal asymptote is y = 3/2 (the ratio of the leading coefficients).

Front

What is the equation of the vertical asymptote for the function f(x) = 1/(x - 1)?

Back

The vertical asymptote is at x = 1.

Front

What is the equation of the horizontal asymptote for the function f(x) = 5/(x + 3)?

Back

The horizontal asymptote is y = 0.

Front

What is the significance of asymptotes in graphing functions?

Back

Asymptotes help to understand the end behavior of functions and indicate where the function may not be defined.

Front

How do you determine if a function has an oblique asymptote?

Back

A function has an oblique asymptote if the degree of the numerator is exactly one greater than the degree of the denominator.