Asymptotes

Front

What is an asymptote?

Back

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

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.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the function's value approaches infinity or negative infinity.

Front

How do you find horizontal asymptotes for rational functions?

Back

For rational functions, compare the degrees of the numerator and denominator. If the degree of the numerator is less, the horizontal asymptote is y=0. If they are equal, divide the leading coefficients.

Front

What is the horizontal asymptote of the function f(x) = 3x^2 + 2 / 4x^2 + 1?

Back

y = 3/4, since the degrees of the numerator and denominator are equal.

Front

What is the equation of the vertical asymptote for the function f(x) = 1/(x-2)?

Back

x = 2, since the function approaches infinity as x approaches 2.

Front

What does it mean if a function has no horizontal asymptote?

Back

It means that as x approaches infinity or negative infinity, the function does not approach a specific y-value.

Front

What is the significance of asymptotes in graphing functions?

Back

Asymptotes help identify the behavior of a function at extreme values and guide the sketching of the graph.

Front

What is the difference between a removable and non-removable discontinuity?

Back

A removable discontinuity occurs when a factor cancels in a rational function, while a non-removable discontinuity occurs at vertical asymptotes.

Front

How can you determine if a line is an asymptote of a graph?

Back

You can determine if a line is an asymptote by analyzing the limits of the function as it approaches that line.