HW 15 - Ch 2.6 Homework(PC)

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

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

Back

For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). If the degree of the numerator is greater, there is no horizontal asymptote.

Front

What is a vertical asymptote?

Back

A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the function is undefined.

Front

How do you find vertical asymptotes in a rational function?

Back

Vertical asymptotes can be found by setting the denominator of a rational function equal to zero and solving for x.

Front

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

Back

If a function has no horizontal asymptote, it means that as x approaches positive 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 to understand the behavior of a function at extreme values and provide guidance on how to sketch the graph accurately.

Front

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

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 respectively.

Front

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

Back

The horizontal asymptote is y = 0, since the degree of the numerator is less than the degree of the denominator.

Front

What are the vertical asymptotes of the function f(x) = 1/(x^2 - 25)?

Back

The vertical asymptotes are x = 5 and x = -5, found by setting the denominator x^2 - 25 equal to zero.

Front

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

Back

There is no horizontal asymptote, as the degree of the numerator is greater than the degree of the denominator.