Understanding Slant Asymptotes in Rational Functions

This video tutorial explains how to determine slant asymptotes of rational functions. It assumes prior knowledge of vertical and horizontal asymptotes. The video covers the conditions under which a rational function has a slant asymptote and demonstrates how to find it using long division. Two examples are provided: the first focuses on identifying all asymptotes and graphing the function, while the second includes a hole in the function. The video concludes with a reminder to check for holes when using graphing software.

What is a slant asymptote in the context of rational functions?

• A line that the graph never approaches

• A vertical line that the graph approaches

• A slanted line that the graph approaches

• A horizontal line that the graph approaches

Under what condition does a rational function have a slant asymptote?

• When the degree of the numerator is equal to the degree of the denominator

• When the degree of the numerator is one less than the degree of the denominator

• When the degrees of the numerator and denominator are both zero

• When the degree of the numerator is one more than the degree of the denominator

What is the first step in determining all asymptotes of a rational function?

• Factor the given function

• Find the y-intercept

• Perform long division

• Graph the function

In the first example, what is the vertical asymptote of the function?

• x = 2

• x = 1

• x = -1

• x = 0

What is the equation of the slant asymptote in the first example?

• y = -1/2x + 1

• y = 2x - 1

• y = 1/2x + 1

• y = x + 1

In the second example, what is identified as a hole in the function?

• x = 1

• x = 2

• x = 0

• x = -1

What is the equation of the slant asymptote in the second example?

• y = 2x

• y = x - 1

• y = x

• y = -x

What is the importance of factoring in determining asymptotes?

• It determines the slope of the asymptote

• It simplifies the function for easier division

• It is not important

• It helps in finding the y-intercept

Why might software not show holes in the graph of a function?

• Because holes do not exist in rational functions

• Because the screen lacks enough definition

• Because the function is not simplified

• Because the software is outdated

What should you check on your graphing calculator to identify holes?

• The graph

• The equation

• The table

• The settings