Understanding Asymptotes in Rational Functions

The video tutorial explains how to find the equations of asymptotes for a rational function, including vertical, horizontal, and slant asymptotes. It begins with factoring the numerator and denominator to check for common factors, which indicate holes rather than asymptotes. The tutorial then details the process of finding vertical asymptotes by setting the denominator to zero, horizontal asymptotes by comparing the degrees of the numerator and denominator, and slant asymptotes through long division. Finally, the video verifies these findings using a graph.

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

• Graph the function

• Perform long division

• Factor the numerator and denominator

• Set the numerator equal to zero

How do you find the vertical asymptotes of a rational function?

• Perform long division

• Find the common factors

• Set the denominator equal to zero

• Set the numerator equal to zero

What is the result of solving 2x^2 - 3 = 0 for x?

• x = ±√3/2

• x = ±√4/3

• x = ±√6/2

• x = ±√2/3

When does a rational function have no horizontal asymptote?

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

• When the degree of the numerator is greater than the degree of the denominator

• When the function is undefined

• When the degrees of the numerator and denominator are equal

What is the horizontal asymptote if the degrees of the numerator and denominator are equal?

• y = 0

• There is no horizontal asymptote

• y = 1

• The ratio of the leading coefficients

What condition leads to a slant asymptote in a rational function?

• The degree of the numerator is less than the degree of the denominator

• The function has common factors

• The degrees of the numerator and denominator are equal

• The degree of the numerator is one higher than the degree of the denominator

What is the equation of the slant asymptote found in the video?

• y = -2x - 4

• y = 2x + 4

• y = 2x - 4

• y = -2x + 4

What is the remainder when performing long division to find the slant asymptote?

• 8x - 19

• 8x + 19

• 19x + 8

• 19x - 8

What does the graph of the rational function confirm about the asymptotes?

• There is one vertical asymptote and one horizontal asymptote

• There is one vertical asymptote and one slant asymptote

• There are two vertical asymptotes and one horizontal asymptote

• There are two vertical asymptotes and no horizontal asymptote

What is the approximate value of the vertical asymptote x = √6/2?

• 1.73

• 1.41

• 1.22

• 0.61