Understanding Asymptotes in Rational Functions

The video tutorial explains how to find the equations of vertical and slant asymptotes for rational functions. It emphasizes the importance of factoring the numerator and denominator to identify common factors that create holes, not vertical asymptotes. The tutorial demonstrates the process of finding vertical asymptotes by setting the denominator to zero and solving for x. It also covers slant asymptotes, which occur when the numerator's degree is one higher than the denominator's, and explains how to use long division to find their equations. The video concludes by verifying the results with a graph.

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

• Perform long division.

• Factor the numerator and denominator.

• Set the numerator equal to zero.

• Find the degree of the numerator.

Why do common factors in the numerator and denominator not result in vertical asymptotes?

• They simplify the function to a constant.

• They produce holes in the function.

• They result in slant asymptotes.

• They create horizontal asymptotes.

In the example provided, what is the vertical asymptote of the rational function?

• x = 0

• x = -2

• x = -1

• x = 3/4

What condition must be met for a rational function to have a slant asymptote?

• The numerator and denominator have common factors.

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

• The degree of the numerator is equal to the degree of the denominator.

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

What mathematical process is used to find the equation of a slant asymptote?

• Factoring

• Long division

• Synthetic division

• Completing the square

As x approaches infinity, what happens to the remainder in the division used to find the slant asymptote?

• It approaches zero.

• It becomes infinite.

• It remains constant.

• It oscillates.

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

• y = 4x - 3

• y = 2x + 1

• y = 3x - 1

• y = x + 1

Why does the rational function not have a horizontal asymptote?

• The degree of the numerator is greater than the degree of the denominator.

• The numerator and denominator have common factors.

• The degree of the numerator is equal to the degree of the denominator.

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

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

• There is no vertical asymptote.

• There is no slant asymptote.

• The calculated asymptotes are correct.

• The function has a horizontal asymptote.

What is the significance of excluding the vertical asymptote value from the domain?

• It prevents division by zero.

• It creates a horizontal asymptote.

• It ensures the function is continuous.

• It simplifies the function.