Understanding Horizontal and Vertical Asymptotes

This video tutorial explains how to determine the horizontal and vertical asymptotes of rational functions. It covers the definition of rational functions, their domains, and how to graph them. The tutorial provides detailed steps for identifying vertical asymptotes and holes by factoring polynomials and analyzing zeros. It also explains how to find horizontal asymptotes by comparing the degrees of the numerator and denominator. The video includes examples to illustrate these concepts and demonstrates graphing techniques using a calculator.

What is a rational function?

• A function that can be described as a quotient of two polynomials

• A function that is always linear

• A function that can be expressed as a sum of polynomials

• A function that has no variables

Which of the following is true about vertical asymptotes?

• They are always at x = 0

• They are lines that the graph crosses

• They occur where the numerator is zero

• They occur where the denominator is zero and not a common factor with the numerator

What happens at a hole in the graph of a rational function?

• The graph is undefined at that point

• The graph has a vertical asymptote

• The graph crosses the x-axis

• The graph has a horizontal asymptote

How do you determine the horizontal asymptote if the degree of the numerator is greater than the degree of the denominator?

• The horizontal asymptote is y = 1

• The horizontal asymptote is y = 0

• The horizontal asymptote is the ratio of the leading coefficients

• There is no horizontal asymptote

If the degrees of the numerator and denominator are equal, what is the horizontal asymptote?

• y = 1

• The ratio of the leading coefficients

• There is no horizontal asymptote

• y = 0

What is the horizontal asymptote when the degree of the denominator is higher than the degree of the numerator?

• y = 0

• There is no horizontal asymptote

• y = 1

• The ratio of the leading coefficients

In graphing rational functions, why are asymptotes useful?

• They help determine the x-intercepts

• They show where the graph crosses the y-axis

• They provide a framework for sketching the graph

• They indicate where the graph is undefined

What should you do if a zero of the denominator is also a zero of the numerator?

• Draw a horizontal asymptote

• Draw a vertical asymptote

• Ignore it

• Indicate a hole in the graph

When graphing a rational function, what is the first step?

• Plot random points

• Determine the y-intercept

• Factor the numerator and denominator

• Find the x-intercepts

What is the horizontal asymptote of a function where both the numerator and denominator have the same degree?

• y = 0

• There is no horizontal asymptote

• y = 1

• The ratio of the leading coefficients