Exploring Asymptotes in Rational Functions

The video tutorial explains how to graph a rational function, focusing on understanding its asymptotic behavior. It covers horizontal and vertical asymptotes, points where the function is undefined, and the general shape of the graph. The tutorial uses examples to illustrate how the function behaves as x approaches infinity and near specific points, providing a comprehensive understanding of graphing rational functions.

What is a rational function?

• A function with only constants

• A function with a polynomial in the numerator and a constant in the denominator

• A function with a polynomial in both the numerator and the denominator

• A function with a constant in the numerator and a polynomial in the denominator

What happens to the value of the function as x approaches positive or negative infinity?

• It approaches 1

• It approaches infinity

• It remains constant

• It approaches 0

Why does the function approach the same value as x approaches positive or negative infinity?

• Because the function is undefined

• Because the denominator is larger than the numerator

• Because the numerator is larger than the denominator

• Because the numerator and denominator are both squared

What is a horizontal asymptote?

• A line that the graph touches at one point

• A line that the graph never approaches

• A line that the graph crosses

• A line that the graph approaches but never touches

What is the horizontal asymptote for the function y = x^2 / (x^2 - 16)?

• y = 1

• y = -2

• y = 2

• y = 0

What happens to the function at x = 4 and x = -4?

• The function equals 1

• The function equals 0

• The function is undefined

• The function is defined

What is a vertical asymptote?

• A line that the graph approaches but never touches

• A line that the graph crosses

• A line that the graph never approaches

• A line that the graph touches at one point

How do you identify vertical asymptotes?

• By finding where the numerator equals 0

• By finding where both the numerator and denominator equal 0

• By finding where the function is undefined

• By finding where the denominator equals 0

What is the behavior of the function as x approaches 4 from the positive direction?

• The function approaches infinity

• The function approaches 1

• The function remains constant

• The function approaches 0

What is the general shape of the graph for the function y = x^2 / (x^2 - 16)?

• It has only vertical asymptotes

• It has no asymptotes

• It has both horizontal and vertical asymptotes

• It has only horizontal asymptotes