Understanding Rational Functions and Asymptotes

The video tutorial covers graphing rational functions, focusing on identifying and understanding different types of asymptotes: vertical, horizontal, and oblique. It explains how to determine these asymptotes based on the degrees of the numerator and denominator. The tutorial also discusses point discontinuities, or holes, in graphs. Several examples are provided to illustrate how to graph rational functions, determine their domain and range, and interpret the results.

What is a rational function characterized by?

• A variable only in the numerator

• A variable only in the denominator

• A variable in both the numerator and the denominator

• No variables at all

When does a vertical asymptote occur in a rational function?

• When the degrees of numerator and denominator are equal

• When both numerator and denominator are zero

• When the denominator is zero

• When the numerator is zero

How is a horizontal asymptote determined when the degrees of the numerator and denominator are equal?

• By the ratio of the leading coefficients of the numerator and denominator

• By the leading coefficient of the denominator

• There is no horizontal asymptote in this case

• By the leading coefficient of the numerator

What condition must be met for an oblique asymptote to exist?

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

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

• The degrees of the numerator and denominator are equal

• The degree of the denominator is greater than the numerator

What is a point discontinuity in a graph?

• A point where the graph crosses the y-axis

• A point where the graph crosses the x-axis

• A hole in the graph where factors cancel out

• A point where the graph has a vertical asymptote

What is the first step in graphing a rational function?

• Identify the x-intercepts

• Identify the y-intercepts

• Identify the degrees of the numerator and denominator

• Identify the asymptotes

How do you determine the domain of a rational function?

• All real numbers except where both numerator and denominator are zero

• All real numbers except where the numerator is zero

• All real numbers except where the denominator is zero

• All real numbers

In the context of the field goal problem, what does the horizontal asymptote represent?

• The maximum number of field goals

• A 0% field goal percentage

• The minimum number of field goals

• A 100% field goal percentage

What does the vertical asymptote represent in the field goal percentage problem?

• The point where the field goal percentage is maximum

• The point where the field goal percentage is zero

• The point where the field goal percentage is minimum

• The point where the field goal percentage is undefined

Why can't the field goal percentage reach 100% in the given problem?

• Because the player has already missed some field goals

• Because the player has not made any field goals

• Because the player has not attempted enough field goals

• Because the player has made all field goals