Understanding Asymptotes and Graphing Rational Functions

The video tutorial explains how to determine and graph asymptotes for a rational function. It begins by identifying vertical asymptotes, emphasizing the importance of factoring and simplifying the function. The concept of holes is introduced, occurring at common factors of the numerator and denominator. The tutorial then covers horizontal asymptotes, using limits to determine their equations. Finally, a T table is used to find additional points for graphing, ensuring the function passes through these points and approaches the asymptotes.

What is the first step in determining vertical asymptotes for a rational function?

• Finding the zeros of the numerator

• Simplifying the function

• Finding the zeros of the denominator

• Graphing the function

What happens when a zero is common to both the numerator and the denominator?

• The function becomes undefined

• A hole is formed

• A horizontal asymptote is formed

• A vertical asymptote is formed

How do you determine the horizontal asymptote of a rational function?

• By finding the zeros of the numerator

• By evaluating the limit as x approaches infinity

• By simplifying the function

• By finding the zeros of the denominator

What is the shortcut method to determine the horizontal asymptote when the degree of the denominator is greater than the numerator?

• The limit is undefined

• The limit is 0

• The limit is the ratio of leading coefficients

• The limit is 1

What is the purpose of using a T-table in graphing rational functions?

• To simplify the function

• To find additional points for the graph

• To determine the asymptotes

• To find the zeros of the function

What is the y-intercept of the function when x = 0?

• -1/3

• 0

• 1/3

• -1

What should be done at x = -3 on the graph?

• Shade the area under the curve

• Draw a horizontal asymptote

• Draw a vertical asymptote

• Mark an open point to indicate a hole

What is the function value when x = 2?

• -1

• 0

• 1

• 2

What is the function value when x = 4?

• -1

• 2

• 1

• 0

What is the importance of knowing the asymptotes and additional points when graphing a function?

• To determine the domain of the function

• To simplify the function

• To accurately sketch the graph

• To find the range of the function