Rational Functions: Holes and Asymptotes

This video tutorial explains how to find points of discontinuity and vertical asymptotes in rational functions. It begins by discussing domain restrictions due to variables in the denominator, which can lead to division by zero. The process involves factoring the denominator to find values that make it zero, identifying domain restrictions, and determining whether these lead to points of discontinuity or vertical asymptotes. The video also covers simplifying fractions to identify holes in the graph and addresses the possibility of multiple graphical features.

What is the primary cause of points of discontinuity and vertical asymptotes in rational functions?

• Variables in the denominator

• Variables in the numerator

• Constant terms

• Exponential terms

How do you determine the domain restrictions of a rational function?

• By integrating the function

• By finding the derivative

• By setting the denominator equal to zero

• By setting the numerator equal to zero

What is the next step after finding the domain restrictions in a rational function?

• Calculating the integral

• Finding the derivative

• Simplifying the function

• Graphing the function

Which factor in the numerator simplifies with the denominator in the given example?

• X + 3

• X - 2

• X + 2

• X - 4

What is the x-coordinate of the hole in the graph?

• -2

• -4

• 4

• 2

How is the y-coordinate of the hole calculated?

• By setting the denominator to zero

• By substituting the x-coordinate into the original function

• By substituting the x-coordinate into the simplified function

• By setting the numerator to zero

What is the y-coordinate of the hole in the graph?

• 1/3

• -1/3

• 1/6

• -1/6

What happens to domain restrictions that remain after simplification?

• They become horizontal asymptotes

• They are ignored

• They become vertical asymptotes

• They become points of discontinuity

Where is the vertical asymptote located in the given example?

• x = 2

• x = 4

• x = -2

• x = -4

Can a rational function have more than one hole or vertical asymptote?

• No, it can only have multiple holes

• Yes, if more than one factor cancels

• No, it can only have one of each

• Yes, but only if the function is linear