Understanding Rational Functions and Asymptotes

The video tutorial explains how to graph a rational function by factoring the denominator to find vertical asymptotes and analyzing the degrees of the numerator and denominator to determine horizontal asymptotes. It includes a step-by-step guide on plotting points and verifying the graph using a calculator.

What is the first step in graphing the rational function y = x / (x^2 - x - 6)?

• Find the horizontal asymptotes

• Factor the numerator

• Factor the denominator

• Calculate the y-intercept

How do you determine the vertical asymptotes of a rational function?

• Set the numerator equal to zero

• Set the denominator equal to zero

• Calculate the x-intercept

• Find the highest degree term

What happens to the function as x approaches infinity if the degree of the denominator is higher than the numerator?

• The function remains constant

• The function approaches infinity

• The function approaches zero

• The function oscillates

Why is there a horizontal asymptote at y = 0 for the function y = x / (x^2 - x - 6)?

• The denominator grows faster than the numerator

• The degrees of the numerator and denominator are equal

• The numerator grows faster than the denominator

• The numerator has a higher degree than the denominator

What is the value of y when x = 0 for the function y = x / (x^2 - x - 6)?

• 1

• Undefined

• 0

• -1

What is the behavior of the function as it approaches the vertical asymptote from the right?

• It approaches negative infinity

• It remains constant

• It approaches positive infinity

• It approaches zero

What is the y-value when x = 4 for the function y = x / (x^2 - x - 6)?

• 3/4

• 1/3

• 2/3

• 1/2

What is the significance of plotting points on the graph of a rational function?

• To determine the function's domain

• To understand the function's behavior near asymptotes

• To verify the asymptotes

• To find the x-intercepts

How does the graphing calculator help in understanding the function?

• It provides exact values for all points

• It automatically finds all asymptotes

• It shows the general shape and behavior of the function

• It eliminates the need for manual calculations

What is the overall shape of the graph of y = x / (x^2 - x - 6) as verified by the calculator?

• A parabolic curve

• A hyperbolic shape with asymptotes

• A circular shape

• A straight line