Understanding Rational Functions and Asymptotes

The video tutorial covers graph sketching with a focus on rational functions and asymptotes. It explains how to find limits involving infinity, determine vertical and horizontal asymptotes, and graph rational functions. The tutorial provides methods for finding limits at infinity and discusses the behavior of graphs near asymptotes. Practical examples are used to illustrate these concepts, including using a graphing calculator to plot points and verify limits.

What is the primary focus of the lesson on graph sketching?

• Exploring exponential growth

• Learning about trigonometric functions

• Studying asymptotes and rational functions

• Finding the roots of polynomials

Which of the following best describes a rational function?

• A function with a constant numerator

• A function with a quadratic numerator

• A function with a linear denominator

• A function with a polynomial numerator and denominator

What is a vertical asymptote?

• A horizontal line that a graph approaches

• A vertical line that a graph approaches but never touches

• A diagonal line that a graph crosses

• A line that a graph intersects at infinity

What is the significance of zeros in the denominator of a rational function?

• They are the points where vertical asymptotes occur

• They have no significance

• They determine the horizontal asymptotes

• They indicate the function's maximum points

How can you determine a horizontal asymptote when the degrees of the numerator and denominator are equal?

• The horizontal asymptote is always y = 0

• The horizontal asymptote is the sum of the coefficients

• There is no horizontal asymptote

• The horizontal asymptote is the ratio of the leading coefficients

What happens to the limit of a rational function as x approaches infinity if the degree of the numerator is less than the degree of the denominator?

• The limit approaches zero

• The limit approaches a constant value

• The limit approaches infinity

• The limit does not exist

What is the horizontal asymptote of the function 2x - 5 / x - 3 as x approaches infinity?

• y = 3

• y = 1

• y = 0

• y = 2

How can you verify a vertical asymptote using limits?

• By checking if the function is continuous at the asymptote

• By finding the integral of the function

• By calculating the derivative at the asymptote

• By finding the limit as x approaches the asymptote from both sides

What is the first step in sketching the graph of a rational function?

• Determining the y-intercepts

• Calculating the derivative

• Identifying the asymptotes

• Finding the x-intercepts

Why is it important to plot additional points when graphing a rational function?

• To calculate the function's range

• To determine the function's domain

• To ensure the graph is accurate

• To find the function's derivative