Algebra 92 - Rational Functions and Holes

This lecture explores rational functions, focusing on zeros of numerators and denominators, and their impact on graph characteristics like vertical asymptotes and holes. It explains how to simplify rational functions by eliminating identical factors and provides examples to illustrate these concepts. The lecture emphasizes the importance of understanding function domains and graphing techniques.

What are the zeros of a rational function's numerator called?

• Vertical asymptotes

• Numerator zeros

• Denominator zeros

• Holes

What happens to a rational function's graph when the denominator is zero?

• The graph has a hole

• The function is undefined

• The graph has an X-intercept

• The graph is continuous

What is a vertical asymptote in the context of rational functions?

• A point where the graph is undefined

• A line the graph approaches but never touches

• A point where the graph has a hole

• A point where the graph crosses the X-axis

When does a hole occur in the graph of a rational function?

• When the denominator is zero

• When the numerator is zero

• When numerator and denominator zeros are identical

• When the function is continuous

How can simplifying a rational function help in graphing?

• It changes the function's domain

• It adds more zeros to the function

• It eliminates vertical asymptotes

• It simplifies the process of sketching the graph

What is the first step in graphing a rational function?

• Finding the X-intercepts

• Factoring the numerator and denominator

• Calculating the function's values

• Identifying vertical asymptotes

What does a zero in the numerator of a rational function indicate?

• An X-intercept

• A hole in the graph

• A point of discontinuity

• A vertical asymptote

In a rational function, what does a zero in the denominator indicate?

• A point of continuity

• A hole in the graph

• An X-intercept

• A vertical asymptote

What happens when multiple identical factors are present in a rational function?

• They add more zeros to the function

• They change the function's domain

• They can be eliminated to simplify the function

• They create multiple holes

How does a vertical asymptote differ from a hole in a graph?

• A hole is a line, while a vertical asymptote is a point

• A vertical asymptote is a line, while a hole is a point

• Both are points

• Both are lines