Understanding Rational Functions

The video tutorial explains how to analyze and graph rational functions. It covers determining the domain by identifying values that make the denominator zero, factoring the function, and simplifying it. The tutorial also discusses vertical asymptotes and holes, using a graphing calculator to complete a table of values, and identifying horizontal asymptotes. Finally, it demonstrates how to sketch the graph, considering all these features.

What is the primary reason for excluding certain x-values from the domain of a rational function?

• They make the function positive.

• They make the function negative.

• They make the denominator zero.

• They make the numerator zero.

Why is it helpful to factor the numerator and denominator of a rational function?

• To find the x-intercepts.

• To simplify the function and identify holes and vertical asymptotes.

• To determine the y-intercept.

• To make the function continuous.

What is the significance of a hole in the graph of a rational function?

• It shows where the function is undefined due to a common factor in the numerator and denominator.

• It marks the y-intercept.

• It represents the x-intercept.

• It indicates a vertical asymptote.

When creating a table of values, why is it important to choose values around the vertical asymptote?

• To determine the x-intercept.

• To simplify the function.

• To find the y-intercept.

• To understand the behavior of the function near the asymptote.

What tool is used to efficiently complete the table of values for a rational function?

• A compass

• A protractor

• A graphing calculator

• A ruler

What is the y-intercept of a function?

• The point where the graph has a hole.

• The point where the graph crosses the y-axis.

• The point where the graph crosses the x-axis.

• The point where the graph is undefined.

What does a horizontal asymptote represent in the graph of a rational function?

• The point where the function is undefined.

• The point where the function has a hole.

• The x-value the function approaches as x increases or decreases.

• The y-value the function approaches as x increases or decreases.

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

• By identifying the vertical asymptote.

• By finding the y-intercept.

• By comparing the degrees of the numerator and denominator.

• By finding the x-intercept.

What happens to the y-value of a rational function as x increases or decreases?

• It approaches the horizontal asymptote.

• It approaches infinity.

• It approaches zero.

• It becomes undefined.

What is the relationship between the degrees of the numerator and denominator in determining the horizontal asymptote?

• The horizontal asymptote is the difference of the degrees.

• The horizontal asymptote is the sum of the degrees.

• The horizontal asymptote is the ratio of the leading coefficients.

• The horizontal asymptote is the product of the degrees.