Vertical Asymptotes and Discontinuities

Nancy explains how to find vertical asymptotes in rational functions using three steps: factoring the numerator and denominator, canceling common factors, and setting the denominator to zero. She provides examples, including cases with no common factors, canceling factors, and no vertical asymptotes. Graphical representations are used to illustrate the concept, and additional resources are offered for further learning.

What is a vertical asymptote in the context of a graph?

• A diagonal line that the graph follows

• A horizontal line that the graph intersects

• An invisible line that the graph approaches but never touches

• A visible line that the graph crosses

Which of the following is the first step in finding vertical asymptotes of a rational function?

• Find the horizontal asymptotes

• Graph the function

• Factor the numerator and denominator

• Set the numerator equal to zero

After factoring the numerator and denominator, what should you do next to find vertical asymptotes?

• Add the factors together

• Multiply the factors

• Cancel any common factors

• Divide the factors

What is the final step in finding vertical asymptotes after simplifying the rational function?

• Graph the function

• Find the domain of the function

• Set the numerator equal to zero

• Set the denominator equal to zero

In the example where the denominator is x^2 - 4, what are the factors?

• (x + 1)(x - 1)

• (x + 2)(x + 2)

• (x + 2)(x - 2)

• (x - 2)(x - 2)

What happens when a factor cancels out in the numerator and denominator?

• It creates a vertical asymptote

• It results in a horizontal asymptote

• It leads to a removable discontinuity

• It has no effect on the graph

What is a removable discontinuity?

• A vertical asymptote

• A point where the graph crosses the x-axis

• A point where the graph is undefined due to a canceled factor

• A horizontal asymptote

What is the result when solving x^2 + 1 = 0 for vertical asymptotes?

• No vertical asymptote

• x = 0

• x = -1

• x = 1

If a rational function has a denominator of x^2 + 1, what can be concluded about its vertical asymptotes?

• It has two vertical asymptotes

• It has a horizontal asymptote

• It has one vertical asymptote

• It has no vertical asymptotes

What is the significance of a horizontal asymptote in a graph?

• It is a point of intersection with the y-axis

• It is where the graph crosses the x-axis

• It shows the end behavior of the graph

• It indicates where the graph is undefined