Vertical Asymptotes and Discontinuities

Nancy explains how to find vertical asymptotes of rational functions using three steps: factor the numerator and denominator, cancel common factors, and set the denominator to zero. She provides examples with different scenarios, including cases with no common factors, with common factors, and where no vertical asymptotes exist. The video concludes with a brief mention of horizontal asymptotes and additional resources for further learning.

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

• A point where the graph intersects the x-axis.

• A line that the graph crosses at infinity.

• A vertical line that the graph approaches but never touches.

• A horizontal line that the graph approaches but never touches.

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

• Set the numerator equal to zero.

• Factor the numerator and denominator.

• Graph the function.

• Find the horizontal asymptotes first.

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

• Cancel any common factors.

• Multiply the factors.

• Add the factors together.

• Divide the factors.

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

• Graph the function.

• Set the numerator equal to zero.

• Find the domain of the function.

• Set the denominator equal to zero.

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

• (x + 2)(x - 2)

• (x - 2)(x - 2)

• (x + 2)(x + 2)

• (x + 1)(x - 1)

What is the significance of a vertical asymptote on a graph?

• It represents the average rate of change.

• It marks the maximum value of the function.

• It shows where the function is zero.

• It indicates where the function is undefined.

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

• The graph becomes undefined.

• The graph has a hole or removable discontinuity.

• The graph has a horizontal asymptote.

• The graph has a vertical asymptote.

What is a removable discontinuity in a graph?

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

• A horizontal line that the graph approaches.

• A hole in the graph where a factor cancels out.

• A vertical line that the graph approaches.

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

• x = 0

• x = -1

• x = 1

• No real solution, hence no vertical asymptote.

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

• It has a horizontal asymptote.

• It has two vertical asymptotes.

• It has one vertical asymptote.

• It has no vertical asymptotes.