Vertical Asymptotes and Rational Functions

The video tutorial by Nancy explains how to find vertical asymptotes of rational functions. It introduces the concept of vertical asymptotes as invisible lines that graphs approach but never touch. Nancy outlines three steps to find vertical asymptotes: factor the numerator and denominator, cancel common factors, and set the denominator to zero. She provides examples, including cases with no common factors, canceling factors, and no vertical asymptotes. 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 a graph?

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

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

• A line that the graph crosses multiple times.

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

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

• Factor the numerator and denominator.

• Find the horizontal asymptotes first.

• Set the numerator equal to zero.

• Graph the function.

After factoring the numerator and denominator, what should you do next?

• Add the factors together.

• Cancel any common factors.

• Multiply the factors.

• Divide the factors.

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 + 3)

• (x + 2)(x - 2)

What happens when you cancel out a factor that appears in both the numerator and denominator?

• It creates a vertical asymptote.

• It has no effect on the graph.

• It creates a horizontal asymptote.

• It creates a hole in the graph.

What is a removable discontinuity?

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

• A horizontal asymptote that can be removed.

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

• A vertical asymptote that can be removed.

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

• x = 1

• x = -1

• No real solution, hence no vertical asymptote.

• x = 0

Why might a rational function not have a vertical asymptote?

• Because the function is linear.

• Because the function is quadratic.

• Because the numerator is a constant.

• Because the denominator has no real zeros.

What does a vertical asymptote indicate about the function at that point?

• The function is undefined and discontinuous.

• The function is defined and continuous.

• The function has a minimum value.

• The function has a maximum value.

What should you do if you cannot factor the denominator of a rational function?

• Graph the function to find asymptotes.

• Factor the numerator instead.

• Set the denominator equal to zero and solve for x.

• Assume there are no vertical asymptotes.