Vertical Asymptotes

Vertical asymptotes are vertical lines that a graph approaches but does not touch. They occur when the denominator of a rational function becomes zero while the numerator does not. In other words, a vertical asymptote is a vertical line that the graph of a function approaches as the input values get larger and larger, or smaller and smaller, but never actually touches.

Finding Vertical Asymptotes

To find the vertical asymptotes of a rational function, set the denominator equal to zero and solve for the variable. The values that make the denominator zero are the x-values of the vertical asymptotes.

Example:

Find the vertical asymptotes of the function f(x) = (x^2 - 4) / (x - 2).

To find the vertical asymptotes, set the denominator (x - 2) equal to zero:
x - 2 = 0
x = 2
Therefore, the function has a vertical asymptote at x = 2.

Graphing Vertical Asymptotes

When graphing a rational function with a vertical asymptote, draw a dashed vertical line at the x-value of the vertical asymptote. This line represents the boundary that the graph approaches but does not cross. You can also plot points on either side of the vertical asymptote to show the behavior of the graph as it approaches the asymptote.

Example:

Graph the function f(x) = (x^2 - 4) / (x - 2).

To graph the function, start by plotting points on both sides of the vertical asymptote at x = 2. This will help you visualize how the graph approaches the vertical asymptote without crossing it.

After plotting the points, draw a dashed vertical line at x = 2 and draw the graph approaching the vertical asymptote without crossing it.

Key Points to Remember

• Vertical asymptotes occur when the denominator of a rational function becomes zero while the numerator does not.
• To find the vertical asymptotes, set the denominator equal to zero and solve for the variable.
• When graphing a rational function with a vertical asymptote, draw a dashed vertical line at the x-value of the vertical asymptote and plot points on either side to show the behavior of the graph.

Understanding vertical asymptotes is essential for graphing rational functions and analyzing their behavior as the input values approach certain x-values. Practice finding and graphing vertical asymptotes to gain a deeper understanding of their significance in the behavior of functions.

Good luck with your studies!