# 7.8 – Vertical Asymptotes

## Key Terms

• No key terms for this section.

## Notes

Vertical Asymptotes
• A vertical asymptote is a vertical line that the curve of a function’s graph approaches but never touches.
• A function has a vertical asymptote at any x-values where it is undefined.
• Near a function’s vertical asymptotes, its values become very large positive or negative numbers.
• Graph of $f(x)=\frac{1}{x}$
• As x becomes a smaller positive number, F(x) becomes a larger positive number.
• As x becomes a smaller negative number, F(x) becomes a larger negative number.
• As x gets closer to zero, F(x) gets closer to being undefined.
• When x = 0, F(x) is undefined.
• So, x = 0 is a vertical asymptote of F(x).
• How to Find Vertical Asymptotes
1. Factor the denominator of the function’s equation.
2. Write each factor in an equation equal to zero.
3. Solve each equation for x.
4. Each of those values of x is a vertical asymptote.

• Ex. $f(x)=\frac{1}{x^{2}-x-2}$
1. Factor the denominator: (x + 1)(x – 2)
2. Write each factor in an equation equal to zero: x + 1 = 0, x – 2 = 0
3. Solve: x = -1, x = 2

Rational Functions
• To make a sketch of any rational function whose numerator is a number and whose denominator is a factored polynomial, use the following rule and steps:
• These functions have a vertical asymptote at every x-value where its denominator is zero, and you can make a table for each vertical asymptote to find out what happens to the function there.
• Graphing Rational Functions
1. Find any vertical asymptotes.
2. Make a table of values to find points on the graph on both sides of the asymptote(s).
3. Use a dashed line to draw the vertical asymptote on the xy-plane.
4. Plot the points and connect them with smooth curves.
5. Do not let the curve(s) touch the asymptote(s).
• Graph: $f(x)=\frac{1}{x^{2}}$
• The denominator = 0 when x = 0
• 1. So, x = 0 is a vertical asymptote.
• 2. Make a table:

• 3. Draw the asymptote(s)
• Plot points
• Connect the curves

• The number of variables in the denominator determines the number of curves on the graph.

## Examples

 Ex 1. At what value of x does the graph of the following function F(x) have a vertical asymptote? $f(x)=\frac{2}{x-2}$ Answer: 2 Ex 2. At what value of x does the graph of the following function F(x) have a vertical asymptote? $f(x)=\frac{4x}{3x-6}$ Set the denominator equal to zero, then solve: 3x – 6 = 0 Add 6 to both sides, then divide by 3. Answer: 2 Ex 3. What is the rational functions graphed below? Does the graph cross the x-axis? If not, put a 1 in the numerator. Where is the asymptote located? It is shown at x = 6; so, for the denominator, write the factor: x – 6. Answer: $f(x)=\frac{1}{x-6}$ Ex 4. Which of the following rational functions is graphed below? Does the graph cross the x-axis? If not, put a 1 in the numerator. Where is the asymptote located? It is shown at x = -1; so, for the denominator, write the factor: (x + 1). There are 2 graphs that never touch that asympotote, so the asymptote works for both. The denominator actually has 2 of these: (x + 1)(x + 1), or you can write it as: $(x+1)^{2}$ Answer: $f(x)=\frac{1}{(x+1)^{2}}$ Ex 5. For the function $f(x)=\frac{1}{x+1}$, whose graph is shown below, what is the relative value of F(x) when the value of x is close to -1? Answer: Either a very large positive or very large negative number Ex 6. Given the graph of the function F(x) below, what happens to F(x) when x is a number between 0 and 1? Answer: F(x) is a very large negative number. Ex 7. How many vertical asymptotes does the graph of this function have? $f(x)=\frac{2}{3(x-7)}$ There is 1 x-factor in the denominator, so: Answer: 1 Ex 8. How many vertical asymptotes does the graph of this function have? $f(x)=\frac{3}{(x-11)(x+4)}$ There are 2 x-factors in the denominator, so: Answer: 2 Ex 9. At which values of x does the function F(x) have a vertical asymptote? $f(x)=\frac{9}{(x-1)(x-8)}$ The asymptotes are like roots, the opposite signs of the factors; so: Answers: 1 and 8 Ex 10. At which values of x does the function F(x) have a vertical asymptote? $f(x)=\frac{3}{x(x-5)(x+1)}$ Answers: 0, 5, and -1 Ex 11. What is the rational function graphed below? Notice that there are 2 asymptotes and the graph never crosses the x-axis. The numerator will be 1. The denominator will be the roots of all the x-factors in the denominator. Answer: $f(x)=\frac{1}{(x-1)(x+4)}$ Ex 12. What is the rational function graphed below? Answer: $f(x)=\frac{1}{(x+3)(x-7)}$