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7.8 – Vertical Asymptotes
Key Terms
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Notes
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 xvalue 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
 Find any vertical asymptotes.
 Make a table of values to find points on the graph on both sides of the asymptote(s).
 Use a dashed line to draw the vertical asymptote on the xyplane.
 Plot the points and connect them with smooth curves.
 Do not let the curve(s) touch the asymptote(s).

 Graph:
 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?


 Ex 2. At what value of x does the graph of the following function F(x) have a vertical asymptote?

 Set the denominator equal to zero, then solve: 3x – 6 = 0
 Add 6 to both sides, then divide by 3.

 Ex 3. What is the rational functions graphed below?
 Does the graph cross the xaxis? 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:

 Ex 4. Which of the following rational functions is graphed below?
 Does the graph cross the xaxis? 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:
 Answer:

 Ex 5. For the function , 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?

 There is 1 xfactor in the denominator, so:
 Answer: 1

 Ex 8. How many vertical asymptotes does the graph of this function have?

 There are 2 xfactors in the denominator, so:
 Answer: 2

 Ex 9. At which values of x does the function F(x) have a vertical asymptote?

 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?


 Ex 11. What is the rational function graphed below?
 Notice that there are 2 asymptotes and the graph never crosses the xaxis.
 The numerator will be 1. The denominator will be the roots of all the xfactors in the denominator.
 Answer:

 Ex 12. What is the rational function graphed below?
 Answer:

Important!
Practice (Apex Study 7.8)
 Try practice problems on Pg 10
 See how graphs and equations change on Pg 16, 18
 Mandatory: write and answer problems on Pg 11, 19
 2 Quizzes
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