1

2.5 Graphing Rational Functions

Since rational functions are fractions, the denominator cannot be zero. Division by zero is undefined.

Therefore, the functions are not defined when the denominator equals zero. In part a, the function is not defined when x=–5. In part b., the function is not defined when x=2 or x=–2. In part 3, the function is not defined when x=1/2 or x = 3. In the last example, there are no real numbers which make the function undefined.

p(x)A function of the form: f(x) , q(x) 0 where p(x) and q(x)are polynomials,

q(x)i rationas cal l funled a . ction

Here are some examples:

2

a. f(x)x 5

2

2

xb. f(x)

x 4

2

x 3c. f(x)

2x 5x 3

2

xd. f(x)

x 9

"Problem areas" such as these undefined values create "boundaries" of a sort. When we begin to graph, these boundaries are called asymptotes.Being able to locate an asymptote is essential to graphing rational functions

These values that make the denominator zero were found by setting the denominator equal to zero and solving the equation.

2

x 3c. f(x)

2x 5x 3

22x 5x 3 0

(2x 1)(x 3) 0

1

x , x 32Therefore x cannot equal ½ or –3.

2

2.5 Graphing Rational Functions

(Do not graph.)Example 1. Find the vertical asymptotes of the function:2

3f(x)

x 4

Solution: 2x 4 0

2x 4x 2

x = –2 and x = 2 are the vertical asymptotes.Answer:

Vertical AsymptotesWhen graphing rational functions, values that cause the denominator to become undefined create a "problem area" and must be excluded.  The graph of a rational function will approach the values that cause the denominator to be zero but will never touch or cross those values.  For this reason we often draw a dotted vertical line at those locations.  These lines are called vertical asymptotes. 

Procedure: To Find a Vertical Asymptote

1. Set the denominator equal to zero and solve. The solutions (if any) are the vertical asymptotes.

Your Turn Problem #1

Find the vertical asymptotes of the function: 2

4xf(x)

x 4x 21

Answer: The vertical asymptotes are x = –3 and x = 7.

3

2.5 Graphing Rational Functions

3Let's graph this function by plotting points: f(x)

x 5

1. Find the vertical asymptote and draw it in using a vertical dashed line.2. Because the graph does not cross the vertical asymptote, we have two regions to consider. Let’s

first pick values for x to the left of –5. Not all of these will be shown or need to be shown. Many of the following points are listed to obtain an understanding of the graph. The graph gets close to the x axis but never touches it.

x = –5

x

x y

-6 -3-7 -1 ½ -8 -1-9 -3/4 -10 -3/5-11 -1/2-12 -3/7-100 -3/95-5.5 -6-5.1 -30

3. Now we will choose plenty of points to the right of the vertical asymptote.

Right now, we are picking plenty of points to get an understanding of

what rational functions look like. Don’t worry, later, we will only need to plot a couple of points on each side.

x y

-4 3-3 1 ½ -2 1-1 3/4 0 3/510 1/3100 1/35-4.5 6-4.9 30

4. Notice again that the graph gets close to the x axis on both sides but will never reach it. In

this problem, the x axis is a horizontal asymptote.

Next Slide

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2.5 Graphing Rational Functions

Horizontal AsymptotesLocating the horizontal asymptote is essential to graphing a rational function. Finding the horizontal asymptote is not as simple as finding a vertical asymptote. Recall, to find a vertical asymptote, set the denominator equal to zero and solve. Let’s consider the general form of a rational function where the numerator and denominator are in descending order where the degree of the numerator is n and the degree of the denominator is m.n n 1

n n 1 1 0n mm m 1

m m 1 1 0

a x a x a x af(x) , a 0, b 0

b x b x b x b

Procedure: To Find a Horizontal Asymptote

1. If n < m , the horizontal asymptote is y = 0 (which is the x-axis).

2. If n = m, the horizontal asymptote is the line n

m

ay .

b

3. If n > m , there is no horizontal asymptote. This case will be discussed in the next section.

Let’s say that again:The degree of a polynomial is the largest exponent. The leading coefficient is the coefficient of the term with the largest exponent.

2 5xExample: 2x 3 The degree of this polynomial is 2 (largest exponent), and the leading coefficient is 5.

If the degree of the numerator is less than the degree of the denominator, then the H.A. is automatically y = 0.

If the degree of the numerator equals the degree of the denominator, then the H.A. is y = a/b where a and b are the leading coefficients of numerator and denominator.

5

2.5 Graphing Rational Functions

H.A. y 0

2H.A. y ,

81

y4

H.A. y 0

Example 2. Find the horizontal asymptotes (if any) of the functions:(Do not graph.)

Answer:

3

a. f(x)x 4

2x 9

b. f(x)8x 5

2

3x 2c. f(x)

4x 9

Note: In example 2a and 2c, the horizontal asymptote is y = 0 because the degree of the numerator is less than the degree of the denominator. In example 2b, the horizontal asymptote is the y = 1/4 because the degree of the numerator equals the degree of the denominator.

Your Turn Problem #2

2x 9

a. f(x)5x 8

2

x 2b. f(x)

x 5x 6

2x 1c. f(x)

2x 3

Find the horizontal asymptotes of the functions (if any): Answers:

2h.a. y

5

h.a. y 0

No horizontal asymptote

6

2.5 Graphing Rational Functions

Answer:

Example 3. Find all asymptotes of the functions:

5

a. f(x)2x 3

(Do not graph.)

4 x

b. f(x)x 7

2

x 6c. f(x)

x 2x 24

H.A. y 0

3

V.A. x2

H.A. y 1

V.A. x 7

H.A. y 0

V.A. x 4 and x 6

Your Turn Problem #3

Find all asymptotes of the functions:

7

a. f(x)2x 5

2

x 2b. f(x)

x 9

2

2

4xc. f(x)

x 1

5

v.a. x2

h.a. y 0

Answer: v.a. x 3 or x 3

h.a. y 0

v.a. none

h.a. y 4

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2.5 Graphing Rational Functions

1. Find the vertical asymptotes by setting the denominator equal to zero.

2. Find the horizontal asymptote (if there is one). If the deg p(x)< deg q(x), the horizontal asymptote is y = 0. If the deg p(x) = deg q(x), the horizontal asymptote is y = a/b where a and b are the leading coefficients.3. Plot enough points in each region to obtain an accurate description of the

graph.

Next Slide

4. Draw in the graph showing the function approaching the asymptotes.

Strategy for Graphing a Rational Function

p(x)Suppose that f(x) ,

q(x)

where p(x) and q(x) are polynomial functions with no common factors.

8

2.5 Graphing Rational Functions

3 Graph fExampl (x)e .

44

x

1. Find the vertical asymptote.

x = 4

f(x)

x

2. Find the horizontal asymptote.

3. Plot a few points to the left and right of the vertical asymptote.

y = 0, (degree of numerator < degree of denominator)

4. Draw the graph showing the function approaching the asymptotes

Next Slide

x y

3 -34 -1 ½0 -¾ 5 36 1 ½

(5,3)

(6,1½)

(2,-1½)

(3,-3)

(0,-3/4)

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2.5 Graphing Rational Functions

Your Turn Problem #4

2

Graph f(x)x 1

Show the any asymptotes that exist.

V.A. x = –1

f(x)

x

H.A. y = 0 x y

-2 -2-3 -1-4 -2/3 0 2 1 1

(1,1)(0,2)

(-4,-2/3)

(-3,-1)

(-2,-2)

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2.5 Graphing Rational Functions

2

2 Graph f(x)Example 5.

x 2x 3

f(x)

x

Plot 3 points in the center region and 2 points on the outer regions.

x y

-4 2/5-5 1/6-2 -2/3-1 -1/2 0 -2/3 2 2/5 3 1/6

Draw the graph showing the function approaching the asymptotes

Horizontal asymptote: y = 0.

Vertical asymptotes: x = -3 and x = 1

2

2 2f(x)

(x 3)(x 1)x 2x 3

Find the asymptotes.

(-4,2/5)(-5,1/6) (3,1/6)

(2,2/5)

(0,-2/3)

(-1,-1/2)

(-2,-2/3)

Next Slide

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2.5 Graphing Rational Functions

Your Turn Problem #5

2

4Graph f(x)

x 4

Show any asymptotes that exist.

(-4,1/3)

(-1,-4/3)

f(x)

x

V.A. x = 2 and x = –2 H.A. y = 0

x y

-3 4/5-4 1/3 -1 -4/3 0 -1 1 -4/3 3 4/5 4 1/3

(4,1/3)(3,4/5)

(1,-4/3)

(0,-1)(-3,4/5)

Note: You may recognize that this graph exhibits symmetry with respect to the y axis. This will save time with picking points. You can prove this graph is symmetric by showing f(-x) = f(x).

12

2.5 Graphing Rational Functions

2

2

3x Graph f(x)Exampl

1e 6.

x

Draw the graph showing the function approaching the asymptotes

Horizontal asymptote: y = 3

Vertical asymptotes: none f(x)

x

Find the asymptotes.

(2,2 2/5)

(1,1 1/2)

(0,0)

(2,2 7/10)

x y

0 0 1 1 1/2 2 2 2/5 3 2 7/10 4 2 14/17

Next Slide

Since the function is symmetric, we only need to find points on one side then reflect across y-axis.

The graph is symmetric to the y-axis because:

2

2

3( x)f( x) f(x)

( x) 1

13

2.5 Graphing Rational Functions

Your Turn Problem #6

2

5Graph f(x)

x 1

f(x)

xV.A. none

H.A. y = 0

Notice, this graph is symmetric to y-axis. You only need to pick points on one side and reflect across y-axis for the complete graph. Even if you don’t consider this functions symmetry, just choose enough values for x.

x y

0 5 1 2 1/2 2 1 3 1/2 4 5/17

(1,2 1/2)

(0,5)

(2,1)

(4,5/17)

(3,1/2)

There may be functions whose graph is a little different from the examples in this lesson. Keep in mind the basic procedure. Check for symmetry. Find all asymptotes. Plot enough points. The End.

B.R.7-9-03