Rational Functions

Rational Functions

Transformations and Graphing

1y

x

2

1y

x

Parent Function for Simple Rational Functions

Parent function:

xxf

1)(

Shape: ______________

Parts are called: ______________

Domain: _______________________

Range: _______________________

hyperbola

branches

All reals EXCEPT 0 or x ≠ 0

All reals EXCEPT 0 or y ≠ 0

Example• Identify vertical and horizontal asymptotes of the

following rational functions.

1y

x

Vertical Asymptote (V.A.) _______

Horizontal Asymptote(H. A.)________

Domain__________________

Range ___________________

x = 0

y = 00 & 0x x

All Real except x can’t equal zero

0 & 0y y All Real except y can’t equal zero

Example• Identify vertical and horizontal asymptotes of the

following rational functions.

2

1y

x

Vertical Asymptote (V.A.) _______

Horizontal Asymptote(H. A.)________

Domain__________________

Range ___________________

x = 0

y = 00 & 0x x

All Real except x can’t equal zero

0 y

Transformations

• Forms1

( )f x a kx h

2

1( )f x a k

x h

a = vertical stretch/compression = reflection over the x-axis

h = horizontal shifttranslates it right or leftvertical asymptote

k = vertical shifttranslates it up or down horizontal asymptote

Translating Rational Functions

khx

axf

)(

1. _______the asymptotes of _______ and ______.

2. _______ the points to the left and to the right of the __________ asymptote. Make a table to help!

3. _______ the two ____________ of the hyperbola so that they pass through the plotted points and approach the asymptotes.

DRAW x = h

Three steps to graphing rational functions in the form of

y = k

PLOTvertical

DRAW branches

Ex 1: Graph Simple Rational Functions

Graph the function . Compare the graph with the parent function.

xy

6

Step 2: Draw the asymptotes.

x = 0 and y = 0

Step 1: Graph the parent function.

Step 3: Plot points on either side of the asymptotes. Look at the table on your calculator.

Step 4: Connect your points!

Compare and contrast the graphs.

Ex 2: Translate Simple Rational Functions

Graph . State the domain and range.

12

4

x

y

Step 1: Determine the asymptotes. Draw them in!

Step 3: Connect your points!

x = -2 and y = -1Remember that x = h and y = k

Calc: y = ( -4 / (x + 2)) -1

Domain: ___________________________

Range: ____________________________

All reals except x ≠ -2

All reals except y ≠ -1

Step 2: Plot points on either side of the asymptotes. Look at the table on your calculator.

ExampleState the Domain, Range, the vertical and

horizontal asymptote.

2( ) 2

1f x

x

State the Domain, Range, the vertical and horizontal asymptote. Also, be able to state all the transformations that take place from the parent function.

1. 3.

2. 4.

6( ) 5

3f x

x

2

3( ) 4

( 2)f x

x

5( ) 2f x

x

2

2( ) 4

( 3)f x

x

Vertical Asymptotes

• Vertical (VA)– The graph of f(x) has a vertical asymptote

corresponding to each root of .– Set each factor in the denominator equal to zero.

Meaning you might have to factor, and most of the time will!!! ( denominator = 0) Solve for x. This will give the location of the vertical asymptote.

– If there is not a real solution of h(x)=0, then there is no vertical asymptote.

Let be a rational function in lowest terms with the degree of at least 1

)(

)()(

xh

xgxf

0)( xh

Find the Vertical Asymptotes

1. 2. 6( )

3f x

x

2

3( )

1f x

x

3. 4. 2

1( )

3 4f x

x x

2

5( )

9f x

x

V.A. x = -3 V.A. x = 1 & x = -1

V.A. x = 4 & x = -1 V.A.= noneh(x)= no real solutions

Horizontal Asymptotes

–Numerator and denominator must be in standard form. –m and n are the degrees of the polynomials• Degree is the highest power of x when in standard

form• The polynomial 2x3-4x+3 has a degree of _______.

Let be a rational function in lowest terms with the degree of at least 1

)(

)()(

xh

xgxf

3m

n

axy

bx

Look at these equations and see what value of f(x) is approaching as it goes to infinity.

1. 2. 3.

4. 5. 6.

5

7

x 3

2

5

x

x

3

5

3 4

9

x x

x x

5

x 37

3

x

x

4

2

2

8 3

x

x x

Graphing General Rational Functions

x

xy

42

How do we find the HORIZONTAL ASYMPTOTE(S)?

To do this, you will have to look at the highest powers in your numerator and denominator. You have three options:

If the numerator power is bigger, then there is ________________________!NO HORIZONTAL asymptote!

If the denominator power is bigger, then the ___________________________! HORIZONTAL asymptote is y = 0

If the powers are the SAME, then the asymptote is ____________________________! the RATIO of the coefficients

3

24

x

xy

5

5

4

3

x

xy so y = ¾

m

n

axy

bx

Asymptotes (HA-3 Cases)• Horizontal (Case 1)

1. m and n are the degrees of the polynomials• If the degree of g(x) is less than the degree

of h(x), the x-axis is a horizontal asymptote.– If m < n, then the graph has a horizontal

asymptote at y = 0.

m

n

axy

bx

2

6

5

3

xy

x

. . 0H A y


2. If the degree of g(x) equals the degree of h(x), then the horizontal asymptote is determined by the ratio of the leading coefficients.– If m = n, then the graph of the function

has a horizontal asymptote at .ay

b

m

n

axy

bx

2

2

3

7 9

x xy

x

3

. .7

H A y


3. m > n If the degree of g(x) is greater than the

degree of h(x), then there is no horizontal asymptote.

m

n

axy

bx

3

25

xy

x . .H A none

BOBO BOTN

• Bigger on Bottom the asymptote is at y = 0• Bigger on Top there is no asymptote!• The third case occurs when the degrees are

the same. In this case the asymptote is at the ratio of the coefficients.

Review

• If m < n then _____________________

• If m > n then _____________________

• If m = n then _____________________

BOBO: HA is at y = 0

BOTN: No HA

HA is at the ratio

Rational Functions—HOLE(S)

With rational functions, there are sometimes when a hole (or several) will appear in the graph.

How do we find these holes???

Hopefully you remember how to factor, because it is a big part.

Directions for finding holes. Factor the numerator AND denominator if possible. Leave them in factored form. If there are

factors that are the same, then set it equal to 0 and solve for x.

1

12 2

x

xxy

22 1 ( 1)(2 1)x x Factored x x

How do we find the HOLE(S)?

Factor the numerator AND denominator if possible. Leave them in factored form. If there are factors that are the same, then set it equal to 0 and solve for x.

Example: Factor the numerator

( 1)(2 1)

1

x xy

x

Put your factors back into the fraction.

Are there any factors that are the same?YES! (x + 1)

011 x Set the factor equal to 0 and solve. This will give you the hole!1x

Factor and Simplify• Factor the numerator and denominator & Simplify

( 1) ( 1)( 1)( 1)( )

( 1)

x xx xf x

x

( 1)x 1x

2

2

1( )

3 4

xf x

x x

2

12 24( )

2

xf x

x x

12 ( 2)12( 2)

( )( 2)

xxf x

x x

( 2)x x 12

x

( 1) ( 1)( 1)( 1)( )

( 4)( 1)

x xx xf x

x x

( 4) ( 1)x x ( 1)

( 4)

x

x

2 2 1( )

1

x xf x

x

Hole x = -2

Hole x = 1

Hole x = -1

Factor and Simplify• Enter the function into y1 and the answer into y2

• Are they two the same?

2

12 24 12( )

2

xf x

x x x

1212( 2)( )

( 2)

( 2)xf

x x

xx

( 2)x x

12

x We have a

hole!

No

Putting it All Together…

2

432

x

xxy

For the following equation, find the vertical and horizontal asymptotes and the holes, if they exist.

Example 1:

Find the vertical asymptote(s): 02 x

2

)1)(4(

x

xxy

Find the horizontal asymptote(s):

2x

Since the power on the top is bigger, there is no horizontal asymptote!

Find the hole(s):

There are no holes since we cannot “cancel” out any factors!


3

152 2

x

xxy

03 x


Example 2:

Find the vertical asymptote(s):

3

)3)(52(

x

xxy

Find the horizontal asymptote(s): Since the power on the top is bigger, there is no horizontal asymptote so it is a LINE!

Find the hole(s):

Since the common factor is (x - 3), there is a hole when x = 3

3x


3

152 2

x

xxy


Example 2:

Can you have an asymptote AND a hole at the same value? In this case, when x = 3??? NO!!!

When you graph the function, what does it look like?

Lines do NOT have asymptotes! Therefore, there is a hole when x = 3.

SO…if you “cancel” out the common factors and a hole and asymptote exist at the same value, check to see if your equation reduces to a line!

a line

Looking at the Symmetry… • Odd-Rotational about the origin.

• Even-Symmetric about the y-axis.

Example• Odd or Even?

2

1( )

( 3)

xf x

x

2

12 24( )

2

xf x

x x

2 2 1( )

1

x xf x

x

Odd

Even

Odd

Practice the Parent Functions• Graph the following on your graphing calculator and

determine whether each is Even, Odd or Neither.

1. ( ) 4f x 2. ( )f x x

10. ( ) lnf x x

111. ( )f x

x

6. ( )f x x

34. ( )f x x5. ( )f x x

9. ( ) xf x e

23. ( )f x x

2

112. ( )f x

x

Even

Even

Even

Odd

Odd Odd

EvenNeither

Neither

Neither

37. ( )f x x Odd

8. ( ) logf x x Neither

Documents

Rational Functions