Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.

Unit 7—Rational Functions

Rational Expressions• Quotient of 2 polynomials

0,

91

132

3

52

15

3 2

3

2

qq

pso

xory

xxor

y

xor

xy

yx

Things to Consider

Graphing Rational Functions

1. Factor2. Determine where discontinuities would occur in

the graph3. Graph any asymptotes on the graph and pick

points on both sides to locate the branches of the function

• If factors cancel, then you have a point of discontinuity (hole)

• If factors remain in the denominator, you have a vertical asymptote(s)

Horizontal Asymptotes

1. If the degree of the numerator is bigger than the degree of the denominator, then there is NO H.A. (top-heavy)

2. If the degree of numerator is smaller than the degree of the denominator, then the HA is at y=0 (bottom heavy)

3. If the degree of numerator is equal to degree of the denominator, then the HA is equal to the leading coefficients (equal weight)


209

1272

2

xx

xxy


5

10133 2

x

xxy


2

322

2

xx

xxy


322

xx

xy


3

92

x

xy


1

92

2

x

xy

To simplify a rational expression

• Look for common factors

3

3.

9124

94.

)1)(2(

)2(.

9

27.

2

2

4

3

x

xex

xx

xex

xx

xex

yx

yxex

To simplify a rational expression

• Look for common factors

654

36.

25

5.

2

2.

2

2

xx

xex

x

xex

x

xex

Multiply Rational Expressions

• Factor, Reduce common factors first, then multiply

3

4

127

)3(.

3

44

4

9.

2

22

2

2

x

x

xx

xex

x

xx

x

xex

Multiply Rational Expressions

• Factor, Reduce common factors first, then multiply

5

25

9

27.

8

16

4

4.

2

2

32

2

3

x

x

x

xex

a

b

bb

abex

Dividing Rational Expressions

• Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply

1

3

12.

123

65

2412

63.

22

2

2

2

x

x

xx

xex

x

xx

x

xex


• Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply

2

2

4

322

9

16

3

4.

84

9

2

93.

y

x

y

xex

x

x

x

xxex


77

5

4

22

56

25.

2

2

2

2

3

x

xx

x

x

xx

xxex


bybxayax

bybxayax

bybxayax

bybxayaxex

.


1077856

149

2

2

2

2

xxxxxxxx


12 11

1

1 x

x

x

x

x

x

Adding and Subtracting Rational Expressions

• To add fractions, you must have a common denominator

• To determine the LCD, list any common factor that occurs in two or more of the denominators only once in the LCD and then include all other factors that are not common.


1

4

1

52.

44

1.

22

x

x

x

xex

x

x

x

xex


9

4

3

35.

52

4

13

6.

2

x

x

x

xex

x

x

x

xex


)1)(1(

442.

168

3

16

4.

222

xxx

x

xx

xex

xxxex


2

3

2

87.

1

3

1

5

2

1.

22

yyyy

yex

xx

x

xex


324

7

245

23.

25

5

25

102.

2222

yyyy

yex

x

x

x

xex

Complex Rational Expressions

x

yx1

1

11


x

xyx

11

24

2


263

1235

mm

mm


yx

yx11

11


6

3x

x

xx

Solving Rational Equations

1. Factor all denominators2. Multiply both sides of equation by LCD3. Solve4. Eliminate any solution that would make the

denominator zero5. Check remaining solutions


xxxx 3

8

3

452


x

x 42

5

1


45103

22

x

x

xx

x


3

2

3

1

yy

y


3

2

1

3

mm


xxx

4

2

5050


2

5

4

2

2

32

yy

y

y

Solving Rational Inequalities

1. State the excluded values2. Solve the related equation3. Use those values on a number line and test

the values


3)3(2

2

x

x

x

x


12

4

c


Possible Graphs:

Direct and Inverse Variation

• Direct Variation can be expressed in the form y=kx

• K is the constant of variation

• Equation of variation—equation representing the relationship between the variable but substitute the value of k

Ex. Y=2x (if k=2)


10y when x Find

3. x12,y When ith x.directly w varies

y


1y x when Find

5. x25,y When ith x.directly w varies

y


4p when q Find

.3p 8,q When p. of squareith directly w varies

Q


3m when L Find

.2m ,2

1L When m. of cubeith directly w varies

L

Inverse Variation

• Expressed as y=k/x


4

3y when x Find

.5

2 x3,y When with x.inversely varies

y


4y when x Find

.2

1 x2,y When x.of square with theinversely varies

y


8y x when Find

.3- x6,y When with x.inversely varies

y

Joint Variation

• When one quantity varies directly with the product of two or more other quantities

• Combination Variation—when one quantity varies directly with another quantity and inversely with the other quantity

Joint or Combination Variation

68y when x Find

.32 x4,y When z. of square theandith x directly w varies

zand

zandy

Joint or Combination Variation

65y when x Find

.14 x20,y When z. of square the

withinversely andith x directly w varies

zand

zand

y

Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.

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Transcript of Unit 7—Rational Functions Rational Expressions Quotient of 2 polynomials.