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)

Graphing Rational Functions

209

1272

2

xx

xxy

Graphing Rational Functions

5

10133 2

x

xxy

Graphing Rational Functions

2

322

2

xx

xxy

Graphing Rational Functions

322

xx

xy

Graphing Rational Functions

3

92

x

xy

Graphing Rational Functions

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

Dividing Rational Expressions

• 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

Dividing Rational Expressions

77

5

4

22

56

25.

2

2

2

2

3

x

xx

x

x

xx

xxex

Dividing Rational Expressions

bybxayax

bybxayax

bybxayax

bybxayaxex

.

Dividing Rational Expressions

1077856

149

2

2

2

2

xxxxxxxx

Dividing Rational Expressions

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.

Adding and Subtracting Rational Expressions

1

4

1

52.

44

1.

22

x

x

x

xex

x

x

x

xex

Adding and Subtracting Rational Expressions

9

4

3

35.

52

4

13

6.

2

x

x

x

xex

x

x

x

xex

Adding and Subtracting Rational Expressions

)1)(1(

442.

168

3

16

4.

222

xxx

x

xx

xex

xxxex

Adding and Subtracting Rational Expressions

2

3

2

87.

1

3

1

5

2

1.

22

yyyy

yex

xx

x

xex

Adding and Subtracting Rational Expressions

324

7

245

23.

25

5

25

102.

2222

yyyy

yex

x

x

x

xex

Complex Rational Expressions

x

yx1

1

11

Complex Rational Expressions

x

xyx

11

24

2

Complex Rational Expressions

263

1235

mm

mm

Complex Rational Expressions

yx

yx11

11

Complex Rational Expressions

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

Solving Rational Equations

xxxx 3

8

3

452

Solving Rational Equations

x

x 42

5

1

Solving Rational Equations

45103

22

x

x

xx

x

Solving Rational Equations

3

2

3

1

yy

y

Solving Rational Equations

3

2

1

3

mm

Solving Rational Equations

xxx

4

2

5050

Solving Rational Equations

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

Solving Rational Inequalities

3)3(2

2

x

x

x

x

Solving Rational Inequalities

12

4

c

Graphing Rational Functions

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)

Direct and Inverse Variation

10y when x Find

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

y

Direct and Inverse Variation

1y x when Find

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

y

Direct and Inverse Variation

4p when q Find

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

Q

Direct and Inverse Variation

3m when L Find

.2m ,2

1L When m. of cubeith directly w varies

L

Inverse Variation

• Expressed as y=k/x

Direct and Inverse Variation

4

3y when x Find

.5

2 x3,y When with x.inversely varies

y

Direct and Inverse Variation

4y when x Find

.2

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

y

Direct and Inverse Variation

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