Advanced Algebra Chapter 9

1
Advanced Algebra Chapter 9

• Rational Equations and Functions

2
Inverse and Joint Variation9.1
3
Direct Variation

• Two variables x and y vary directly iff
• If x and y vary directly and y6 when x3, write
  the general direct variation equation

4
Inverse Variation

• x and y vary inversely if they are related by
• k is our constant of variation

5
Direct or Inverse?

• How do we tell the difference?
• Direct
• Constant and x are multiplied
• Inverse
• Constant is divided by x
• x is the denominator

6
Writing Equations

• x and y vary inversely, and y6 and x1.5
• Find y when

7
Joint Variation

• When a quantity varies directly as the product of
  two or more other variables
• However, there are other possibilities of joint
  variation

8
Variation

• y varies directly with x
• Y varies inversely with x

9
Variation

• z varies jointly with x and y
• y varies inversely with the square of x

10
Variation

• z varies directly with y and inversely with x
• y varies inversely with x and z

11
p.53721-24, 29-31, 35-36, 39-40, 45-47
12
Graphing Simple Rational Functions9.2
13
Domain and Range

• Domain Any and all numbers that can be plugged
  into a function
• X-values
• Range All output values of a function
• Y-values

14
Rational Functions

• Any function composed of the quotient of two
  functions

15
Graphing Rational Functions
16
Hyperbolas

• The x-axis is the horizontal asymptote
• The y-axis is the verticalasymptote
• Domain All values except 0
• Range All values except 0
• Contains two symmetrical parts called branches

17
Hyperbolas--Shifting

• All functions in the form
  are hyperbolas
• Vertical asymptote at
• Horizontal asymptote at

18
Graphing
19
Graphing
20
Other Hyperbolas

• All functions of form
  are also hyperbolas
• Vertical asymptote
• Horizontal asymptote

21
Graphing
22
Graphing
23
p.54312-18 Even, 26, 27, 35, 36
24
Graphing Other Rational Functions9.3
25
Graphs of Rational Functions

• The graph of the of the functionhas the
  following
• The x-intercepts of the graph are the real zeros
  of
• The vertical asymptote occurs at each real zero
  of

26
Graphs of Rational Functions

• The graph of the of the functionhas the
  following
• The graph has at most 1 horizontal asymptote
• If , the line is the hor.
  asym.
• If , the line is the hor.
  asym.
• If , the graph has no hor. asym. The
  graphs end behavior is that of the line

27
Graphing
28
Graphing
29
Graphing
30
p.55023-31
31
Multiplying and Dividing Rational Expressions9.4
32
Rational Expressions

• A rational expression is in simplest form iff the
  numerator and denominator share no common factors
  (other than 1)

33
Simplifying Expressions

• Two Step process
• Factor the numerator and denominator completely
• Divide out any common factors

34
Simplifying Expressions
35
Simplifying Expressions
36
Simplifying Expressions
37
Simplifying Expressions
38
Simplifying Expressions
39
Simplifying Expressions
40
Simplifying Expressions
41
Simplifying Expressions
42
Simplifying Expressions
43
p.55816-18, 28, 30, 36, 37, 44,45
44
Addition, Subtraction, and Complex Fractions9.5
45
Addition and Subtraction

• With any fraction
• When adding or subtracting must have a common
  denominator
• Example

46
Addition and Subtraction
47
Addition and Subtraction
48
Addition and Subtraction
49
Addition and Subtraction
50
Addition and Subtraction
51
Addition and Subtraction
52
Addition and Subtraction
53
Addition and Subtraction
54
Complex Fractions

• A fraction where the numerator and denominator
  are themselves fractions
• Fractions within fractions

55
Simplifying Complex Fractions

• Combine the numerator into a single fraction
• Combing the denominator into a single fraction
• Multiply by the reciprocal
• Simplify

56
Complex Fractions
57
Complex Fractions
58
Complex Fractions
59
p.56626-30, 38-40
60
Solving Rational Equations9.6
61
Solving Equations

• Thoughts on Fractions??
• Maybe not
• Multiply all parts by the LCD
• Least Common Denominator
• OrIf only 1 term on each side
• Cross Multiply (Baseball)

62
Solving Rational Equations
63
Solving Rational Equations
64
Solving Rational Equations
65
Solving Rational Equations
66
Solving Rational Equations
67
(No Transcript)