Define Inverse Variation #3 Give a real life example.

Define Inverse Variation

#3

Give a real life example

•The PRODUCT of two variables will always be

the same (constant).• Example:

–The speed, s, you drive and the time, t, it takes for you to get to Rochester.

#3

State the General Form of an inverse variation

equation.

Draw an example of a typical inverse variation

and name the graph.#4

xy = k or . x

ky

HYPERBOLA (ROTATED)

#4

FUNCTIONSBLUE CARD

Define Domain

Define Range

#9

• DOMAIN - List of all possible x-values

(aka – List of what x is allowed to be).

• RANGE – List of all possible y-values.

#9

Test whether a relation (any random equation) is a FUNCTION or not?

#10

Vertical Line Test• Each member of the

DOMAIN is paired with one and only one member of the

RANGE.

#10

Define 1 – to – 1 Function

How do you test for one?

#11

1-to-1 Function: A function whose inverse is also a

function.

Horizontal Line Test

#11

How do you find an INVERSE Function…

ALGEBRAICALLY?

GRAPHICALLY?

#12

Algebraically:Switch x and y…

…solve for y.Graphically:

Reflect over the line y=x

#12

What notation do we use for Inverse?

If point (a,b) lies on f(x)…

#13

)(1 xf

…then point (b,a) lies on )(1 xf

Notation:

#13

f(-x)

•Identify the action

•Identify the result

#17

•Action: Negating x

•Result: Reflection over the y-axis

#17

-f(x)•Identify the action

•Identify the result

#18

•Action: negating y

•Result: Reflection over the x-axis

#18

Exponents

When you multiply…

the base and

the exponents

#46

• KEEP (the base)

• ADD (the exponents)

#46

853 222

baba xxx

When dividing… the base&

the exponents.

#47

• Keep (the base)

• SUBTRACT (the exponents)

#47

67

33

3

bab

a

xx

x

Power to a power…

#48

• MULTIPLY the exponents

#48

22

4

1

4

2

14

2

1

xxxx

xx abba

Negative Exponents…

#49

• Reciprocate the base

#49

666

66

1)(

22

baab

bb

Ground Hog Rule

#50

4

34 3 xx

xx n

mn m

#50

Exponential Equations

y = a(b)x

Identify the meaning of a & b#51

• Exponential equations occur when the exponent contains a variable

• a = initial amount

• b = growth factor

b > 1 Growth

b < 1 Decay#51

Name 2 ways to solve an

Exponential Equation

#52

1. Get a common base, set the exponents equal

2. Take the log of both sides

5log

7log

7log5log

75

x

x

x

3

22

823

x

x

x

#52

A typical EXPONENTIAL GRAPH looks like…

#53

Horizontal asymptote y = 0y = 2^x

#53

Logarithms

Expand

1) Log (ab)

2) Log(a+b)

#55

1. log(a) + log (b)

2. Done!

#55

Expand

1. log (a/b)

2. log (a-b)

#56

1. log(a) – log(b)

2. DONE!!

#56

Expand

1. logxm

#57

m log x

#57

Convert exponential to log form

23 = 8

#58

#58

Convert log form to exponential form

log28 = 3

#59

Follow the arrows.

823 #59

Log Equations

1. every term has a log

2. not all terms have a log

#60

1. Apply log properties and knock out all the logs

2. Apply log properties condense log equationconvert to exponential and solve

112)4)(32(

)112log()4log()32log(2

2

xxx

xxx

xx

xx

xx

89

1)8)((log

1)8(loglog

21

9

99

#60

What does a typical logarithmic graph look

like?

#61

Vertical asymptote at x = 0

#61

Change of Base Formula

What is it used for?

#62

Used to graph logs

a

xxa log

loglog

#62

EXACT TRIG VALUES

sin 30or

sin #66

6

2

1

#66

sin 60orsin

#67

3

#67

2

3

sin 45orsin

#68

4

#68

2

2

sin 0

#69

0

#69

sin 90or sin

#70

2

1

#70

sin 180or

sin #71

0

#71

sin 270or sin 2

3

#72

-1

#72

sin 360or sin

#73

2

0

#73

cos 30or cos 6

#74

2

3

#74

cos 60or

cos 3

#75

2

1

#75

cos 45or cos 4

#76

2

2

#76

cos 0

#77

1

#77

cos 90or cos 2

#78

0

#78

cos 180 or cos

#79

-1

#79

cos 270 or cos 2

3

#80

0

#80

cos 360or cos 2

#81

1

#81

tan 30or tan 6

#82

3

3

#82

tan 60or tan 3

#83

#83

3

4

tan 45or tan

#84

1

#84

tan 0

#85

0

#85

tan 90or tan 2

#86

D.N.E.or

Undefined

#86

tan 180or tan

#87

0

#87

tan 270or

tan 2

3

#88

D.N.E.

Or

Undefined#88

tan 360or tan 2

#89

0

#89

Trigonometry Identities

Reciprocal Identity

sec =#90

cos

1

#90

Reciprocal Identity

csc =

#91

sin

1

#91

cot =

Reciprocal Identity

#92

sin

cos

tan

1or

#92

Quotient Identity

tan#93

cos

sin

#93

Trig Graphs

Amplitude

#94

Height from the midline

y = asin(fx)y = -2sinxamp = 2

a

#94

Frequency

#95

How many complete cycles between 0 and 2

#95

Period

#96

How long it takes to complete one full cycle

Formula:

fperiod

2

#96

y = sinx

a) graph b) amplitudec) frequency

d) periode) domain

f) range #97

a)

b) 1c) 1d)e) all real numbersf)

2

1

2

11 y

x

y

#97

y = cosx

a) graph b) amplitudec) frequency

d) periode) domain f) range

#98

a)

b) 1c) 1d)e) all real numbersf)

2

1

2

x

y

11 y

#98

y = tan x

a) graphb) amplitude

c) asymptotes at…

#99

a)

b) No amplitude

c) Asymptotes are at odd multiplies of

x

y

2

Graph is always increasing

#99

y = csc x• A) graph

• B) location of the asymptotes

#100

b) Asymptotes are multiples of

x

y

Draw in ghost sketch

#100

y = secx

• A) graph

• B) location of the asymptotes

#101

x

y

• B) asymptotes are odd multiples of 2

Draw in ghost sketch

#101

y=cotx

• A) graph

• B) location of asymptotes

#102

x

y

• B) multiplies of • Always decreasing

#102