Define Inverse Variation #3 Give a real life example
Dec 29, 2015
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