define inverse variation #3 give a real life example
TRANSCRIPT
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Define Inverse Variation
#3
Give a real life example
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•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
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State the General Form of an inverse variation
equation.
Draw an example of a typical inverse variation
and name the graph.#4
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xy = k or . x
ky
HYPERBOLA (ROTATED)
#4
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FUNCTIONSBLUE CARD
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Define Domain
Define Range
#9
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• DOMAIN - List of all possible x-values
(aka – List of what x is allowed to be).
• RANGE – List of all possible y-values.
#9
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Test whether a relation (any random equation) is a FUNCTION or not?
#10
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Vertical Line Test• Each member of the
DOMAIN is paired with one and only one member of the
RANGE.
#10
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Define 1 – to – 1 Function
How do you test for one?
#11
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1-to-1 Function: A function whose inverse is also a
function.
Horizontal Line Test
#11
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How do you find an INVERSE Function…
ALGEBRAICALLY?
GRAPHICALLY?
#12
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Algebraically:Switch x and y…
…solve for y.Graphically:
Reflect over the line y=x
#12
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What notation do we use for Inverse?
If point (a,b) lies on f(x)…
#13
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)(1 xf
…then point (b,a) lies on )(1 xf
Notation:
#13
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f(-x)
•Identify the action
•Identify the result
#17
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•Action: Negating x
•Result: Reflection over the y-axis
#17
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-f(x)•Identify the action
•Identify the result
#18
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•Action: negating y
•Result: Reflection over the x-axis
#18
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Exponents
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When you multiply…
the base and
the exponents
#46
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• KEEP (the base)
• ADD (the exponents)
#46
853 222
baba xxx
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When dividing… the base&
the exponents.
#47
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• Keep (the base)
• SUBTRACT (the exponents)
#47
67
33
3
bab
a
xx
x
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Power to a power…
#48
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• MULTIPLY the exponents
#48
22
4
1
4
2
14
2
1
xxxx
xx abba
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Negative Exponents…
#49
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• Reciprocate the base
#49
666
66
1)(
22
baab
bb
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Ground Hog Rule
#50
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4
34 3 xx
xx n
mn m
#50
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Exponential Equations
y = a(b)x
Identify the meaning of a & b#51
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• Exponential equations occur when the exponent contains a variable
• a = initial amount
• b = growth factor
b > 1 Growth
b < 1 Decay#51
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Name 2 ways to solve an
Exponential Equation
#52
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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
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A typical EXPONENTIAL GRAPH looks like…
#53
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Horizontal asymptote y = 0y = 2^x
#53
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Logarithms
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Expand
1) Log (ab)
2) Log(a+b)
#55
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1. log(a) + log (b)
2. Done!
#55
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Expand
1. log (a/b)
2. log (a-b)
#56
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1. log(a) – log(b)
2. DONE!!
#56
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Expand
1. logxm
#57
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m log x
#57
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Convert exponential to log form
23 = 8
#58
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#58
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Convert log form to exponential form
log28 = 3
#59
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Follow the arrows.
823 #59
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Log Equations
1. every term has a log
2. not all terms have a log
#60
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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
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What does a typical logarithmic graph look
like?
#61
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Vertical asymptote at x = 0
#61
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Change of Base Formula
What is it used for?
#62
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Used to graph logs
a
xxa log
loglog
#62
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EXACT TRIG VALUES
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sin 30or
sin #66
6
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2
1
#66
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sin 60orsin
#67
3
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#67
2
3
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sin 45orsin
#68
4
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#68
2
2
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sin 0
#69
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0
#69
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sin 90or sin
#70
2
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1
#70
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sin 180or
sin #71
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0
#71
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sin 270or sin 2
3
#72
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-1
#72
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sin 360or sin
#73
2
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0
#73
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cos 30or cos 6
#74
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2
3
#74
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cos 60or
cos 3
#75
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2
1
#75
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cos 45or cos 4
#76
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2
2
#76
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cos 0
#77
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1
#77
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cos 90or cos 2
#78
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0
#78
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cos 180 or cos
#79
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-1
#79
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cos 270 or cos 2
3
#80
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0
#80
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cos 360or cos 2
#81
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1
#81
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tan 30or tan 6
#82
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3
3
#82
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tan 60or tan 3
#83
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#83
3
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4
tan 45or tan
#84
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1
#84
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tan 0
#85
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0
#85
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tan 90or tan 2
#86
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D.N.E.or
Undefined
#86
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tan 180or tan
#87
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0
#87
![Page 99: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/99.jpg)
tan 270or
tan 2
3
#88
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D.N.E.
Or
Undefined#88
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tan 360or tan 2
#89
![Page 102: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/102.jpg)
0
#89
![Page 103: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/103.jpg)
Trigonometry Identities
![Page 104: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/104.jpg)
Reciprocal Identity
sec =#90
![Page 105: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/105.jpg)
cos
1
#90
![Page 106: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/106.jpg)
Reciprocal Identity
csc =
#91
![Page 107: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/107.jpg)
sin
1
#91
![Page 108: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/108.jpg)
cot =
Reciprocal Identity
#92
![Page 109: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/109.jpg)
sin
cos
tan
1or
#92
![Page 110: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/110.jpg)
Quotient Identity
tan#93
![Page 111: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/111.jpg)
cos
sin
#93
![Page 112: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/112.jpg)
Trig Graphs
![Page 113: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/113.jpg)
Amplitude
#94
![Page 114: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/114.jpg)
Height from the midline
y = asin(fx)y = -2sinxamp = 2
a
#94
![Page 115: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/115.jpg)
Frequency
#95
![Page 116: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/116.jpg)
How many complete cycles between 0 and 2
#95
![Page 117: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/117.jpg)
Period
#96
![Page 118: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/118.jpg)
How long it takes to complete one full cycle
Formula:
fperiod
2
#96
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y = sinx
a) graph b) amplitudec) frequency
d) periode) domain
f) range #97
![Page 120: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/120.jpg)
a)
b) 1c) 1d)e) all real numbersf)
2
1
2
11 y
x
y
#97
![Page 121: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/121.jpg)
y = cosx
a) graph b) amplitudec) frequency
d) periode) domain f) range
#98
![Page 122: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/122.jpg)
a)
b) 1c) 1d)e) all real numbersf)
2
1
2
x
y
11 y
#98
![Page 123: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/123.jpg)
y = tan x
a) graphb) amplitude
c) asymptotes at…
#99
![Page 124: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/124.jpg)
a)
b) No amplitude
c) Asymptotes are at odd multiplies of
x
y
2
Graph is always increasing
#99
![Page 125: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/125.jpg)
y = csc x• A) graph
• B) location of the asymptotes
#100
![Page 126: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/126.jpg)
b) Asymptotes are multiples of
x
y
Draw in ghost sketch
#100
![Page 127: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/127.jpg)
y = secx
• A) graph
• B) location of the asymptotes
#101
![Page 128: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/128.jpg)
x
y
• B) asymptotes are odd multiples of 2
Draw in ghost sketch
#101
![Page 129: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/129.jpg)
y=cotx
• A) graph
• B) location of asymptotes
#102
![Page 130: Define Inverse Variation #3 Give a real life example](https://reader036.vdocuments.mx/reader036/viewer/2022062308/56649e7c5503460f94b7deb5/html5/thumbnails/130.jpg)
x
y
• B) multiplies of • Always decreasing
#102