Download - Exponential Graphs
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Exponential Graphs
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Warm UpSolve:
3 2( ) 3 10f x x x x
Find the Vertex: 2( ) 4 24 5f x x x
2,3
5,0
0253
0103 2
xxx
xxx
xxx
342
24
2
a
bx 31532434 2 y
31,3 V
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Definition
In an exponential function, the base is fixed and the exponent is a variable.
xbxf Exponent
base
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Exploration
Using your GDC, graph the following exponential functions on the same screen:
xxx yyy 5.1,2,3 321
?INTERSECTalltheydo
POINTwhatAt
.1,0POINTthethrough
passayform
theoffunctionsAllx
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Exploration
What do you observe about the function as the base gets larger, and the exponent remains positive?
.
,
,0
grows
functionthefasterthe
basetheLARGERthe
xforthatObserve
.ModelGrowth
acalledisThis
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Exploration
Using your GDC, graph the following exponential functions on the same screen:
xxx yyy 5.1,2,3 321
?whyandgraphs
previousthefromdiffer
graphsthesedoHow
.righttheof
insteadlefttheonrise
tofunctiontheforce
ExponentsNegative
.ModelDecay
acalledisThis
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Exploration…
Using your GDC, graph the following exponential functions on the same screen:
xx yy
3
1,3 21
?relatedpair
eachofgraphs
theareHow
.axisytheacrossMODELS
GROWTHtheofreflectiona
areMODELSDECAYThe
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Continued….
?relatedpairsthe
ofbasestheareHow
xx yy
3
1,3 21
.othereachof
sreciprocalareThey
.33
1: x
x
yassametheisyNOTE
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Graph: 2xy
x y
-2 0.25
-1 0.5
0 1
1 2
2 4
HA: y = 0
Domain:
Range:
,
,0
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Graph:
2 xy
x y
-2 4
-1 2
0 1
1 0.5
2 0.25
Decreasing!
Domain:
Range:
,
,0
HA: y = 0
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Graph:32xy
.
3
functionparentthefrom
leftthetomoved
beenhasgraphThe
Domain:
Range:
,
,0
HA: y = 0
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Graph: 2 2xy
.
2
functionparent
thefromupmoved
beenhasgraphThe
HA: y = 2
Domain:
Range:
,
,2
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Graph:42 3xy
FunctionParent
3
4
Down
Right
HA: y = -3
Domain:
Range:
,
,3
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Graph:12 5xy
FunctionParent
versed
Down
Left
Re
5
1
52 11 xy
Domain:
Range:
,
,5
HA: y = -5
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Graph:42 2xy
Domain:
Range:
Parent Function
Right 4Up 2
HA: y = 2
,
,2
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Natural exponential function
( ) xf x e
2.718281828...e
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Graph:1( ) 3xf x e
Domain:
Range:
Left 1Down 3
,
,3
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Logarithmic Function
It’s the inverse of the exponential function
log ya x y a x
Switch the x’s and the y’s!
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Graph:2( ) logf x x
Domain:
Range:
Is the inverse of xy 2
Domain:
Range:
,
,0
xy 2
yx 2
,0 ,
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Graph:2( ) 3 logf x x
Domain:
Range:
Up 3 from previous example!
,0 ,
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Graph:2( ) log ( 4)f x x
Domain:
Range:
Left 4 from Original Example!
,4
,
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Graph:2( ) log ( 2)f x x
Domain:
Range:
Right 2 from Original Example!
,2 ,
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Graph:2( ) log ( )f x x
Domain:
Range:
Reflected over y-axis.
0, ,
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Graph:2( ) logf x x
Domain:
Range:
Reflected over x-axis.
,0 ,
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Compound Interest
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An infectious disease begins to spread in a small city of population 10,000. After t days, the number of persons who have succumbed to the virus is modeled by the function:
How many infected people are there initially?
How many people are infected after five days?
0.97
10,000( )
5 1245 tv t
e
81250
000,10
12455
000,100
097.0
e
v
1.67812455
000,100 597.0
e
v
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Compound Interest
P = Principal
r = rate
t = time in years
n = number of times it’s compounded per year
( ) 1nt
rA t P
n
Compounded: annually n = 1
quarterly n = 4
monthly n = 12
daily n = 365
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Find the Final Amount: $8000 at 6.5% compounded quarterly for 8 years
nt
n
rPtA
1
09.134004
065.18000
84
tA
4n
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Find the Final Amount: $600 at 9% compounded daily for 20 years
98.3628365
09.01600
20365
tA
365n
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Find the Final Amount: $300 at 6% compounded annually for 25 years
tn
n
rPtA
1
56.12871
06.01300
251
tA
1n
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Compounded Continuously:
P = Principal
r = rate
t = time in years
E = 2.718281828…
( ) rtA t Pe
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Find the Final Amount: $2500 at 4% compounded continuously for 25 years
rtPetA
70.67952500 2504.0 etA
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Suppose your are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? How much will you make?
One million dollars at the end of the month.
Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, 2n cents on the nth day.
48.474836,21$
21474836482
231
A
centsA
centsA nMore Profitable