7RITE AND 'RAPH %XPONENTIAL
'ROWTH &UNCTIONS
+ 7RITE AND GRAPH EXPONENTIAL GROWTH MODELS
'OAL
9OUR .OTES
6/#!"5,!29
%XPONENTIAL FUNCTION Êv՘V̈œ˜ÊœvÊ̅iÊvœÀ“Ê ÞÊÊ>L ÝÊ
܅iÀiÊ>ÊpÊä]ÊLʀÊä]Ê>˜`ÊLÊpÊ£
%XPONENTIAL GROWTH ʵÕ>˜ÌˆÌÞÊ̅>Ìʈ˜VÀi>ÃiÃÊLÞÊ̅iÊ
Ã>“iÊ«iÀVi˜ÌʜÛiÀÊiµÕ>Ê̈“iÊ«iÀˆœ`Ã
#OMPOUND INTEREST ˜ÌiÀiÃÌÊi>À˜i`ʜ˜ÊLœÌ…Ê>˜Êˆ˜ˆÌˆ>Ê
ˆ˜ÛiÃ̓i˜ÌÊ>˜`ʜ˜Ê«ÀiۈœÕÏÞÊi>À˜i`ʈ˜ÌiÀiÃÌ
7RITE A FUNCTION RULE
%XAMPLE 7RITE A RULE FOR THE FUNCTION
X
Y
]z
]z
3OLUTION
3TEP 4ELL WHETHER THE FUNCTION IS EXPONENTIAL (ERE THE
YVALUES ARE MULTIPLIED BY Ê ÎÊ FOR EACH INCREASE
OF IN X SO THE TABLE REPRESENTS AN EXPONENTIAL
FUNCTION OF THE FORM Ê ÞÊÊ>L ÝÊ WHERE Ê LÊÊÎÊ 3TEP &IND THE VALUE OF A BY FINDING THE VALUE OF Y WHEN
X 7HEN X Y Ê >LäÊ Ê >Ê+Ê£Ê Ê >Ê 4HE VALUE OF Y WHEN X IS Ê ÓÊ SO Ê >ÊÊÓ
3TEP 7RITE THE FUNCTION RULE ! RULE FOR THE FUNCTION IS
Y Ê ÓÊ+ÊÎÝÊ #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
'RAPH AN EXPONENTIAL FUNCTION
%XAMPLE 'RAPH THE FUNCTION Y X )DENTIFY ITS DOMAIN AND RANGE
3OLUTION
3TEP -AKE A TABLE BY CHOOSING A
FEW VALUES FOR X AND FINDING
THE VALUES OF Y 4HE DOMAIN
IS Ê >ÊÀi>Ê˜Õ“LiÀÃÊ X
Y
£
Ê]
Ê£Êz Ê Ê]
Êz
™
Î
ÊÎÊ
Ê£Ê
ʙÊ
3TEP 0LOT THE POINTS
3TEP $RAW A SMOOTH CURVE THROUGH THE POINTS &ROM
EITHER THE TABLE OR THE GRAPH YOU CAN SEE THAT THE
RANGE IS Ê >Ê«œÃˆÌˆÛiÊÀi>Ê˜Õ“LiÀÃÊ #OMPARE GRAPHS OF EXPONENTIAL FUNCTIONS
%XAMPLE 'RAPH Y + X #OMPARE THE GRAPH WITH THE GRAPH OF
Y X
3OLUTION
4O GRAPH EACH FUNCTION MAKE
A TABLE OF VALUES PLOT THE
POINTS AND DRAW A SMOOTH
CURVE THROUGH THE POINTS
X
Y X
Y + X
Ê Êz
]
£
™
Ê Êz
]
Ê Êz
]
£
Î
Ê Êz
]
Ê £Ê
Ê ÓÊ
Ê ÎÊ
Ê ÈÊ
Ê ™Ê
Ê £nÊ
Ó
™
Ó
Î
"ECAUSE THE YVALUES FOR Y + X ARE Ê ÓÊ̈“iÃÊ THE
CORRESPONDING YVALUES FOR Y X THE GRAPH OF Y + X
IS A Ê ÛiÀ̈V>ÊÃÌÀiÌV…Ê OF THE GRAPH OF Y X
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
7RITE A RULE FOR THE FUNCTION
Ê
X
Y
]
z ]
z Ê ÞÊÊ£Ê+Ê{ÝÊ
'RAPH Y X )DENTIFY ITS
DOMAIN AND RANGE
/…iÊ`œ“>ˆ˜ÊˆÃÊ>ÊÀi>ÊÊ
˜Õ“LiÀðÊ/…iÊÀ>˜}iʈÃÊÊ
>Ê«œÃˆÌˆÛiÊÀi>Ê˜Õ“LiÀð
'RAPH Y + X #OMPARE
THE GRAPH WITH THE GRAPH OF
Y X
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
/…iÊ}À>«…ÊœvÊ ÞÊÊÓÊ+ÊÎÝÊÊ
ˆÃÊ>ÊÛiÀ̈V>ÊÃÌÀiÌV…Ê>˜`Ê>ÊÊ
ÀiviV̈œ˜Êˆ˜Ê̅iÊ Ý‡>݈ÃʜvÊÊ
̅iÊ}À>«…ÊœvÊ ÞÊÊÎÝ°
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%80/.%.4)!, '2/74( -/$%,
Y A RT
A IS THE Ê ˆ˜ˆÌˆ>Ê>“œÕ˜ÌÊ R IS THE Ê }ÀœÜ̅ÊÀ>ÌiÊ R IS THE Ê }ÀœÜ̅Êv>V̜ÀÊ T IS THE Ê Ìˆ“iÊ«iÀˆœ`Ê %XAMPLE 3OLVE A COMPOUND INTEREST PROBLEM
)NVESTMENT 9OU PUT IN A SAVINGS ACCOUNT THAT EARNS
ANNUAL INTEREST COMPOUNDED YEARLY 9OU DO NOT
MAKE ANY DEPOSITS OR WITHDRAWALS (OW MUCH WILL YOUR
INVESTMENT BE WORTH IN YEARS
3OLUTION
4HE INITIAL AMOUNT IS Ê fÓxäÊ THE INTEREST RATE IS Ê {¯Ê OR Ê ä°ä{Ê AND THE TIME PERIOD IS Ê £äÊÞi>ÀÃÊ 7RITE EXPONENTIAL GROWTH
MODEL
Y A RT
Ê ÓxäÊ Ê ä°ä{Ê Ê £äÊ
3UBSTITUTE Ê ÓxäÊ FOR A
Ê ä°ä{Ê FOR R AND Ê £äÊ FOR T
Ê £°ä{Ê 3IMPLIFY
y Ê ÎÇä°äÈÊ
5SE A CALCULATOR
9OU WILL HAVE Ê fÎÇä°äÈÊ IN YEARS
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
)N %XAMPLE SUPPOSE THE ANNUAL INTEREST RATE IS
(OW MUCH WILL YOUR INVESTMENT BE WORTH IN
YEARS
Ê >LœÕÌÊf{äÇ°ÓÓ
(OMEWORK
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY