7RITE AND 'RAPH %XPONENTIAL
$ECAY &UNCTIONS
'OAL
9OUR .OTES
+ 7RITE AND GRAPH EXPONENTIAL DECAY FUNCTIONS
6/#!"5,!29
%XPONENTIAL DECAY ʵÕ>˜ÌˆÌÞÊ̅>ÌÊ`iVÀi>ÃiÃÊLÞÊ̅iÊ
Ã>“iÊ«iÀVi˜ÌʜÛiÀÊiµÕ>Ê̈“iÊ«iÀˆœ`Ã
'RAPH AN EXPONENTIAL FUNCTION
%XAMPLE X
'RAPH THE FUNCTION Y |]z AND IDENTIFY ITS DOMAIN AND
RANGE
3OLUTION
3TEP -AKE A TABLE OF VALUES
4HE DOMAIN IS Ê >ÊÀi>ÊÊ
˜Õ“LiÀÃÊ X
Y
ʙÊ
ÊÎÊ
£
Ê£Ê Ê ]
Ê£Êz Ê Ê]
Êz
Î
3TEP 0LOT THE POINTS
™
3TEP $RAW A SMOOTH CURVE THROUGH THE POINTS &ROM
EITHER THE TABLE OR THE GRAPH YOU CAN SEE THAT THE
RANGE IS Ê >Ê«œÃˆÌˆÛiÊÀi>Ê˜Õ“LiÀÃÊ #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
#OMPARE GRAPHS OF EXPONENTIAL FUNCTIONS
%XAMPLE X
'RAPH Y + |]z #OMPARE THE GRAPH WITH THE GRAPH
X
OF Y |]z 3OLUTION
X
Y |]z
X
X
Y + |]z
Ê ™Ê
Ê £nÊ
Ê ÎÊ
Ê ÈÊ
Ê £Ê
Ê ÓÊ
£
Ê]
Êz
Î
Ó
Ê]
Êz
Î
Ê Êz
]
£
™
Ó
Ê]
Êz
™
X
"ECAUSE THE YVALUES FOR Y + |]
z ARE Ê ÓÊ̈“iÃÊ
X
THE CORRESPONDING YVALUES FOR Y |]
z THE GRAPH OF
X
Y + |]
z IS A Ê ÛiÀ̈V>ÊÃÌÀiÌV…Ê OF THE GRAPH OF
X
Y |]
z #HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
X
'RAPH Y + |]
z #OMPARE THE GRAPH WITH THE
X
GRAPH OF |]
z Ý
Ê /…iÊ}À>«…ÊœvÊÞÊÊÓÊ+ÊÊ|]
Ê£Êz Ê Ê
Î
Ê
Ê ˆÃÊ>ÊÛiÀ̈V>ÊÃÌÀiÌV…Ê>˜`Ê>Ê
Ê
Ê ÀiviV̈œ˜Êˆ˜Ê̅iÊ݇>݈ÃʜvÊ
Ê
Ê Ì…iÊ}À>«…ÊœvÊÞÊÊÊ|]
Ê£ÊzÊ °
,ESSON s !LGEBRA .OTETAKING 'UIDE
Ý
Î
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE #LASSIFY AND WRITE RULES FOR FUNCTIONS
4ELL WHETHER THE GRAPH REPRESENTS EXPONENTIAL GROWTH OR
EXPONENTIAL DECAY 4HEN WRITE A RULE FOR THE FUNCTION
3OLUTION
4HE GRAPH REPRESENTS
Ê iÝ«œ˜i˜Ìˆ>Ê`iV>ÞÊ
Y AB X WHERE B 4HE YINTERCEPT IS Ê xÊ SO
A Ê xÊ &IND THE VALUE
OF B BY USING THE POINT
AND A Ê xÊ Y AB X
7RITE FUNCTION
Ê £Ê Ê xÊ + B Ê £Ê
Ê ä°ÓÊ B
3UBSTITUTE
3OLVE
! FUNCTION RULE IS Ê ÞÊÊx­ä°Ó®ÝÊ %80/.%.4)!, '2/74( !.$ $%#!9
%XPONENTIAL 'ROWTH
%XPONENTIAL $ECAY
Y AB X A AND B Y AB X A AND B %80/.%.4)!, $%#!9 -/$%,
Y A RT
A IS THE Ê ˆ˜ˆÌˆ>Ê>“œÕ˜ÌÊ R IS THE Ê `iV>ÞÊÀ>ÌiÊ R IS THE Ê `iV>ÞÊv>V̜ÀÊ T IS THE Ê Ìˆ“iÊ«iÀˆœ`Ê #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE 5SE THE EXPONENTIAL DECAY MODEL
0OPULATION 4HE POPULATION OF A CITY DECREASED FROM TO BY ANNUALLY )N THERE WERE ABOUT
PEOPLE LIVING IN THE CITY 7RITE A FUNCTION THAT
MODELS THE CITYgS POPULATION SINCE 4HEN FIND THE
POPULATION IN 3OLUTION
,ET 0 BE THE POPULATION OF THE CITY IN THOUSANDS AND
LET T BE THE TIME IN YEARS SINCE 4HE INITIAL VALUE
IS Ê ÎxÇÊ AND THE DECAY RATE IS Ê ä°ä£xÊ 0 A RT
7RITE EXPONENTIAL DECAY
MODEL
Ê ÎxÇÊ Ê ä°ä£xÊ T
3UBSTITUTE Ê ÎxÇÊ FOR A
AND Ê ä°ä£xÊ FOR R
Ê Îxǭ䰙nx®ÌÊ
3IMPLIFY
4O FIND THE POPULATION IN Ê nÊ YEARS AFTER SUBSTITUTE Ê nÊ FOR T
0 Ê Îxǭ䰙nx®nÊ
3UBSTITUTE Ê nÊ FOR T
y Ê Î£È°ÎÊ
5SE A CALCULATOR
4HE CITYgS POPULATION WAS ABOUT Ê Î£È]ÎääÊ IN #HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
4HE GRAPH OF AN EXPONENTIAL FUNCTION PASSES THROUGH
THE POINTS AND 'RAPH THE FUNCTION 4ELL WHETHER
THE GRAPH REPRESENTS EXPONENTIAL
GROWTH OR EXPONENTIAL DECAY
4HEN WRITE A RULE FOR THE FUNCTION
Ê
(OMEWORK
Ê Ý«œ˜i˜Ìˆ>Ê}ÀœÜ̅ÆÊ
ÞÊÊ{­Ó°x®Ý
)N %XAMPLE SUPPOSE THAT THE DECAY RATE OF THE
CITYgS POPULATION REMAINS THE SAME BEYOND 7HAT WILL BE THE POPULATION IN Ê
Ê >LœÕÌÊÓ{{]Çää
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY