'RAPH %XPONENTIAL $ECAY
&UNCTIONS
'OAL
9OUR .OTES
+ 'RAPH AND USE EXPONENTIAL DECAY FUNCTIONS
6/#!"5,!29
%XPONENTIAL DECAY FUNCTION Êv՘V̈œ˜ÊœvÊ̅iÊvœÀ“ÊÊ
ÞÊÊ>L ÝÊ܅iÀiÊ>ʀÊäÊ>˜`ÊäʐÊLʐʣ
$ECAY FACTOR ˜Ê>Êv՘V̈œ˜ÊœvÊ̅iÊvœÀ“Ê ÞÊÊ>L Ý]Ê̅iÊ
L>ÃiÊLʈÃÊ̅iÊ`iV>ÞÊv>V̜À°
0!2%.4 &5.#4)/. &/2 %80/.%.4)!, $%#!9
&5.#4)/.3
4HE FUNCTION Y B X WHERE B IS THE PARENT
FUNCTION FOR THE FAMILY OF EXPONENTIAL DECAY FUNCTIONS
WITH BASE B 4HE GENERAL SHAPE OF THE GRAPH OF Y B X
IS SHOWN BELOW
4HE DOMAIN OF Y B X IS Ê >ÊÀi>Ê˜Õ“LiÀÃÊ 4HE RANGE
IS Ê ÞʀÊäÊ %XAMPLE 'RAPH Y AB X FOR B 'RAPH THE FUNCTION Y |]z X
Î
Ó
0LOT Ê ÓÊ AND | z]
z Êz 4HEN
FROM RIGHT TO LEFT DRAW A CURVE THAT
BEGINS JUST Ê LiœÜÊ THE XAXIS
PASSES THROUGH THE TWO POINTS AND
MOVES Ê `œÜ˜Ê TO THE LEFT
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,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
'RAPH Y AB X H K FOR B %XAMPLE 'RAPH Y |]z
X
3TATE THE DOMAIN AND RANGE
3OLUTION
"EGIN BY SKETCHING THE GRAPH OF
Y |]
z WHICH PASSES THROUGH Ê ÓÊ X
AND | È
]Êz 4HEN TRANSLATE THE GRAPH
x
Ê Àˆ}…ÌÊ£Ê՘ˆÌÊ AND Ê Õ«Ê£Ê՘ˆÌÊ .OTICE
THAT THE GRAPH PASSES THROUGH Ê ÎÊ ££
AND | ]ÊÊz x
4HE GRAPHS ASYMPTOTE IS THE LINE Ê ÞÊÊ£Ê 4HE DOMAIN
IS Ê >ÊÀi>Ê˜Õ“LiÀÃÊ AND THE RANGE IS Ê ÞÊ€Ê£Ê #HECKPOINT 'RAPH THE FUNCTION 3TATE THE DOMAIN AND
RANGE
X
Y |]
z
Ê
Ê `œ“>ˆ˜\Ê>ÊÀi>Ê˜Õ“LiÀÃ]Ê
À>˜}i\Ê ÞʐÊä
z
Y |]
X
,ESSON s !LGEBRA .OTETAKING 'UIDE
`œ“>ˆ˜\Ê>ÊÀi>Ê˜Õ“LiÀÃ]Ê
À>˜}i\ÊÞʀÊÎ
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9OUR .OTES
3OLVE A MULTISTEP PROBLEM
%XAMPLE 4ELEVISIONS ! NEW TELEVISION COSTS 4HE VALUE OF
THE TELEVISION DECREASES BY EACH YEAR 7RITE AN
EXPONENTIAL DECAY MODEL GIVING THE TELEVISIONS VALUE Y
IN DOLLARS AFTER T YEARS %STIMATE THE VALUE AFTER YEARS
'RAPH THE MODEL 5SE THE GRAPH TO ESTIMATE WHEN THE
VALUE OF THE TELEVISION WILL BE 3OLUTION
A 4HE INITIAL AMOUNT IS A Ê £ÓääÊ AND THE PERCENT
DECREASE IS R Ê ä°Ó£Ê 3O THE MODEL IS
Y A RT
Ê £Óää­£ÊÊä°Ó£®ÌÊ
Ê £Óää­ä°Ç™®ÌÊ
7RITE EXPONENTIAL DECAY FORMULA
3UBSTITUTE FOR A AND R
3IMPLIFY
B 4HE GRAPH PASSES THROUGH
THE POINTS Ê £ÓääÊ AND Ê ™{nÊ )T HAS THE
Ê Ì‡>݈ÃÊ AS ITS ASYMPTOTE
0LOT A FEW OTHER POINTS
4HEN DRAW A SMOOTH
CURVE THROUGH THE POINTS
7HEN T THE TELEVISIONS VALUE IS
Y Ê fÇ{n°™ÓÊ C 5SING THE GRAPH YOU CAN ESTIMATE THAT THE VALUE OF THE
TELEVISION WILL BE AFTER ABOUT Ê ÈÊ YEARS
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
(OMEWORK
2EWORK %XAMPLE WITH A DECREASE EACH YEAR
ÞÊÊ£Óää­ä°nn®ÌÆÊ>LœÕÌÊ££ÊÞi>ÀÃ
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,ESSON s !LGEBRA .OTETAKING 'UIDE