growthdecay. 8.2 exponential decay goal 1: i will graph exponential decay functions. goal 2: i will...
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8.2 Exponential Decay
Goal 1: I will graph exponential decay functions.
Goal 2: I will use exponential decay functions to model real-life examples.
Vocabularyexponential decay function:involves the equation
where b is the base, and a 00 b 1and a is the initial value of y when x = 0, and b is that growth factor.
y = abx
Exponential Factors
• If the factor b is greater than 1, then we call the relationship exponential growth.
• If the factor b is less than 1, we call the relationship exponential decay.
practice
Goal 1: Graph exponential decay functions
State whether the function is an exponential growth or exponential decay function.
a. f (x) 52
3
x
b. f (x) 83
2
x
growthdecay
c. f (x) 10 3 x
101
3
x
decayd. f (x) 4
3
8
x
decaye. f (x) 3
4
3
x
growth
Graph the function
x f (x) 3g
1
4
x y
1f ( 1) 3g
1
4
1
120 f (0) 3g
1
4
0
31 f (1) 3g
1
4
1 3
4
2 f (2) 3g
1
4
2 3
16
Exponential Decay ModelsWhen a real-life quantity (e.g. value of car) decreases byfixed percent each time period, the amount y of the quantity after t time periods is:
y C r t ( )1
Where
C = initial amount
r = growth rate (percent written as a decimal)
t = time where t 0
(1 - r) = decay factor where 1 - r < 1
Value of CarYou buy a new car for $23,000. The value decreases by 15% each year. Write an exponential decay model for the car’s value. Use the model to estimate the value after 3 years.
a 23,000 r 15% 0.15
y 23000(1 0.15)t
y 23000(0.85)t
f (3) 23000(0.85)3 $14,125
y C r t ( )1
practice
An adult takes 400 mg of ibuprofen. Each hour h, the amount i of ibuprofen in the person’s system decreases by about 29%.
Write an exponential decay model that describes the situation.
How much is left in the system after 4 hours?
f (4) 101.6 mg
y C(1 r)t
y 400(1 0.29)t
y 400(0.71)t