7.7 EXPONENTIAL GROWTH AND DECAY: Exponential Decay: An equation that decreases. Exponential Growth:...

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Transcript of 7.7 EXPONENTIAL GROWTH AND DECAY: Exponential Decay: An equation that decreases. Exponential Growth:...
7.7 EXPONENTIAL GROWTH AND DECAY:
Exponential Decay: An equation that decreases.
Exponential Growth: An equation that increases.
Growth Factor: 1 plus the percent rate of change which is expressed as a decimal.
Decay Factor: 1 minus the percent rate of change expressed as a decimal.
GOAL:
Definition:An EXPONENTIAL FUNCTION is a function
of the form:
𝑦=𝑎 ∙𝑏𝑥
BaseExponentWhere a ≠ 0, b > o, b ≠ 1,
and x is a real number.
Constant
GRAPHING: To provide the graph of the equation we can go back to basics and create a table.
Ex: What is the graph of y = 3 2∙ x?
GRAPHING:X y = 3 2∙ x y
2 3 2∙ (2) = 𝟑𝟒
3 2∙ (1) = 𝟑𝟐1
0 3 2∙ (0) 3 = 3 1 ∙
1 3 2∙ (1) 6 = 3 2 ∙
2 3 2∙ (2) 12 = 3 4 ∙
GRAPHING: X y
2 𝟑𝟒𝟑𝟐1
0 3
1 6
2 12
This graph grows fast = Exponential Growth
YOU TRY IT:
Ex: What is the graph of y = 3∙x?
GRAPHING:X y = 3∙x y
2 3 ∙ (2) 12
61
0 3 = 3 1 ∙
1
2
=3 (2)∙ 2
3 ∙ (1) =3 (2)∙ 1
3 ∙ (0)
3 ∙ (1) =3 ∙𝟑𝟐𝟑𝟒
3 ∙ (2) =3 ∙
GRAPHING: X y
2
𝟑𝟒
𝟑𝟐
1
0 3
1
6
2
12
This graph goes down = Exponential Decay
YOU TRY IT:
Ex: What are the differences and
similarities between:
y = 3 2∙ x
and y = 3∙x?
y = 3 2∙ x
Base = 2 Exponential growth y intercept (x=0) = 3
y = 3∙x
Base = Exponential Decay y intercept (x=0) = 3
MODELING: We use the concept of exponential growth in the real world:Ex: Since 2005, the amount of money spent at restaurants in the U.S. has increased 7% each year. In 2005, about 36 billion was spend at restaurants. If the trend continues, about how much will be spent in 2015?
EVALUATING: To provide the solution we must know the following formula:
y = a∙bx
y = totala = initial amount b = growth factor (1 + rate)x = time in years.
SOLUTION:
Y= total:
Since 2005, … has increased 7% each year. In 2005, about 36 billion was spend at restaurants…. about how much will be spent in 2015?
$36 billion Initial:
Growth: 1 + 0.07
Time (x): 10 years(20052015)
unknown y = a∙bx
y = 36∙(1.07)10
y = 36∙(1.967)
y = 70.8 b.
BANKING: We also use the concept of exponential growth in banking:
A = P(1+)nt
A = total balanceP = Principal (initial) amount
n = # of times compound interestt = time in years.
r = interest rate in decimal form
MODELING GROWTH:Ex:
You are given $6,000 at the beginning of your freshman year. You go to a bank
and they offer you 7% interest. How much money will you have after
graduation if the money is:a) Compounded annuallyb) Compounded quarterlyc) Compounded monthly
COMPOUNDED ANNUALLY:
A = P(1+)nt
A = ?P = $6000
n = 1t = 4 yrs
r = 0.07
A = 6000(1+)1(4)
A = 6000(1.07)4
A = 6000(1.3107)A = $7864.77
COMPOUNDED QUARTERLY:
A = P(1+)nt
A = ?P = $6000
n = 4 timest = 4 yrs
r = 0.07
A = 6000(1+)4(4)
A = 6000(1.0175)16
A = 6000(1.3199)A = $7919.58
COMPOUNDED MONTHLY:
A = P(1+)nt
A = ?P = $6000
n = 12 timest = 4 yrs
r = 0.07
A = 6000(1+)12(4)
A = 6000(1.0058)48
A = 6000(1.3221)A = $7932.32
MODELING DECAY:Ex:
Doctors can use radioactive iodine to treat some forms of cancer. The halflife
of iodine131 is 8 days. A patient receives a treatment of 12 millicuries (a unit of radioactivity) of iodine131. How much iodine131 remains in the patient
after 16 days?:
DECAY: To provide the solution we g back to the following formula:
y = a∙bx
y = totala = initial amount b = decay factor (1  rate)x = time in years.
SOLUTION:
Y= total:
The halflife of iodine131 is 8 days. A patient receives a treatment of 12 millicuries (a unit of radioactivity) of iodine131. How much iodine131 remains in the patient 16 days later?:
12Initial:
Growth: 1 1/2
Time (x): 16/8 = 2
unknown y = a∙bx
y = 12∙(1/2)2
y = 12∙(.25)
y = 3
VIDEOS: ExponentialFunctions
Growthhttps://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/exponentialgrowthfunctionsGraphing
https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/graphingexponentialfunctions
VIDEOS:ExponentialFunctions
Decay
https://www.khanacademy.org/math/trigonometry/exponential_and_logarithmic_func/exp_growth_decay/v/wordproblemsolvingexponentialgrowthanddecay
CLASSWORK:
Page 450452:
Problems: As many as needed to master the concept.