Transcript 7_7 Exponential Growth - Decay
7.7 EXPONENTIAL GROWTH AND DECAY: Exponential Growth: An equation that increases.
Exponential Decay: An equation that decreases.
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:
π¦ = π β
π π₯
Constant
Base
Exponent
Where a β 0, b > o, b β 1, and x is a real number.
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 = 3β2 x ?
GRAPHING: X -2 -1 0 1 2 y = 3β2 x 3β2 (-2)
π
=
π π
3β2 (-1)
π
=
π π
3β2 (0) = 3β1 3β2 (1) = 3β2 3β2 (2) = 3β4 y
π π π π
3 6 12
GRAPHING: X -2 -1 0 1 2 y
π π π π
3 6 12 This graph grows fast = Exponential Growth
YOU TRY IT:
Ex:
What is the graph of y = 3β
π π
x ?
GRAPHING: X -2 -1 0 1 2 y = 3β 3β
π π
x
π π
(-2) =3β(2) 2 3β 3β
π π π π
(-1) =3β(2) 1 (0) = 3β1 3β 3β
π π π π
(1) =3β (2) =3β
π π π π
12 6 y 3
π π π π
GRAPHING: X -2 y 12 -1 6 0 1 2 3
π π π π
This graph goes down = Exponential Decay
YOU TRY IT:
Ex:
What are the differences and similarities between: y = 3β2 x and
π
y = 3β
π
x ?
y = 3β2 x
ο
Base = 2
ο ο
Exponential growth y- intercept (x=0) = 3
y = 3β
ο ο π
x
π
Base =
π π ο
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 β b x
y
= total a = initial amount b = growth factor (1 + rate) x = time in years.
SOLUTION: Since 2005 , β¦ has increased 7% each year. In 2005, about 36 billion was spend at restaurantsβ¦. about how much will be spent in 2015?
Y= total: unknown
y =
a
β b
x Initial: $36 billion
y =
36
β (1.07)
10 Growth: 1 + 0.07
y =
36
β (1.967)
Time (x): 10 years (2005-2015)
y = 70.8 b.
BANKING: We also use the concept of exponential growth in banking: A = P (1 +
π π
) n t
A
= total balance P = Principal (initial) amount r = interest rate in decimal form n = # of times compound interest t = time in years.
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 annually b) Compounded quarterly c) Compounded monthly
A
P r COMPOUNDED ANNUALLY:
A =
P
(1 +
π π
)
n t = ?
= $6000 = 0.07
A A =
6000
(1 +
π.ππ
)
1 (4)
π
= 6000(1.07)
4 n = 1 t = 4 yrs
A = 6000(1.3107) A = $7864.77
A
P r COMPOUNDED QUARTERLY:
A =
P
(1 +
π π
)
n t = ?
= $6000 = 0.07
A A =
6000
(1 +
π.ππ
)
4 (4)
π
= 6000(1.0175)
16 n = 4 times t = 4 yrs
A = 6000(1.3199) A = $7919.58
COMPOUNDED MONTHLY: r
A
P
A =
P
(1 +
π π
)
n t = ?
= $6000
A
= 0.07
A =
6000
(1 +
π.ππ ππ
) = 6000(1.0058)
12 (4) 48 t n = 12 times
A
= 4 yrs
A = 6000(1.3221) = $7932.32
MODELING DECAY:
Ex:
Doctors can use radioactive iodine to treat some forms of cancer. The half-life of iodine-131 is 8 days. A patient receives a treatment of 12 millicuries (a unit of radioactivity) of iodine-131. How much iodine-131 remains in the patient after 16 days?:
DECAY: To provide the solution we g back to the following formula: y = a β b x
y
= total a = initial amount b = decay factor (1 - rate) x = time in years.
SOLUTION: The half-life of iodine-131 is 8 days . A patient receives a treatment of 12 millicuries (a unit of radioactivity) of iodine-131. How much iodine-131 remains in the patient 16 days later?: Y= total: Initial: unknown 12 Growth: 1- 1/2 Time (x): 16/8 = 2
y =
a
β b
x
y =
12
β (1/2)
2
y =
12
β (.25)
y = 3
VIDEOS:
Exponential Functions Growth
https://www.khanacademy.org/math/trigonometry/expon ential_and_logarithmic_func/exp_growth_decay/v/expone ntial-growth-functions
Graphing
https://www.khanacademy.org/math/trigonometry/expon ential_and_logarithmic_func/exp_growth_decay/v/graphi ng-exponential-functions
VIDEOS:
Exponential Functions Decay
https://www.khanacademy.org/math/trigonometry/expon ential_and_logarithmic_func/exp_growth_decay/v/word problem-solving--exponential-growth-and-decay
CLASSWORK:
Page 450-452: Problems: As many as needed to master the concept.