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:

𝑦 = 𝑎 ∙

𝑏 𝑥

Constant

Base

Exponent

Where a ≠ 0, b > o, b ≠ 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 = 3∙2 x ?

GRAPHING: X -2 -1 0 1 2 y = 3∙2 x 3∙2 (-2)

𝟑

=

𝟐 𝟐

3∙2 (-1)

𝟑

=

𝟐 𝟏

3∙2 (0) = 3∙1 3∙2 (1) = 3∙2 3∙2 (2) = 3∙4 y

𝟑 𝟒 𝟑 𝟐

3 6 12

GRAPHING: X -2 -1 0 1 2 y

𝟑 𝟒 𝟑 𝟐

3 6 12 This graph grows fast = Exponential Growth

YOU TRY IT:

Ex:

What is the graph of y = 3∙

𝟏 𝟐

x ?

GRAPHING: X -2 -1 0 1 2 y = 3∙ 3∙

𝟏 𝟐

x

𝟏 𝟐

(-2) =3∙(2) 2 3∙ 3∙

𝟏 𝟐 𝟏 𝟐

(-1) =3∙(2) 1 (0) = 3∙1 3∙ 3∙

𝟏 𝟐 𝟏 𝟐

(1) =3∙ (2) =3∙

𝟏 𝟒 𝟏 𝟐

12 6 y 3

𝟑 𝟐 𝟑 𝟒

GRAPHING: X -2 y 12 -1 6 0 1 2 3

𝟑 𝟐 𝟑 𝟒

This graph goes down = Exponential Decay

YOU TRY IT:

Ex:

What are the differences and similarities between: y = 3∙2 x and

𝟏

y = 3∙

𝟐

x ?

y = 3∙2 x



Base = 2

 

Exponential growth y- intercept (x=0) = 3

y = 3∙

  𝟏

x

𝟐

Base =

𝟏 𝟐 

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 ∙ b x

y

= total a = initial amount b = growth factor (1 + rate) x = time in years.

SOLUTION: Since 2005 , … has increased 7% each year. In 2005, about 36 billion was spend at restaurants…. about how much will be spent in 2015?

Y= total: unknown

y =

a

∙ b

x Initial: $36 billion

y =

36

∙ (1.07)

10 Growth: 1 + 0.07

y =

36

∙ (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 +

𝒓 𝒏

) 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 +

𝒓 𝒏

)

n t = ?

= $6000 = 0.07

A A =

6000

(1 +

𝟎.𝟎𝟕

)

1 (4)

𝟏

= 6000(1.07)

4 n = 1 t = 4 yrs

A = 6000(1.3107) A = $7864.77

A

P r COMPOUNDED QUARTERLY:

A =

P

(1 +

𝒓 𝒏

)

n t = ?

= $6000 = 0.07

A A =

6000

(1 +

𝟎.𝟎𝟕

)

4 (4)

𝟒

= 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 +

𝒓 𝒏

)

n t = ?

= $6000

A

= 0.07

A =

6000

(1 +

𝟎.𝟎𝟕 𝟏𝟐

) = 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 ∙ 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

∙ b

x

y =

12

∙ (1/2)

2

y =

12

∙ (.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.