7-6 & 7-7 Exponential Functions

Evaluate and graph exponential functions

Exponential function

A function in the form of y =

Examples:

Exponential Growth, modeled by the following y = a

𝑦=𝑎 ∙𝑏𝑥

Initial amount

The base & when b>1, called the Growth factor(1 + the percent rate written as a decimal)

exponent

(this is when x = 0)

Exponential Decay

𝑦=𝑎 ∙𝑏𝑥

Initial amount

The base is the decay factor (1 – percent rate written as a decimal)

exponent

(this is when x = 0)

What is the graph of y = ?

-5 -4 -3 -2 -1 0 1 2 3 4 5-2

-1

0

1

2

3

4

5

6

7

8

9

10x y = (x, y)

-2

-1

0

1

2

y =

y =

y =

y =

y =

(-2, )

(-1, 1 )

(0, 3)

(1, 6)

cc

(2, 12)

Does the table or rule represent a linear or an exponential function?

ANSWER: EXPONENTIAL FUNCTION.

B. y = 3x

A.

ANSWER: LINEAR FUNCTION.

Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. The function f(x) = gives the population after x weeks. How many beetles will there be after 56 days?

f(x) =

=

=

= Answer: after 56 days, there will be 7,680 beetles.

What does x represent?

Evaluate the function for the given value.

for x = 3

𝑦=3 ∙4 ()

❑3

𝑦=3 ∙64❑

𝑦=192

Since 2005, the amount of money spent at restaurants in the US has increased about 7% each year. In 2005, about $360 billion was spent at restaurants. If the trend continues, about how much will be spent at restaurants in 2015?

𝑦=𝑎 ∙𝑏𝑥

Let y = The annual amount spent in restaurants (in billions of dollars)

Let a = The initial amount:

Let b = The growth factor:

Let x =

360

(1 + %) or 1 + .07 = 1.07

The number of years since 2005: 10

𝑦=360 ∙1.0710

𝑦=708.174488𝑦=$708billion

Compound interest: When a bank pays interest on both the principal and the interest an account has earned. (it uses the following formula)

A=P(1+ rn)n t

A = The balance

P = the principal (the initial deposit)

r = the annual interest rate----convert from % to a decimal—(move 2 places to the left)

n = the number of times interest is compounded per year

t= the time in years

A=P(1+ rn)n t

Find the balance in the account after the given period:

$12,000 principal earning 4.8% compounded annually, after 7 years

P =r =n =t =

12,000.04817

A=12 ,000(1+ .0481

)1(7)

A = $16,661.35

A=P(1+ rn)n t

Find the balance in the account after the given period:

$20,000 principal earning 3.5% compounded monthly, after 10 years

P =r =n =t =

20,000.0351210

A=20 ,000 (1+ .03512

)12(10)

A = $28,366.90

The kilopascal is unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.5%. What is the approximate pressure at an altitude of 3000 m?

𝑦=𝑎 ∙𝑏𝑥

Let y = The atmospheric pressure (in kilopascals)Let a = The initial amount:

Let b = The decay factor:

Let x =

101

(1 - %) or 1 - .115 = .885

The altitude (in thousands of meters) 3

𝑦=101 ∙.8853

𝑦=70.0085𝑦=70 kilopascals

Pg 457: 9-21 odd & 20pg 464: 9-21 odd (skip 13)