Aim: Exponential Function Course: Math Literacy Aim: How does the exponential model fit into our lives? Do Now: The price of an item increased by 18% due

Aim: Exponential Function Course: Math Literacy

Aim: How does the exponential model fit into our lives?

Do Now:

The price of an item increased by 18% due to high demand. The amount of increase was $95. What was the original price of the item?


Zero Power Property

Properties of Exponents

Product of Powers Property

a0 = 1

Power of Power Property

Power of Product Property

Negative Power Property

Quotients of Powers Property

Power of Quotient Property

am • an = am+n

(am)n = am•n

a-n = 1/an, a 0

(ab)m = ambm

am

an am n , a 0

(a

b)m

am

bm , b 0


Types of exponents

Positive Integer Exponent an = a • a • a • • • • a

n factors

Zero Exponent a0 = 1

Rational Exponent 1/n

a1 n an

Rational Exponent m/n

am n ( an )m

Negative Exponent - m/n

a m n 1

am n

If a 0, am n ( an )m

results in a nonreal number

ex . ( 2)3 2 imaginary number

beware! -23/2 is not the same as (-2)3/2


Exponential Function

y = a • bx

. . have variables as exponents, andwhere a 0, base b > 0, and also b 1.

The x-axis is a horizontal asymptote: ax 0 as x -

The domain (x) is the set of real numbers: (-, )The range (y) is the set of positive real numbers: (0, )

If b > 1, the graph is increasing and continuous

As b increases in value from 1, the slope ofthe graph gets steeper

y = 1 • 2x

y = 2x

When a = 1, graph always goes through (0,1)

(0,1)

What happens as b increases in value?


The b Affect

y = 2xy = (1/2)x

(0,1)

the graph is decreasing - Decay

If b is a positive number other than 1, the graphs of y = bx and y = (1/b)x

are reflections through the y-axis of each other

If b > 1, the graph is increasing - Growth

What if b = 1? horizontal line: y = 1

y = 1

y = a • bx

If 0 < b < 1


The b Affect: b < 0

4

2

-2

-4

-5 5

f x = 2x y = a • bx

Let b = (-2)?

What if b < 0?

if b < 0, no longer the exponential function

y = 1 • (-2)x

graph: table: x = 1table: x = .1


The a Affect

4

2

-2

-4

-5 5

f x = 2x

4

2

-2

-4

-5 5

f x = -2x

Graph f(x) = -(1/2)x = -1 • (1/2)x

Graph f(x) = -6x = -1 • (6)x

4

2

-2

-4

-5 5

g x = -1

2

x

4

2

-2

-4

-5 5g x = -6x (0,-1)

y = a • bx

Graph f(x) = -2x = -1 • 2x


The a Affect

20

18

16

14

12

10

8

6

4

2

-2

2 4

h x = 2x

g x = 22x

f x = 42x

(0,4)

a = 4

(0,2)

a = 2

y = a • bx

(0,1)

a = 1


The a Affect

Graph the exponential functions

11,

5 552

1,6

xx x

yy y

10

5

-5

-10

-10 10

h x = 1

5

x10

5

-5

-10

-10 10

h x = 61

5

x

10

5

-5

-10

-10 10

h x = -21

5

x


Model Problem

The exponential function f(x) = 13.49(0.967)x – 1 describes the number of 0-rings expected to fail, f(x), when the temperature is xoF. On the morning the Challenger was launched, the temperature was 31oF, colder than any previous experience. Find the number of 0-rings expected to fail at this temperature.

( ) 13.39(0.967) 1xf x

xo = 3131(31) 13.39(0.967) 1 3.8 4f

( ) number of 0-rings expected to failf x


Model Problem

Horses were born with eight deformed legs, pigs with no eyes, and eggs contained several yolks. This was part of the grotesque aftermath of the 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union. Nearby cities were abandoned and 335,000 people were permanently displaced from their homes. The explosion sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function f(x) = 1000(0.5)x/30 describes the amount of cesium-137, f(x), in kilograms, remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Will people be able to live in the area 80 years after the accident?


Model Problem

Horses were born with eight deformed legs, pigs with no eyes, and eggs contained several yolks. This was part of the grotesque aftermath of the 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union. Nearby cities were abandoned and 335,000 people were permanently displaced from their homes. The explosion sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function f(x) = 1000(0.5)x/30 describes the amount of cesium-137, f(x), in kilograms, remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyl’s atmosphere, the area is considered unsafe for human habitation. Will people be able to live in the area 80 years after the accident?

f(x) = 1000(0.5)x/30

f(x) = amount of cesium remaining; x = 80 yrs

f(80) = 1000(0.5)80/30

f(80) 157 kilogramshabitation

not possible

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Aim: Exponential Function Course: Math Literacy Aim: How does the exponential model fit into our lives? Do Now: The price of an item increased by 18% due