x y 4 ( 2 ) Growth x y 100 (. 07 ) Decay G RAPHS OF - - PowerPoint PPT Presentation

DAY 109 - EXPLORING EXPONENTIAL MODELS

WHAT IS AN EXPONENTIAL EQUATION?

An exponential equation has the general form

y = abx

1 b and b , where    a

GROWTH FACTOR, DECAY FACTOR

Given the general form

y = abx

 When b > 1, b is the

growth factor

 When 0 < b < 1, b is the

decay factor

GROWTH OR DECAY???

x

y ) 2 . 1 ( 10 

x

y ) 9 (. 5 

x

y ) 54 . 1 ( 50 

x

y ) 70 (. 2 . 5 

x

y ) 2 ( 4 

x

y ) 07 (. 100 

Growth Decay Growth Decay Growth Decay

GRAPHS OF EXPONENTIAL FUNCTIONS

x

y ) 2 ( 10 

WHAT IS AN ASYMPTOTE?

x

y ) 2 ( 10 

“Walking halfway to the wall”

An Asymptote is a line that a graph approaches as x or y increases in absolute value. In this example, the asymptote is the x axis.

GRAPHING EXPONENTIAL FUNCTIONS

X .5x Y=100(.5)x

• 3
• 2
• 1

1 2 3

x

y ) 5 (. 100 

Complete the table using the integers -3 through 3 for x.

LET’S GRAPH ONE TOGETHER

X .5x Y=100(.5)x

• 3

.5-3 800

• 2

.5-2 400

• 1

.5-1 200 .50 100 1 .51 50 2 .52 25 3 .53 12.5

x

y ) 5 (. 100 

LET’S TRY ONE

x

y ) 5 (. 2 

Complete the table using the integers -3 through 3 for x. Then graph the function.

X

.5x y=2(.5)x

• 3

.5-3 16

• 2

.5-2 8

• 1

.5-1 4 .50 2 1 .51 1 2 .52 0.5 3 .53 0.25

LET’S TRY ONE

X

.5x y=2(.5)x

• 3

.5-3 16

• 2

.5-2 8

• 1

.5-1 4 .50 2 1 .51 1 2 .52 0.5 3 .53 0.25

x

y ) 5 (. 2 

LET’S TRY ONE

x

y ) 10 ( 5 

Complete the table using the integers -3 through 3 for x. Then graph the function.

X 10x y=5(10)x

• 3
• 2
• 1

1 2 3

LET’S TRY ONE

X 10x y=5(10)x

• 3

10-3 0.005

• 2

10-2 0.05

• 1

10-1 0.5 100 5 1 101 50 2 102 500 3 103 5000

x

y ) 10 ( 5 

WRITING EXPONENTIAL EQUATIONS

Find the exponential equation passing through the points (3,20) and (1,5).

a b 

3

2

x

ab y 

a b 

3

20

1 3

20 5 b b 

3 1

20 5



 b

3

20 ab 

Start with the general form. Choose a point. Substitute for x and y using (3, 20) Solve for a Substitute x and y using (1, 5) and a using Division property of exponents

WRITING EXPONENTIAL EQUATIONS

 Find the exponential equation passing through

the points (3,20) and (1,5).

2 5 8 20 2 20 20

3 3

    b a

2 4 5 20 20 5 20 5

2 2 2

    



b b b b

Simplify Solve for b Go back to the equation for a; substitute in b and solve for a

WRITING EXPONENTIAL EQUATIONS

 Find the exponential equation passing through

the points (3,20) and (1,5).

x

y ) 2 ( 2 5 

x

ab y 

x

y ) 2 ( 2 5 

Going back to the general form, substitute in a and b

The exponential equation passing through the points (3,20) and (1,5) is

LET’S TRY ONE

 Find the exponential equation passing through

the points (2,4) and (3,16).

a b 

2

4

x

ab y 

a b 

2

4

3 2

4 16 b b 

2 3

4 16



 b

2

4 ab 

Start with the general form. Choose a point. Substitute for x and y using (2, 4) Solve for a Substitute x and y using (3, 16) and a using Division property of exponents

WRITING EXPONENTIAL EQUATIONS

25 . 4 1 4 4

2

   a

1

4 16 b 

4  b

Simplify Solve for b Go back to the equation for a; substitute in b and solve for a

x

ab y 

x

y ) 4 ( 25 . 

Going back to the general form, substitute in a and b

The exponential equation passing through the points (2,4) and (3,16) is

x

y ) 4 ( 25 . 

PUTTING IT ALL TOGETHER . . .

 Find the equation of the exponential function

that goes through (1,6) and (0,2). Graph the function.

3 2 6 6 2 ) ( 6 2 ) 2 , ( 6 6 ) 6 , 1 (

1 1 1 1

           b b b b ab y a b ab ab y

x x

x x

y ab y a ) 3 ( 2 2 3 6    

x

y ) 3 ( 2 

x 3x y=2(3)x

• 3

3-3 0.074

• 2

3-2 0.22

• 1

3-1 0.66 30 2

1

31 6

2

32 18

3

33 54

MODELING GROWTH WITH AN EXPONENTIAL EQUATION

 The growth factor can be found in word problems

using b = 1 + r where r = rate or amount of increase. You can substitute your new b into your general equation to find the exponential function.

EX- a guy puts $1000 into a simple 3%

interest account. What is the exponential equation?

x

y ) 03 . 1 ( 1000 

x

ab y 

r = rate 3% (write as 0.03) b = 1 + r = 1.03 x = time a = amount put into the account ($1,000)

 EX – a colony of 1000 bacteria cells doubles every

• hour. What is the exponential equation?

r = 1 (why not 2?) b = r + 1 = 2 x = time (in hours) a = the original number in the colony (1,000 bacteria )

x

y ) 2 ( 1000 

x

ab y 

b = r + 1, where r is the amount of

• increase. We are increasing by 100%

each time something doubles, so r = 1

 EX- a $15000 car depreciates at 10% a year.

What is the exponential equation? r = - 10% (the car is worth 10% less each year) b = 1 - r = 1 – 0.1 = 0.9 x = time (in years) a = amount put into the account ($15,000)

x

y ) 9 . ( 15000 

x

ab y 