8.1 Exploring Exponential Models

What is an exponential equation?

An exponential equation has the general form

y=abx

1b and 0b ,0 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)2.1(10y

x)9(.5y x)54.1(50y

x)70(.2.5y xy )2(4

x)07(.100y

Growth

Decay

Growth

Decay

Growth

Decay

Graphs of Exponential Functions

x)2(10y

What is an asymptote?

x)2(10y “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)5(.100y X .5x Y=100(.5)x

-3

-2

-1

0

1

2

3

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

Let’s graph one together

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

-3 .5-3 800

-2 .5-2 400

-1 .5-1 200

0 .50 100

1 .51 50

2 .52 25

3 .53 12.5

Let’s try one

xy )5(.2X .5x Y=2(.5)x

-3

-2

-1

0

1

2

3

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

Let’s try one

xy )5(.2X .5x Y=2(.5)x

-3

-2

-1

0

1

2

3

Let’s try one

xy )10(5X 10x Y=5(10)x

-3

-2

-1

0

1

2

3

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

Let’s try one

xy )10(5X 10x Y=5(10)x

-3

-2

-1

0

1

2

3

Writing Exponential Equations Find the exponential equation passing through

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

ab

3

20

13

205 b

b

31205 b

320 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 ab

3

2

Division property of exponents

Writing Exponential Equations Find the exponential equation passing through

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

2

45

20

205

205

2

2

2

b

b

b

b

Simplify

Solve for b

2

5

8

20

2

202033

b

aGo 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).

xaby

xy )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 xy )2(

2

5

Let’s Try One Find the exponential equation passing through

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

ab

2

4

32

416 b

b

23416 b

24 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 ab

2

4

Division property of exponents

Writing Exponential Equations

1416 b4b

Simplify

Solve for b

25.04

1

4

42

aGo back to the equation for a; substitute in b and solve for a

xaby

xy )4(25.0 Going back to the general form, substitute in a and b

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

xy )4(25.0

Putting it all together . . .

Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.

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?

xy )03.1( 1000

r = rate 3% (write as 0.03)

b = 1 + r = 1.03

x = time

a = amount put into the account ($1,000)

xaby

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 )

xy )2( 1000

xaby

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)

xy )9.0( 15000

xaby