exponential growth and decay

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SECTION 6.3 Exponential Growth and Decay

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Exponential Growth and Decay. Section 6.3. Warm-up 1. -8 -6 -4 -2 0 2 4 6 8. As the x-values increase by 1, the y-values increase by 2. Warm-up 2. 7 0 -5 -8 -9 -8 -5 0 7. When x1, as the x-values increase, the - PowerPoint PPT Presentation

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Page 1: Exponential Growth and Decay

SECTION 6 .3

Exponential Growth and Decay

Page 2: Exponential Growth and Decay

Warm-up 1

-8-6-4-2024

68

As the x-values increase by 1, the y-values increase by 2.

Page 3: Exponential Growth and Decay

Warm-up 2

70-5-8-9

-8-507

When x<1, as the x-values increase, the y-values decrease.

When x>1, as the x-values increase, the y-values increase.

Page 4: Exponential Growth and Decay

The Great Divide – 10 minutes to complete

Follow up Questions (With your partner be prepared to answer the

following questions about this activity)Do the graphs represent a constant rate of change?Do the graphs appear quadratic?How do the tables verify your observations?What is the y-intercept for each graph?Where do you find the y-intercept recorded on the

table? (in each problem)What is the ratio between any two successive given y-

values? (in each problem)Is the successive quotient for each pair of y-values

constant?(in each problem)

Page 5: Exponential Growth and Decay

The Sky Is Falling

The data given in the table below is a result of placing a mat on a flat surface and dropping beans onto the mat.

For each bean that landed in a shaded area of the mat, you added one additional bean to your total number of beans.

Shaded Mat

Number of Trials, x 0 1 2 3 4 5 6 7

Number of Beans, y 10 14 20 30 45 67 92 123

Bean Drop Data

Page 6: Exponential Growth and Decay

The Sky is Falling

Create a scatterplot of the data and record your window. Use the Sky Is Falling tables you have been given to write a linear and an exponential model for your data. Round answers to the nearest thousandths.

Number of Trials, x 0 1 2 3 4 5 6 7

Number of Beans, y 10 14 20 30 45 67 92 123

Bean Drop Data

Window for this graph:x-min: 0 x-max: 10 x-scale: 1y-min: 0 y-max: 125 y-scale: 10

Page 7: Exponential Growth and Decay

The Sky Is Falling - TablesA

ttach

th

e S

ky

is F

all

ing

Tab

le i

nto

you

r n

ote

s

Page 8: Exponential Growth and Decay

The Sky Is Falling

On your graphing calculator, graph both the linear and the exponential models you created.

Was there a constant rate of change for your models?

Was there a constant successive quotient for your models?

How did you determine the initial values from your tables?

Which one of the above models is a better fit for the data? Why?

Page 9: Exponential Growth and Decay

The Sky Is Falling - TablesA

ttach

th

e S

ky

is F

all

ing

Tab

le t

o y

ou

r n

ote

s

461015222531

10

16.143

y = 16.143x+10

1.41.4291.51.51.4891.3731.336

10

1.433

y = 10(1.433)x

Page 10: Exponential Growth and Decay

What’s Going On? – Before we begin

What is occurring mathematically in order for a situation to be modeled with a linear function?

What values are necessary to write a linear model?

What is occurring mathematically in order for a situation to be modeled with a exponential function?

What values are necessary to write an exponential model?

What is occurring mathematically in order for a situation to be modeled with neither a linear nor exponential function?

Constant Rate Of Change

Slope and y-intercept

Constant successive quotients

The “a” Initial Value and successive quotients to find the base “b”

Will not have a constant rate of change or constant successive quotients

Page 11: Exponential Growth and Decay

What’s Going On? – 15 minutes to complete

How do we handle problems like 2 and 3 when given a rate, r%, of the growth or decay?

Are the dependent values in a situation increasing or decreasing each time?

How might this effect the base value in our exponential model?

(1r)

Increasing values would show growthDecreasing values would show decay

Page 12: Exponential Growth and Decay

Exponential Growth Model

When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:

 

where a is the initial amount and r is the percent increase expressed as a decimal.

NOTE: b is the growth factor.

Sometimes the equation is written, A=P(1+r)t, where A stands for the balance amount and P stands for the principal, or initial amount.

(1 )

ty a b

b r

(1 )ty a r

Page 13: Exponential Growth and Decay

Exponential Decay Model

When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:

 

where a is the initial amount and r is the percent decrease expressed as a decimal.

NOTE: b is the decay factor.

Sometimes the equation is written, A=P(1 - r)t, where A stands for the balance amount and P stands for the principal, or initial amount.

(1 )

ty a b

b r

(1 )ty a r

Page 14: Exponential Growth and Decay

Example 1:

In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year.

a. Identify the constants a, r, and b:

b. Write an exponential model to represent the situation given.

c. About how many incidents were there in 2003?

a = 2573 r = .92Growth, so b =

(1+r) = (1.92)

1 or

2573(1 .92) or 2573(1.92)

t t

t t

y a r y a b

y y

t = 2003 –1996 = 7 years

72573(1.92) 247,485 virusesy y

Page 15: Exponential Growth and Decay

In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year.

d. Estimate the year when there were about 125,000 computer security incidents.

2573(1.92)t ty a b y

Using your calculator: t is approximately 6, so in the year 2002, there were about 125,000 security incidents.

Example 1: (continued)

Page 16: Exponential Growth and Decay

Example 2: A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.

a. Identify the constants a, r, and b:

b. Write an exponential model to represent the situation given.

c. Estimate the value after 3 years.

1 or

4200(1 .1) or 4200(.9)

t t

t t

y a r y a b

y y

Decay, so b = (1 – r)

a = 4200 r = 0.1

= (0.9)

= $3061.80

34200(.9)y

Page 17: Exponential Growth and Decay

A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.

d. Estimate when the value of the snowmobile will be $2500.

4200(.9)

t

t

y a b

y

Example 2: (continued)

1

2

4200(.9)

2500

ty

y

After about 5 years

MIN MIN

MAX MAX

SCL SCL

1 1

10 4000

1 500

x y

x y

x y

Find the point of

intersection.

Page 18: Exponential Growth and Decay

Absent Students – Notes 6.3

Attach this note page into your notebook

Complete all examples.

Page 19: Exponential Growth and Decay

Sky Is Falling Tables: Complete and Attach to Notes