SECTION 6 .3

Exponential Growth and Decay

Warm-up 1

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

Warm-up 2

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

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

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)

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

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

The Sky Is Falling - TablesA

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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?

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461015222531

10

16.143

y = 16.143x+10

1.41.4291.51.51.4891.3731.336

10

1.433

y = 10(1.433)x

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

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

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

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

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

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)

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

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.

Absent Students – Notes 6.3

Attach this note page into your notebook

Complete all examples.

Sky Is Falling Tables: Complete and Attach to Notes