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Exponential Growth and Decay

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Exponential Growth and Decay

Section 6.3

-8

-6

-4

-2

0

2

4

6

8

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

7

0

-5

-8

-9

-8

-5

0

7

When x<1, as the x-values increase, the

y-values decrease.

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

y-values increase.

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 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

Bean Drop Data

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.

Bean Drop Data

Window for this graph:

x-min: 0 x-max: 10 x-scale: 1

y-min: 0 y-max: 125 y-scale: 10

Attach the Sky is Falling Table into your notes

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?

Attach the Sky is Falling Table to your notes

1.4

1.429

1.5

1.5

1.489

1.373

1.336

4

6

10

15

22

25

31

10

10

1.433

16.143

y = 16.143x+10

y = 10(1.433)x

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

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?

Increasing values would show growth

Decreasing values would show decay

(1r)

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.

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.

- 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.
- Identify the constants a, r, and b:
- Write an exponential model to represent the situation given.
- About how many incidents were there in 2003?

a = 2573 r = .92

Growth, so b = (1+r)

= (1.92)

t = 2003 –1996 = 7 years

Example 1: (continued)

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.

Using your calculator: t is approximately 6, so in the year 2002,

there were about 125,000 security incidents.

Example 2:

- A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year.
- Identify the constants a, r, and b:
- Write an exponential model to represent the situation given.
- Estimate the value after 3 years.

a = 4200 r = 0.1

Decay, so b = (1 – r)

= (0.9)

= $3061.80

Example 2: (continued)

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.

Find the point of intersection.

After about 5 years

Absent Students – Notes 6.3

Attach this note page into your notebook

Complete all examples.