*Constructing a linear model: Adding a constant amount each year.

Assuming linear growth, our function is:

P = b + mt

( b is the starting value,

m is the rate or the slope, positive for growth, negative for decay)

change in population

m= ——————————

change in time

This linear model tells the original population, the increasing by the constant amount each year.

Identifying Exponential Functions in Data Table

Both of these two models are used to make predictions, to tell the stories of the explosive change in a long period of time.

A Linear vs. an Exponential Model through Two Points

**5.3 Comparing Linear and Exponential Functions**

A

linear function

represents a quantity to which a constant amount is added for each unit increase in the input .

An

exponential function

represents a quantity that is multiplied by a constant factor for each unit increase in the input.

Linear functions

y = b + mx

y = initial value + (rate of change) • x

Exponential functions

y = C • aˆx

y = initial value + (base)^x

How can we identify Exponential Functions in a Data Table ?

If the difference in consecutive input values is constant, the data represent a linear function if the

difference

between consecutive output values is constant, whereas the data represent an exponential function if the

ratio

of consecutive output values is constant.

Example 1

Do the data represent a linear or an exponential function?

x f(x) g(x)

0

2

4

6

8

10

25

50

75

100

125

150

100

196

384.16

752.95

1475.79

2892.55

Solution

The function f(x) is a linear function. Each time x increase by 2, f(x) increases by 25.

The function g(x) is exponential since the ratio of consecutive output values is constant at 1.96 when x = 2 .

Conclusion

For a Table Describing

y

as a Function of

x

, Where

x

Is Constant

If the

difference

between consecutive y values is constant, the function is linear.

If the

ratio

of consecutive y values is constant, the function is exponential.

Example 1:

t = numbers of years

P = population size

Example 2: Making predictions with the models

a. Town population

—————————————————————————

Year Linear Model Exponential Model

P = 20,000 + 800t P = 20,000 .(1.0371)^t

—————————————————————————

0 20,000 20,000

5 24,000 24,000

10 28,000 28,790

15 32,000 34,540

20 36,000 41,440

25 40,000 49,720

—————————————————————————

*Constructing an exponential model:

Multiplying by a constant factor each year

Assuming exponential growth, our function is:

P = x . a^t

( P is the population

x is the original amount

a is the base or the multiplier.

a > 1 for growth, 0< a <1 for decay

t is the number of years )

For each additional year, the initial population is multiplied by the growth factor, a. After t years, the initial population of x is multiplied by at to get P

CONCLUSION

Both of the models should be considered as generating only crude future estimates.

The further out we try to predict, the more unreliable the estimates become

Linear growth continues to increase at the same rate, whereas exponential growth increases at an expanding rate.

**Summary**

Linear functions: y=mx+b

Exponent functions: y = c*a^x

For a Table Describing

y=f(x)

, delta

x

constant

Difference between consecutive y values constant linear.

If the ratio of consecutive y values is constant, the function is exponential.

Linear Growth: b+mt

Exponential Growth: x*a^t

In the long run, Exponential Growth will always outpace Linear Growth.

Comparing the Average Rates of Change

Examining

average rates of change

between linear and exponential functions

Thank you

for your attention !

Comparing Average - Rate -of-Change Calculations

In the long run, Exponential Growth will always outpace Linear Growth

The Average Rate of Change function

describes the average rate at which one quantity is changing with respect to something else changing.