Lesson 3.8Solving Problems Involving

Exponential FunctionsStandard: F.LE.2

Exponential GrowthExponential growth occurs when a quantity increases by the same percent r in each time period t.

The percent of increase is 100r2

3.4.2: Graphing Exponential Functions

Initial value

𝑦=𝐶 (1+𝑟 )𝑡

𝑓 (𝑥 )=𝑎 ·𝑏𝑥

Growth factor Time Period

Growth rate

Exponential Growth Exponential Decay

Exponential Money Growth

Step 1: Identify the components needed for your exponential growth or decay formula.

Step 2: Substitute your found quantities into your formula.

Step 3: Evaluate the formula.

Step 4: Interpret your answer.

𝑦=𝐶 (1+𝑟 )𝑡 𝑦=𝐶 (1−𝑟 )𝑡

𝐴=𝑃 (1+𝑟 )𝑛

Example 1

A population of 40 pheasants is released in a wildlife preserve. The population doubles each year for 3 years. What is the population after 4 years?

Step 1: Identify the components needed for your exponential growth formula

initial value = C = 40

growth factor = 1 + r = 2 (doubles); r = 1

years = t = 4

Example 1 (continued)

initial value = C = 40

growth factor = 1 + r = 2 (doubles); r = 1

years = t = 4

Step 2: Substitute your found quantities into your formula.

Step 3: Evaluate.

Step 4: Interpret your answer.

After 4 years, the population will about 640 pheasants.

PracticeUse the exponential growth model to answer the question.

𝑦=𝐶 (1+𝑟 )𝑡1

Exponential GrowthWhen dealing with money, they change the letters used for the variables slightly. A stands for account balance, P stands for the initial value, while n stands for number of years.

The percent of increase is 100r7

3.4.2: Graphing Exponential Functions

Initial value

𝐴=𝑃 (1+𝑟 )𝑛

𝑓 (𝑥 )=𝑎 ·𝑏𝑥

Growth factor Time Period

Growth rate

Exponential Growth Exponential Decay

Exponential Money Growth

Step 1: Identify the components needed for your exponential growth or decay formula.

Step 2: Substitute your found quantities into your formula.

Step 3: Evaluate the formula.

Step 4: Interpret your answer.

𝑦=𝐶 (1+𝑟 )𝑡 𝑦=𝐶 (1−𝑟 )𝑡

𝐴=𝑃 (1+𝑟 )𝑛

Example 2

A principal of $600 is deposited in an account that pays 3.5% interest compounded yearly. Find the account balance after 4 years.

Step 1: Identify the components needed for your exponential growth formula

initial value = P = $600

growth rate = r = 3.5% = .035

years = n = 4

Example 2 (continued)

initial value = P = $600

growth rate = r = 3.5% = .035

years = n = 4

Step 2: Substitute your found quantities into your formula.

Step 3: Solve.

Step 4: Interpret your answer.

The balance after 4 years will be about $688.51.

PracticeUse the exponential growth model to find the account balance.𝐴=𝑃 (1+𝑟 )𝑛

2

3

Exponential DecayExponential decay occurs when a quantity decreases by the same percent r in each time period t.

The percent of decrease is 100r12

3.4.2: Graphing Exponential Functions

Initial value

𝑦=𝐶 (1−𝑟 )𝑡

𝑓 (𝑥 )=𝑎 ·𝑏𝑥

Decay factor Time Period

Decay rate

Exponential Growth Exponential Decay

Exponential Money Growth

Step 1: Identify the components needed for your exponential growth or decay formula.

Step 2: Substitute your found quantities into your formula.

Step 3: Evaluate the formula.

Step 4: Interpret your answer.

𝑦=𝐶 (1+𝑟 )𝑡 𝑦=𝐶 (1−𝑟 )𝑡

𝐴=𝑃 (1+𝑟 )𝑛

Example 3

You bought a used truck for $15,000. The value of the truck will decrease each year because of depreciation. The truck depreciates at the rate of 8% per year. Estimate the value of your truck in 5 years.

Step 1: Identify the components needed for your exponential growth formula

initial value = C = $15,000

decay rate = r = 8% = .08

years = t = 5

Example 3 (continued)

initial value = C = $15,000

decay rate = r = 8% = .08

years = t = 5

Step 2: Substitute your found quantities into your formula.

Step 3: Solve.

9,886.22

Step 4: Interpret your answer.

The value of your truck in 5 years will be about $9,886.22

PracticeUse the exponential growth model to find the account balance.𝑦=𝐶 (1−𝑟 )𝑡

3

3

4

5

Annual Percent of Increase/Decrease

The annual percent of increase or decrease comes from the Growth and Decay factors of the exponential formulas

The percent of increase or decrease is 100r.

𝑦=𝐶 (1+𝑟 )𝑡

𝑦=𝐶 (1−𝑟 )𝑡

Annual Percent of Increase or Decrease

Exponential Growth Exponential Decay

Step 1: Identify if the function is a growth or a decay.

Step 2: Look at the growth or decay factor from the exponential formulas above and set it equal to the base. Growth: 1 + r = base Decay: 1 – r = base

Step 3: Solve the formula for r.

Step 4: The percent of increase or decrease is 100r. So multiply your found value for r from step 3 by 100.

𝑦=𝐶 (1+𝑟 )𝑡 𝑦=𝐶 (1−𝑟 )𝑡Growth factor Decay factor

Annual Percent of Increase

Example: Find the annual percent of increase or decrease that f(x) = 2(1.25)x models

Step 1: Identify if it’s a growth or a decay.

Since the base (1.25) is greater than 1, it’s a

growth.

Step 2: Look at the growth factor from the exponential formula: 1 + r and set it equal to the base

1 + r = 1.25

Step 3: Solve the formula for r --- r = .25

Step 4: The percent of increase is 100r, so substitute r for .25

The percent of increase is 25%

Annual Percent of Decrease

Example: Find the annual percent of increase or decrease that f(x) = 3(0.80)x models

Step 1: Identify if it’s a growth or a decay.

Since the base (0.80) is less than 1, it’s a

decay.

Step 2: Look at the decay factor from the exponential formula: 1 – r and set it equal to the base

1 – r = 0.80

Step 3: Solve the formula for r --- r = .20

Step 4: The percent of decrease is 100r, so substitute r for .20

The percent of increase is 20%

PracticeFind the annual percent of increase or decrease that the given exponential functions model.

6.

7.

8.