7.3: Exponential Growth and Decay

7.3: Exponential Growth and Decay

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Steps for Exponential Problem Solving

1: Decide if the problem is describing exponential growth or decay. ("increase" = growth, "decrease" = decay)

2: Convert the growth or decay percentage to a decimal. This decimal is your r value. Ex: 23% = .23 and 5% = .05

3: Write your exponential equation. a is your starting value.

GROWTH: y = a(1 + r)^x

DECAY: y = a(1 - r)^x

4: Use the equation to solve the problem. Remember that x is input and y is output.

Multiple Choice

Example: Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year.

Is this problem describing exponential growth or decay?

• Growth

• Decay

Fill in the Blank

Example: Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year.

Convert the growth percentage to a decimal.

Fill in the Blank

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year.

We now know that our r value is .05. What is our a value? (a is the starting value)

Fill in the Blank

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year.

Okay, now we know this is growth, r = .05, and a = 46,000. Put it all together in an exponential equation. Type it in this format, filling in the values and simplifying: y=a(1+r)^x

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year.

y = 46000(1.05)^x

Type this equation into Desmos. Zoom out until you see the curve.

Multiple Choice

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year. We have modeled this situation with the equation y=46000(1.05)^x.

What does x (input) represent?

• time in years

• salary in dollars

Multiple Choice

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year. We have modeled this situation with the equation y=46000(1.05)^x.

What does y (output) represent?

• time in years

• salary in dollars

Fill in the Blank

Anne currently brings in an annual salary of $46,000 and anticipates a raise of 5% every year. What will her salary be in 4 years?

Using a calculator or Desmos, answer the question by plugging in 4 for x. Round your answer to the nearest dollar: $_____

Great job! Let's try another example.

Multiple Choice

Example: A car bought for $13,000 depreciates at 12% annually.

Is this problem describing exponential growth or decay?

• Growth

• Decay

Fill in the Blank

Example: A car bought for $13,000 depreciates at 12% annually.

Convert the decay percentage to a decimal.

Fill in the Blank

A car bought for $13,000 depreciates at 12% annually.

We now know that our r value is .12. What is our a value? (a is the starting value)

Fill in the Blank

A car bought for $13,000 depreciates at 12% annually.

Okay, now we know this is decay, r = .12, and a = 13,000. Put it all together in an exponential equation. Type it in this format, filling in the values and simplifying: y=a(1-r)^x

Example: A car bought for $13,000 depreciates at 12% annually.

y = 13000(0.88)^x

Type this equation into Desmos. Zoom out until you see the curve.

Multiple Choice

A car bought for $13,000 depreciates at 12% annually. We have modeled this situation with the equation y=13000(0.88)^x.

What does x (input) represent?

• time in years

• car's value in dollars

Multiple Choice

A car bought for $13,000 depreciates at 12% annually. We have modeled this situation with the equation y=13000(0.88)^x.

What does y (output) represent?

• time in years

• car's value in dollars

Fill in the Blank

A car bought for $13,000 depreciates at 12% annually. What will the car be worth after 10 years?

Using a calculator or Desmos, answer the question by plugging in 10 for x. Round your answer to the nearest dollar: $_____

Awesome! Use these steps to solve the problems on your 7.3 Assignment. Don't forget to use Desmos as a resource.