Modeling Exponential Growth and Decay

Exponential Growth and Decay

Two types of exponential situations

- Decay

-Growth

Growth Formula

•  $y=a\left(1+r\right)^t$  

• a is initial value

• r is the rate of growth expressed as a decimal

• t is the time passed

• y is the final amount

Here are more examples of exponential functions that model "growth":

•  $f\left(x\right)=4^x$  

•  $g\left(x\right)=\frac{1}{2}\left(2\right)^x$  

•  $y=12\left(50\right)^x$  

•  $f\left(x\right)=2\left(1.5\right)^x$  

Multiple Choice

Is the pictured graph growth, decay, or linear or none?

• Exponential Growth

• Exponential Decay

• Linear

• None

Multiple Choice

Which of the following functions shows an initial amount of $15 and an increase of 35% each year?

• y = 15(35)^x
• y = 15(1.35)^x
• y = 15(0.35)^x
• y = 35(1.15)^x

Multiple Choice

What is r, the growth rate, for the function: f(x) = 300(1.16)^x?

• 300

• 1.16

• .16

• x

Multiple Choice

Is y = 5(1.04)^x growth or decay?

• Growth

• Decay

Multiple Choice

What does the a in y=a(1+r)^x represent?

• Time

• Rate

• Slope

• Initial Amount

Multiple Choice

James' 70 in. giant peach doubles in size every week. Write an expression that would represent how big the peach is after 5 weeks.

• 70(2)³⁵
• 70(2)⁵
• 2(70)⁵
• 5(70)²

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.

A=10(1.01)³

• Growth

• Decay

Decay Formula

•  $y=a\left(1-r\right)^t$  

• a is initial value

• r is the rate of decrease expressed as a decimal

• t is the time passed

• y is the final amount

Plug in 97 milligrams for the initial amount, 0.05 for the rate of decrease, and 6 hours for the time elapsed.

• a = Plug in 97 milligrams for the initial amount

• r = 0.05 for the rate of decrease

• t = 6 hours for the time elapsed.

Here are more examples of exponential functions that model "decay":

•  $f\left(x\right)=\left(\frac{1}{4}\right)^x$  

•  $g\left(x\right)=2\left(\frac{2}{3}\right)^x$  

•  $y=-12\left(0.9\right)^x$  

•  $y=10\left(\frac{1}{2}\right)^x$  

Multiple Choice

Is this exponential growth or decay?

• Growth

• Decay

Multiple Choice

What is the exponential factor of the following graph?

• Growth

• Deccay

• Linear

• None

Multiple Choice

Suppose a culture of bacteria begins with 5000 cells and dies by 30% each year. Write an equation that represents this situation.

• y=5000(0.7)^x

• y=30(5000)^x

• y=5000(1.3)^x

• y=5000x^x

Multiple Choice

The value of a car is $15,000 and depreciates at a rate of 8% per year. What is the decay factor?

• ^.08

• ^1.08

• .92

• 8

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.

A=1200(.85)⁶

• Growth

• Decay

Now I know when an exponential graph is modeling "growth" or "decay", but how can I tell just from looking at a function?

Let's investigate!

Multiple Choice

Which of the graphs is exponential decay?

Multiple Choice

Is

$$h\left(x\right)=12\left(\frac{1}{3}\right)^x$$  growth or decay? How do you know?

• Decay because it's going downwards from left to right.

• Decay because it's going upwards from left to right.

• Decay because the table of values is increasing, showing a pattern of multiplication.

• Decay because all exponential functions decrease.

Multiple Choice

Is

$$f\left(x\right)=2\left(3\right)^x$$  growth or decay? How do you know?

• Growth because it's going downwards from left to right.

• Growth because it's going upwards from left to right.

• Growth because the table of values is decreasing, showing a pattern of division.

• Growth because all exponential functions grow.

Multiple Choice

Is this exponential growth or decay?

• Growth

• Decay

Multiple Choice

Is this exponential growth or decay?

• Growth

• Decay

Multiple Choice

Which of the following is an example of exponential decay?

• $$y\ =\ 5^x$$  

• $$y\ =\ \left(\frac{1}{2}\right)5^x$$  

• $$y\ \ =\ \left(\frac{1}{3}\right)^x$$  

• $$y\ =\ 8\ \left(4\right)^x$$  

Multiple Choice

Classify the model of an Exponential DECAY.

• $$y=a\left(1+r\right)^t$$  

• $$y=a\left(1-r\right)^t$$  

Multiple Choice

Classify the model of an Exponential GROWTH.

• $$y=a\left(1+r\right)^t$$  

• $$y=a\left(1-r\right)^t$$  

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.

$$A=10(1.01)^3$$  

• Growth  $$y=a\left(1+r\right)^t$$  

• Decay  $$y=a\left(1-r\right)^t$$  

Multiple Choice

Classify the model as Exponential GROWTH or DECAY.

$$A=1200(.85)^6$$  

• Growth  $$y=a\left(1+r\right)^t$$  

• Decay  $$y=a\left(1-r\right)^t$$  

Multiple Choice

Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates at a rate of 5%.

How much will the printer be worth in 8 years?

• $23,219.72

• $136.72

• $51,710.94

• $16,710.94

Multiple Choice

A population of fish starts at 8,000 and increases by 6% per year.

What is the fish population in 10 years?

• $$14,327$$

• $$4,309$$

• $$839$$

• $$7,680$$

Multiple Choice

Some banks charge a fee for a savings account that is left inactive for an extended period of time. 

The equation  $$y=5000(0.98)^x$$   represents the amount remaining, y, of one account that was left inactive for a period of x years. 

What does the number 5000 represent in this situation?

• A fee charged for an inactive account

• The percent of money in the account after x years

• The amount of money in the account initially

• The amount of money in the account after x years

Multiple Choice

An antibiotic is introduced into a colony of 12,000 bacteria during a laboratory experiment. The colony is decreasing by 14.9% per minute.

Which function can be used to model the number of bacteria in the colony after x minutes?

• $$f(x)=12000(1+14.9)^x$$

• $$f(x)=12000(1-14.9)^x$$

• $$f(x)=12000(1+0.149)^x$$

• $$f(x)=12000(1-0.149)^x$$

Multiple Choice

There were 417 cell phones sold at an electronics store in January. Since then, cell phone sales at this store have increased at a rate of 3.75% per month.

At this rate of growth, which function can be used to determine the monthly cell phone sales x months after January?

• $$f(x)=417(1-0.0375)^x$$

• $$f(x)=417(1-3.75)^x$$

• $$f(x)=417(1+0.0375)^x$$

• $$f(x)=417(1+3.75)^x$$

Multiple Choice

A population of 1500 deer decreases by 1.5% per year. At the end of 10 years, there will be approximately 1290 deer in the population.

Which function can be used to determine the number of deer, y, in this population at the end of t years?

• $$y=1500(1-0.015)^t$$

• $$y=1500(0.015)^t$$

• $$y=1500(1+0.015)^t$$

• $$y=1500(1.5)^t$$

Multiple Choice

A population of fish starts at 8,000 and increases by 6% per year.

Which exponential function can be used to find the fish population in 10 years?

• $$y=8000\left(1-0.06\right)^{10}$$

• $$y=8000\left(1+0.06\right)^{10}$$

• $$y=8000\left(1-0.1\right)^6$$

• $$y=8000\left(1+0.1\right)^6$$

Multiple Choice

A population of fish starts at 8,000 and increases by 6% per year.

Which exponential function can be used to find the fish population in 10 years?

• $$y=8000\left(1-0.06\right)^{10}$$

• $$y=8000\left(1+0.06\right)^{10}$$

• $$y=8000\left(1-0.1\right)^6$$

• $$y=8000\left(1+0.1\right)^6$$

Multiple Choice

Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates at a rate of 5%.

Which formula can be used to find out how much the printer will be worth in 8 years?

• $$y=35000\left(1-0.05\right)^8$$

• $$y=35000\left(1-0.08\right)^5$$

• $$y=35000\left(1+0.05\right)^8$$

• $$y=35000\left(1+0.08\right)^5$$

Multiple Choice

If the rate of increase is 5%, what do you plug in for r in the exponential growth?

• 5

• 0.5

• 0.05

• 0.005