Exponential Decay - Math Steps, Examples & Questions

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Exponential decay

Here you will learn about exponential decay, including what it is and how to solve problems involving exponential decay.

Students first learn about exponential decay in seventh grade math when they explore percent of increase and decrease. Students expand their knowledge as they work with exponential functions in Algebra $1$ and Algebra $2.$

What is exponential decay?

Exponential decay is the process of reducing an amount by a consistent percentage rate over a period of time.

Exponential decay, which is sometimes referred to as depreciation, can be modeled using the exponential decay model,

$y=a(1-r)^x \;\;$ OR $\;\; f(t)=a(a-r)^t$

\begin{aligned}& a=\text { initial value } \\\\ & r=\text { rate } \\\\ & x=\text { amount of time } \end{aligned}

Notice how in the parenthesis the rate is being subtracted from $1.$ When a number is multiplied to a decimal less than $1$ , the product is less than the initial value.

The decay factor is the number that shows how much an amount decreases over time. Since the equation models a decay, the answer must be less than the initial amount.

For example, a new car originally costs $\$ 20,000.$ The car’s value has an annual depreciation rate (exponential decay rate) of $10 \%.$ Using the exponential decay formula, find the value of the car after $6$ years.

y=a(1-r)^x

\begin{aligned}a&=\text { initial amount which is } \$ 20,000 \\\\ x&=\text { time which in this case is } 6 \text { years } \\\\ r&=\text { percent of decay which in this case is }10 \% \end{aligned}

Remember, you must change a percent to a decimal or fraction before doing a calculation.

10 \%=0.10

Step-by-step guide: Converting percents to decimals

y=20,000(1-0.10)^6

\begin{aligned}& y=20,000 \times 0.5314 \\\\ & y=10,628.82 \end{aligned}

With a depreciation rate of $10 \%,$ after $6$ years the car will be worth $\$ 10,628.82$

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Exponential graph

Exponential decay functions can be modeled on the coordinate plane.

For example, the function, $y=1\left(\cfrac{1}{2}\right)^x$ can be sketched on the coordinate plane using a table of values.

Notice how the curve of the exponential decay function curves down towards the $x$ -axis but does not cross the $x$ -axis. This is because the $x$ -axis is the horizontal asymptote of the function.

Step-by-step guide: Exponential functions

What is exponential decay?

Common Core State Standards

How does this apply to middle school and high school math?

7th Grade – Ratios and Proportional Relationships (7.RP.A.3)
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

High School Algebra – Seeing Structure in Expressions (HSA.SSE.B.3c)
Use the properties of exponents to transform expressions for exponential functions. For example, the expression $1.15t$ can be rewritten as $(1.151/12)12t \approx 1.01212t$ to reveal the approximate equivalent monthly interest rate if the annual rate is $15 \%.$

High School Functions – Linear, Quadratic and Exponential Models (HSF-LE.A.1c)
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

How to calculate exponential decay

In order to calculate an exponential decay:

Identify the values that are given.
Calculate the exponential decay.
State the solution.

Exponential decay examples

Example 1: finding a value after a depreciation

A boat is bought for $\$ 30,000.$ It depreciates at a rate of $5 \%$ per year. Find its value after $3$ years.

Identify the values of the model that are given.

Using the exponential decay equation,

y=a(1-r)^x

The initial value is the price of the boat, $\$ 30,000,$ so, $a=30,000$ .

The time is $3$ years, so $x=3$ .

The rate of decay is $5 \%,$ so $r=5 \%=0.05$ .

2Use the model to calculate the decay.

The model, $y=a(1-r)^x$ calculates the value of the decay.

In this case, substitute the values into the equation and solve for $y.$

\begin{aligned}& y=30,000(1-0.05)^3 \\\\ & y=30,000(0.95)^3 \\\\ & y=25,721.25 \end{aligned}

3State the solution.

After $3$ years, the value of the boat will be $\$ 25,721.25.$

Example 2: predicting population

In $2019$ , the population of wolves in a certain area of Montana was $820.$ Due to the environment, the wolf population is declining. If the rate at which the population is declining is $4 \%$ per year. Approximately, what will the population of wolves be in $2025$ ?

Identify the values that are given.

The initial population of wolves is $820,$ so $a=820$ .

From $2019$ to $2025$ is $6$ years, so $x=6$ .

The rate is $4 \%$ which is $0.04$ as a decimal, so $r=0.04$ .

Calculate the exponential decay.

Using the model, $y=a(1-r)^x$ , substitute the values into the equation.

$\begin{aligned}& y=820(1-0.04)^6 \\\\ & y=820(0.96)^6 \\\\ & y \approx 641.86 \end{aligned}$

State the solution.

Since $y=641.86,$ the approximate population of wolves in $2025$ will be about $642.$

Example 3: investment

Jill makes an initial investment of $\$ 10,000.$ The investment loses $8.5 \%$ interest per year. How much money does Jill have at the end of $4$ years?

Identify the values that are given.

The initial amount is $\$ 10,000,$ so $a=10,000$ .

The rate is $8.5 \%$ which is $0.85,$ so $r=0.085$ .

The time is $4$ years, so $x=4$ .

Calculate the exponential decay.

Using the model, $y=a(1-r)x ,$ substitute the values into the equation.

$\begin{aligned}& y=10,000(1-0.085)^4 \\\\ & y=10,000(0.915)^4 \\\\ & y=7009.46 \end{aligned}$

State the solution.

After $3$ years, Jill will have $\$ 7009.46$ in her account.

How to determine the rate of decay

In order to identify the rate of decay:

Identify the decay factor.
Subtract the decay factor from $\bf{1}.$
Change the number into a percent.
State the percent.

Example 4: rate of decay

The exponential model, $y=100(0.5)^x$ represents bacteria population over a period of time. What is the decay rate?

Identify the decay factor.

The decay factor is the decimal number in the parenthesis being raised to the $x$ power. In this case the value is $0.5.$

Subtract the decay factor from $\bf{1}.$

1-0.5=0.5

Change the number into a percent.

Multiply the number by $100$ to change it to a percent.

$0.5 \times 100=50$

State the percent.

The percent of decay is $50 \%.$

Example 5: percent of decay

The function, $f(x)=16000\left(\cfrac{1}{3}\right)^x$ represents an exponential decay. What is the percent of decay?

Identify the decay factor.

In this case, the decay factor is the number in the parenthesis being raised to the $x$ power which is $\cfrac{1}{3}.$

Subtract the decay factor from $\bf{1}.$

1-\cfrac{1}{3}=\cfrac{2}{3}

Change the number into a percent.

To convert the fraction to a percent, first change the fraction to a decimal. You can do this by dividing the denominator into the numerator.

$\cfrac{2}{3}=0.666667$

Then multiply the decimal to $100.$

$0.6666667 \times 100=66.67 \%$

Another strategy you could use is to make a proportion,

$\begin{aligned} \cfrac{2}{3}&=\cfrac{x}{100} \\\\ 200&=3 x \\\\ 66.67&=x \\\\ 66.67 &\% \end{aligned}$

State the percent.

The percent of decay is about $66.7 \%.$

Step-by-step guide: Fraction to percent

Example 6

The function, $y=290(0.73)^x ,$ represents an exponential decay. What is the percent of decay?

Identify the decay factor.

In this case, the decay factor is $0.73.$

Subtract the decay factor from $\bf{1}.$

1-0.73=0.27

Change the number into a percent.

In order to change a decimal number to a percent, multiply the number by $100.$

$0.27 \times 100=27 \%$

State the percent.

The percent of decay is $27 \%.$

Teaching tips for exponential decay

Make connections by comparing and contrasting exponential growth functions with exponential decay functions as well as connecting them to real life.

Incorporate real world and cross disciplinary word problems such as half-life and radioactive decay.

Use active learning activities such as scavenger hunts or game playing to have students review skills instead of giving them a worksheet.

Encourage students who are struggling to use digital platforms such as Khan Academy so they can view tutorial videos.

Easy mistakes to make

Using the decay rate without subtraction from $\bf{1}$
For example, when the decay rate is $15 \%,$ using $0.15$ as the decay factor instead of subtracting it from $1.$

When the decay rate is $15 \%,$ you first have to convert that to a decimal number, $0.15,$ and then subtract it from $1.$ So in this case, the decay factor to use is $1-0.15=0.85.$

$0.85$ is the decay factor.

Forgetting to convert percents to decimals before performing calculations
For example, if the decay rate is $13 \%$ , using $13$ instead of $0.13$ to make the calculations.

Practice exponential decay questions

1.04

0.96

0.04

0.94

To find the decay factor, you have to subtract the percent of decay from $1.$ However, before doing that, you have to convert percent to a decimal.

$4 \%=0.04$ (divide $4$ by $100)$

Next, subtract $0.04$ from $1.$

1-0.04=0.96

The decay factor is $0.96.$

0.73

0.27

1.27

0.83

To find the decay factor, you have to subtract the percent of decay from $1.$ However, before doing that, you have to convert percent to a decimal.

$27 \%=0.27$ (divide $27$ by $100)$

Then, subtract $0.27$ from $1.$

1-0.27=0.73

$0.73$ is the decay factor.

\$ 6144

\$ 11,040

\$ 9344.26

\$ 6.14

To calculate the value of the car, use $y=a(1-r)^x$ , where:

\begin{aligned}& a=\text { initial amount } \\\\ & r=\text { percent } \\\\ & x=\text { time } \end{aligned}

In this case,

\begin{aligned}& a=12,000 \\\\ & r=8 \%=0.08 \\\\ & x=3 \end{aligned}

Substitute the values into the equation,

\begin{aligned}& y=12,000(1-0.08)^3 \\\\ & y=12,000(0.92)^3 \\\\ & y=9344.256 \approx 9344.26 \end{aligned}

The value of the car after $3$ years will be $\$ 9344.26$

20,730

10,730

4940

898

To calculate the population, use the model, $y=a(1-r)^x$ where,

\begin{aligned}& a=\text { initial amount } \\\\ & r=\text { percent } \\\\ & x=\text { time } \end{aligned}

In this case,

\begin{aligned}& a=25,670 \\\\ & r=3.5 \%=0.035 \\\\ & x=6(2030-2024=6) \end{aligned}

Substitute the values into the equation to calculate.

\begin{aligned}& y=25670(1-0.035)^6 \\\\ & y=25670(0.965)^6 \\\\ & y=20729.5 \approx 20730 \end{aligned}

0.43 \%

57 \%

0.57 \%

43 \%

The decay factor in this case is $0.43.$ Subtract that value from 1 and then change the decimal answer to a percent.

\begin{aligned}& 1-0.43=0.57 \\\\ & 0.57 \times 100=57 \end{aligned}

The percent of decay is $57 \%$ .

\cfrac{3}{4} \%

75 \%

0.75 \%

25 \%

The decay factor in this case is $\cfrac{3}{4}.$ Subtract that value from $1$ and then change the fraction answer to a percent.

1-\cfrac{3}{4}=\cfrac{1}{4}

\begin{aligned}& \cfrac{1}{4}=0.25 \\\\ & 0.25 \times 100=25 \end{aligned}

The percent of decay is $25 \%.$

Exponential Decay FAQs

What is the growth factor?

The growth factor is similar to the decay factor. It’s the number that shows how much an amount increases over time as opposed to how much a number decreases over time.

What is the growth rate?

The growth rate is similar to the decay rate, it’s the percent of how fast a value of a function is increasing instead of decreasing like the decay rate.

What functions are the focus in an Algebra $\bf{2}$ class?

In an Algebra $2$ class, you study polynomial functions, logarithm functions, exponential functions. quadratic functions, and linear functions.

What is the inverse of an exponential function?

The inverse of an exponential function is a logarithm function.

Are exponential functions the only way to represent growth and decay?

No, differential equations can be used to represent growth and decay.

For example, $\cfrac{dy}{dt}=ky$ is a differential equation representing growth and decay where time $\cfrac{dy}{dt}$ is the rate of change (time $t$ ) and , $k$ the constant of the proportion.

The next lessons are

Compound measures
Types of data
Averages and ranges

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