Exponential Growth and Decay: Definition, Graph, Formula, Examples

Exponential growth and decay are terms used to describe how quickly physical quantities change in value or shape. You might know that the bacterial colony has exponential growth and decay.

But what does that mean? Exponential growth means doubling quantities every second, every hour, or day depending on independent and dependent variables. For instance, the mathematical expression for the exponential growth of a colony after t hours is given by y(t) is:

dy / dt = 2y

This is the first-order equation showing the exponential growth of any quantity. 

Exponential Function – Definition 

An exponential function is one in which the exponent is a variable, the base is positive and not equivalent to one. F (x) =4x, for example, is an exponential function since the exponent is a fixed constant rather than a mutable. f (x) = x3 is a fundamental polynomial function rather than an exponential function. Exponential functions feature uninterrupted curved graphs that never reach a horizontal asymptote. Several practical phenomena are governed by logarithmic or exponential functions.

Exponential Growth

Exponential growth is a mathematical transformation that grows indefinitely using an exponential function. The shift that has occurred can be either positively or negatively directed. The key premise would be that the pace of changes is increasing. When not bound by environmental constraints such as accessible space and nourishment, populations of developing microorganisms, and indeed any expanding population of any species, may be described as an exponential growth function. Another application of an exponential growth function is the growth of savings with compound interest.

Exponential Decay

Exponential decay occurs in mathematical functions when the pace by which changes are occurring are decreasing and must thus reach a limitation, which is the horizontal asymptote of an exponential function. The asymptote is the position on the x-axis at which the speed of changes reached near zero. Exponential decay may be observed in a variety of systems. The reduction in radioactive particles as its fissions and decomposes into some other atoms follows an exponential decay curve. A hot item starts to cool to a constant ambient temperature, or a cold item heat, will demonstrate an exponentially decaying curve. Exponential decay may be used to determine the discharges of an electric capacitor across a resistance. 

Exponential Growth and Decay Formula

The exponential growth formula is used to find compound interest, find the doubling time, and find the population growth. 

Exponential growth is given by,

f (x) = a (1 + r)^x

Where, f (x) = exponential growth function 

a = initial amount

r = growth rate

x = number of time intervals 

In exponential growth, the quantity increases, slowly at first, and then very rapidly. The rate of change increases over time. Hence, the exponential growth graph can be described as 

The amount drops gradually, followed by a quick reduction in the speed of change and increases over time. The exponential decay formula is used to determine the decrease in growth. The exponential decay formula can take one of three forms:

f (x) = ab^x

f (x) = a (1 – r)^x

P = P₀ e^{-k t}

Where,

a (or) P₀ = Initial amount

b = decay factor

e = Euler’s constant

r = Rate of decay (for exponential decay)

k = constant of proportionality

x (or) t = time intervals (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

In exponential decay, the quantity decreases very rapidly at first, and then more slowly. The rate of change decreases over time. The rate of decay becomes slower as time passes. Hence, the exponential decay graph is denoted as 

Understanding the Exponential Growth and Decay Graph  

The graph of exponential growth and decay is not linear. In a straight-line graph, the rate of change is constant, which is not the case in the exponential growth and decay functions. Therefore, the exponential growth and decay graph are not straight lines. 

Observe the graphs based on the functional values a and b. 

x	y = f (x)
-2	2-2 = ¼
-1	2-1 = ½
0	20 = 1
1	21 = 2
2	22 = 4
3	23= 8

Features of the Exponential Growth and Decay Graph

• The domain is all Real numbers.
• The range is all positive real numbers (not zero).
• Graph has a y-intercept at (0,1). Remember any number to the zero power is 1.
• When b > 1, the graph increases. The greater the base, b, the faster the graph rises from left to right.
• When 0 < b < 1, the graph decreases.
• Has an asymptote (a line that the graph gets very, very close to, but never crosses or touches). For this graph the asymptote is the x-axis (y = 0).

How to Calculate Exponential Growth or Decay Rate?

The formula for exponential growth and decay is:

y = a b^x

Where a ≠ 0, the base b ≠ 1 and x is any real number

A show the initial integer in this function, like the initial population or the initial dose amount.

The growth or decay factor is represented by the parameter b. If b is greater than one, the function indicates exponential growth. If the function is 0 < b < 1, it depicts exponential decline.

If a percent of growth or decay is given to you and it is said to calculate the growth/decay factor, add or subtract the percent, expressed in the decimal form, from 1.

Generally, if r is a decimal representation of the growth or decay factor, then:

b = 1 – r Decay Factor

b = 1 + r Growth Factor

The variable x denotes how many times the growth/decay factor is compounded.

Applications of Exponential Growth 

In many everyday tasks, the idea of exponential growth and decay can be seen in action. Let’s look at a few significant uses for exponential growth and decay. 

• Bacterial Development: In many communicable diseases, the first stages of bacterial growth can be seen. The current Covid-19 is a brief illustration of exponential growth in which the illness is extremely contagious and spreads from one to many, then to many more. In the same way that bacteria and viruses multiply exponentially, so do the original bacteria. If the initial germs were to multiply by two per second, they would increase in number to 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. This allows us to see how bacteria exhibit exponential growth. 
• Food Degradation: Exponent growth and decay can also be used to explain food degradation. The meal has a shelf life after which it begins to deteriorate exponentially. The food starts out a little bit stale but quickly goes worse until we realise it’s entirely spoiled. This is also an example of exponential decay, where once the process starts, it progresses quickly until the entire amount of food is fully stale. 
• Human Aging: At the end of a lifespan, any living thing begins to age. This process is characterised by an exponential deterioration. The person lives a regular lifespan of roughly 60 years while remaining hale and hearty. The retirement age for many occupations is established at 60 years for the same reason. After this, the ageing process accelerates to the point that the body is affected exponentially. The earlier years of life’s decline are dramatically followed by years of even greater decline in the standard of living. This can be a result of certain diseases that are already present progressing or specific organs not performing as they should. 

Exponential Growth and Decay Word Problems

Example 1: Carbon-14 has a half-life of 5,730 years. Find the carbon-14, exponential decay model. Please round your answer to the nearest decimal point.

Solution: Use the formula of exponential decay

P = P₀ e^{– k t}

P₀ = initial amount of carbon

Half-life of carbon-14 is 5,730 years,

P = P₀ / 2 = Half of the initial amount of carbon when t = 5, 730.

P₀ / 2 =P₀ e^{– k} (5730)

Divide both sides by P₀

0.5 = e^{– k} (5730)

Take “ln” on both sides,

ln 0.5 = -5730k

Divide both sides by -5730,

k = ln 0.5 / (-5730) ≈ 1.2097

The exponential decay model of carbon-14 is P = P₀ e^{– 1.2097k} 

Example 2: Andrew spent $350,000 on a new couch. The sofa’s worth falls exponentially at a pace of 5% every year. So, how much is the sofa worth after two years? Please round your answer to the nearest decimal point.

Solution: Initial value of Sofa= $350,000

Rate of decay r = 5% = 0.05

Time t = 2 years

Use the exponential decay formula,

A = P (1 – r)t

A = 350000 x (1 – 0.05)2

A = 315,875

The value of the sofa after 2 years = $315,875

Example 3: Maria paid around $20,000 on a fashionable pocketbook. The worth of the pocketbook decreases exponentially (depreciates) at a yearly rate of 8%. So, what is the value of the pocketbook after 5 years? Give your answer to the nearest decimals.

Solution: Initial value P = $20,000.

Rate of decay r = 8% = 0.08.

Time t = 5 years.

Use the exponential decay formula:

A = P (1 – r)^t

A = 20000 x (1 – 0.08)⁵ = 13181.63

The value of the pocketbook after 5 years = $13,181.63.

Frequently Asked Questions on Exponential Growth And Decay

Q1. What Is the Decay Rate of an Exponential Function?

The formula for exponential decay is f(x) = ab^x, where b denotes the decay factor. In the exponential decay function, the decay rate is given as a decimal. The decay rate is expressed as a percentage. We convert it to a decimal by simply reducing the percent and dividing it by 100. Then calculate the decay factor b = 1-r. For instance, if the rate of decay is 25%, the exponential function’s decay rate is 0.25 and the decay factor b = 1- 0.25 = 0.75.

Q2. What exactly is the Exponential Decay Formula?

The amount gradually reduces by a predetermined percentage at regular periods. The exponential decay formula is used to determine this decrease in growth.

f(x) = a (1 – r)^x is the generic form.

Where,

a = The initial value

r = decay rate 

x = time period

Q3. Do we need exponential growth and decay calculator?

The exponential growth and decay calculator is useful when we have to do quick calculations in a generalized manner. However, you must not use it frequently as it can affect your calculation speed to solve problems. You must practice exponential growth and decay word problems on pen and paper to enhance your understanding. 

Q4. How effective is it to practice from exponential growth and decay worksheets? 

The exponential growth and decay worksheet answers three questions for every exponential growth and decay problem – does this function represent exponential growth and decay, what is your initial value, and what is the growth rate or decay rate for the given problem? If these answers are known, then you can master any exponential growth and decay problem.