Recognizing Constant Percent Growth & Decay - Algebra 1

Section 1

Where Did This Idea Come From?

People have been fascinated by quantities that grow "on top of themselves" for centuries. When you earn interest on a savings account, the bank doesn't just add the same dollar amount each month — it adds a percentage of whatever is already there. That means the bigger your balance, the more you earn. This seemingly simple idea — constant percent change — has shaped finance, biology, physics, and technology in ways few other mathematical concepts can match.

~2000 BCE

Ancient Babylonian clay tablets contain problems about compound interest on grain loans. Merchants realized that "interest on interest" makes quantities grow faster than a flat payment ever could.

1614

Scottish mathematician John Napier published his work on logarithms, creating a tool that simplified multiplication by converting exponential relationships into addition. Logarithms are the inverse of exponential growth.

1683

Jacob Bernoulli studied compound interest and discovered the number e ≈ 2.718, which became the base of the natural exponential function — one of the most important constants in all of mathematics.

1798

Thomas Malthus argued that human populations grow by a constant percentage each generation while food supplies grow by a fixed amount, predicting a crisis. His work pushed scientists to study exponential growth seriously.

Today

Constant percent change models are everywhere: smartphone battery drain, virus spread during pandemics, social-media follower growth, radioactive decay, and the depreciation of a car's value the moment you drive it off the lot.

The central question all of these situations share is: How do you tell the difference between something that changes by a constant amount versus something that changes by a constant percentage? Answering that question is exactly what this lesson is about.

Section 2

Core Principles & Definitions

Before we dive in, let's nail down the vocabulary. You already know how to graph a line, and you've probably seen the word "function" in class. Now we're extending those ideas to a new family of functions where the change in a quantity depends on the quantity itself.

Constant Amount (Linear)

A quantity changes by the same number each time period. Example: you save $20 every week. After 5 weeks you have $100 more — no matter how big your balance is. This is a linear function.

Constant Percent (Exponential)

A quantity changes by the same percentage each time period. Example: your bank account earns 5 % interest per year. The dollar amount added each year gets bigger because 5 % of a larger balance is more money. This is an exponential function.

Growth Factor

When a quantity increases by r % each period, you multiply by (1 + r) where r is the rate as a decimal. A 5 % increase has a growth factor of 1.05.

Decay Factor

When a quantity decreases by r % each period, you multiply by (1 − r). A 12 % decrease has a decay factor of 0.88. Values between 0 and 1 always make the quantity shrink.

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✦ Key Takeaway

Think of constant percent change like a copy machine set to enlarge or shrink. If the machine is set to 110 %, every copy is 10 % bigger than the one before it. If it's set to 85 %, every copy is 15 % smaller. The setting never changes, but the actual size keeps compounding — that's what makes it exponential, not linear.

Section 3

Seeing the Difference: Linear vs. Exponential

The graph below shows two functions starting at the same value. The blue line grows by a constant amount each period (linear), while the cyan curve grows by a constant percentage each period (exponential). Watch how the exponential curve starts slowly but eventually races far ahead.

Graph comparing linear growth (straight blue line) and exponential growth (curved cyan line) from year 0 to year 10.

Both functions start at 100. The linear function adds 10 every year, reaching 200 after 10 years. The exponential function grows by 10 % each year, reaching about 259 in the same time. At first, the two paths look almost identical, but by year 5 the exponential curve is already pulling away. That accelerating gap is the signature of constant percent growth — each year's increase is larger than the last because it's calculated on a bigger base.

Section 4

The Mathematical Framework

Now that you can see the difference, let's formalize it. Every exponential situation we'll study in this lesson can be captured by one equation.

Exponential Growth / Decay Model

y = a · bˣ

a = initial value | b = growth or decay factor | x = number of time periods

Let's unpack each piece. The value a is whatever quantity you start with — a population, a dollar amount, or a mass of a radioactive substance. The value b is the multiplier that gets applied each time period. And x counts how many periods have passed (often measured in years, months, or hours).

Growth Factor

b = 1 + r (when r > 0)

r is the percent rate written as a decimal. A 7 % increase → r = 0.07 → b = 1.07

Decay Factor

b = 1 − r (when 0 < r < 1)

r is the percent rate written as a decimal. A 15 % decrease → r = 0.15 → b = 0.85

Here is the critical rule for recognizing exponential situations: if b > 1, the quantity grows; if 0 < b < 1, the quantity decays. Either way, the percent rate stays the same from one period to the next. That constancy is what separates exponential models from other kinds of change.

How do you spot constant percent change in a table of numbers? Divide each value by the one before it. If you get the same ratio every time, you're looking at an exponential function. Contrast that with a linear function, where you subtract consecutive values and get a constant difference.

The Ratio Test

If y₂/y₁ = y₃/y₂ = y₄/y₃ = … = constant, it's exponential

That constant ratio is the growth or decay factor b

Section 5

Growth vs. Decay — A Side-by-Side Look

The diagram below places exponential growth and exponential decay on the same set of axes so you can see how they mirror each other. The growth curve rises steeply, while the decay curve falls toward zero but never actually reaches it.

Exponential growth (green, b = 1.10) and exponential decay (pink, b = 0.80) both starting at 500.

Notice three important things. First, both curves pass through the same starting point at year 0 because they share the same initial value a = 500. Second, the growth curve (b = 1.10) accelerates — it bends upward. Third, the decay curve (b = 0.80) approaches zero but never touches it; in the real world, a decaying quantity gets incredibly small but technically never disappears.

Here is a quick table that shows how to translate real-world language into the factor b:

Real-World Phrase	Type	Rate r	Factor b
"increases by 8 % each year"	Growth	0.08	1.08
"gains 25 % every month"	Growth	0.25	1.25
"loses 6 % of its value per year"	Decay	0.06	0.94
"depreciates by 20 % annually"	Decay	0.20	0.80
"triples every decade"	Growth	2.00	3.00
"retains 90 % each hour"	Decay	0.10	0.90

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✦ Key Takeaway

The factor b is like the "multiplier" on a receipt. If b = 1.08, every period you pay 108 % of what you had — that extra 8 % is growth. If b = 0.85, you keep only 85 % and lose 15 % — that's decay. Anytime you see a quantity being multiplied by the same number over and over, you've found constant percent change.

Section 6

Worked Example

Let's walk through a complete problem from start to finish.

Population Growth Problem

1
Problem

A small town had a population of 12,000 in the year 2020. Researchers found that the population grows by 3 % per year. (a) Write an exponential model for the population. (b) Predict the population in 2028. (c) Explain why this situation is exponential rather than linear.
2
Step 1 — Identify the quantities

The initial value is a = 12,000 (the population in 2020). The rate of growth is r = 3 % = 0.03 per year. The variable x represents the number of years after 2020.
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Step 2 — Find the growth factor

Because the population increases, we use the growth formula: b = 1 + r = 1 + 0.03 = 1.03. Since b > 1, we confirm this is exponential growth.
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Step 3 — Write the model

y = 12,000 × 1.03^x

y = 12,000 × 1.03^x
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Step 4 — Predict the 2028 population

The year 2028 is x = 8 years after 2020. Substitute: y = 12,000 × 1.03⁸. Calculate 1.03⁸. You can do this step by step: 1.03 × 1.03 = 1.0609; then keep multiplying by 1.03 six more times, or use a calculator to get approximately 1.2668.

y ≈ 12,000 × 1.2668 ≈ 15,201
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Step 5 — Interpret the result

The model predicts the town will have about 15,201 people by 2028. This situation is exponential — not linear — because the population grows by the same percentage (3 %) each year, not by the same number of people. In 2021 the town added about 360 people (3 % of 12,000), but by 2027 it adds about 443 people (3 % of a larger population). The actual increase in people gets larger every year, which is the hallmark of exponential growth.

Section 7

Linear vs. Exponential — When to Use Which

One of the most important skills in algebra is choosing the right model. Below is a comparison that highlights when a situation calls for a linear function and when it calls for an exponential function.

Feature	Linear Function	Exponential Function
What stays constant	Amount of change per period (slope)	Percent of change per period (rate)
General form	`y = mx + b`	`y = a · bˣ`
Table test	Consecutive differences are equal	Consecutive ratios are equal
Graph shape	Straight line	Curved (bends up or down)
Real-world clue words	"per," "each," "every" + a fixed dollar/unit amount	"percent," "doubles," "halves," "rate," "factor"
Example	Adding $50 each paycheck to savings	Earning 4 % interest on a bank balance each year

There are also situations where students commonly get tricked. For instance, "a car loses $2,000 in value every year" is linear (constant dollar loss), while "a car loses 15 % of its value every year" is exponential (constant percent loss). The key is always to ask: is the change described as a fixed amount or as a percentage of the current value?

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✦ Key Takeaway

When choosing between linear and exponential, listen for the word "percent." If a problem says "increases by 50 units per day," that's linear. If it says "increases by 5 % per day," that's exponential. The word "percent" nearly always signals constant percent change — which means an exponential function is the right model.

Section 8

Where Does This Lead?

The ideas in this lesson are the foundation for some of the most powerful mathematics you'll encounter later in high school and beyond. Here's a quick look at how constant percent change connects to bigger ideas.

This Lesson	Advanced Version
Growth factor b = 1 + r	In Algebra 2 and Pre-Calculus, you'll use the natural base e and write continuous growth as y = a · e^kt
Recognizing exponential vs. linear from tables	In statistics, you'll use regression analysis to fit exponential models to messy real-world data
Doubling time / half-life intuition	In chemistry and physics, radioactive half-life and pharmacokinetics use the same exponential decay model
y = a · bˣ	In finance, the compound interest formula A = P(1 + r/n)^nt is a direct extension, adding compounding periods

Understanding constant percent change now puts you in an excellent position for all of these topics. You're learning to see the pattern, and that pattern shows up everywhere — from biology (bacteria doubling) to economics (inflation) to computer science (Moore's Law predicting the doubling of transistors on a chip roughly every two years). Once you can recognize the pattern, you can model it, predict it, and make smarter decisions based on it.

Section 9

Practice Problems

Try these five problems. They start simple and build up. Click "Show Answer" when you're ready to check your thinking.

Question 1: Problem 1 — Conceptual

A store sells 200 T-shirts in January, 220 in February, 242 in March, and 266.2 in April. Is the number of shirts sold growing by a constant amount or by a constant percent? How can you tell?

Show answer

Check the differences first: 220 − 200 = 20, 242 − 220 = 22, 266.2 − 242 = 24.2. These are not equal, so it's not linear. Now check the ratios: 220 / 200 = 1.10, 242 / 220 = 1.10, 266.2 / 242 = 1.10. The ratio is constant at 1.10 every month. That means the shirt sales grow by 10 % per month — constant percent growth, which is exponential.

Question 2: Problem 2 — Basic Calculation

A new smartphone costs $900. Its value decreases by 18 % each year. What is the decay factor, and what will the phone be worth after 1 year?

Show answer

The rate is r = 0.18. Since this is a decrease, the decay factor is b = 1 − 0.18 = 0.82. After 1 year: y = 900 × 0.82¹ = 900 × 0.82 = $738. The phone loses $162 in value during the first year.

Question 3: Problem 3 — Intermediate

A scientist counts 800 bacteria in a dish. The colony grows by 25 % every hour. Write an exponential model and find the population after 6 hours.

Show answer

Initial value a = 800. Rate r = 0.25, so b = 1 + 0.25 = 1.25. The model is y = 800 × 1.25ˣ, where x is hours. After 6 hours: y = 800 × 1.25⁶. Calculate 1.25⁶ ≈ 3.8147. So y ≈ 800 × 3.8147 ≈ 3,052 bacteria. The colony nearly quadrupled because the 25 % growth compounds rapidly.

Question 4: Problem 4 — Applied / Multi-Step

You invest $1,500 in an account that earns 6 % interest per year. Your friend puts $1,500 under a mattress and adds $100 cash per year. Write a model for each situation and determine who has more money after 10 years.

Show answer

Your model (exponential): y = 1,500 × 1.06ˣ. After 10 years: y = 1,500 × 1.06¹⁰ ≈ 1,500 × 1.7908 ≈ $2,686.27. Friend's model (linear): y = 1,500 + 100x. After 10 years: y = 1,500 + 100(10) = $2,500. After 10 years, you have about $186 more than your friend. The exponential growth from the 6 % interest eventually overtakes the steady $100 deposits. And the longer you wait, the bigger the gap becomes — that's the power of compounding.

Question 5: Problem 5 — Critical Thinking

The table below shows a car's value over several years. Determine whether the depreciation is linear, exponential, or neither. If it's exponential, state the annual percent rate of decay.

Year 0: $24,000 | Year 1: $20,400 | Year 2: $17,340 | Year 3: $14,739 | Year 4: $12,528.15

Show answer

First, check differences: 20,400 − 24,000 = −3,600; 17,340 − 20,400 = −3,060; 14,739 − 17,340 = −2,601. The differences are not constant, so it's not linear. Now check ratios: 20,400 / 24,000 = 0.85; 17,340 / 20,400 = 0.85; 14,739 / 17,340 = 0.85; 12,528.15 / 14,739 = 0.85. Every ratio is 0.85. This means the car's value follows exponential decay with a factor of b = 0.85. Since b = 1 − r, we get r = 0.15, or a 15 % annual depreciation rate. The model is y = 24,000 × 0.85ˣ. This is a great example of how real-world depreciation often follows constant percent decay rather than a fixed dollar loss.

Summary

Lesson Summary

Summary

In this lesson you learned how to recognize situations involving constant percent change — the defining feature of exponential functions. A quantity exhibits exponential growth or decay when it is multiplied by the same factor b every time period. If b > 1, the quantity grows; if 0 < b < 1, it decays. You can find b by computing 1 + r for growth or 1 − r for decay, where r is the percent rate expressed as a decimal. The general model is y = a · bˣ, and the quickest way to spot this pattern in a table is the ratio test: divide each value by the one before it, and if the ratio stays constant, you're looking at exponential behavior.

Compared to linear functions, which change by a constant amount, exponential functions change by a constant percentage. This distinction shows up in everyday language — look for words like "percent," "doubles," "halves," "rate," and "factor." Mastering this skill prepares you for compound interest, population modeling, radioactive decay, and every other application built on the idea that a quantity's change depends on its current size.