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.
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.
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)
Constant Percent (Exponential)
Growth Factor
Decay Factor
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.
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.
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.
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).
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.
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.
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 |
Worked Example
Let's walk through a complete problem from start to finish.
y = 12,000 × 1.03^xLinear 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?
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 · ekt |
| 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.
Practice Problems
Try these five problems. They start simple and build up. Click "Show Answer" when you're ready to check your thinking.
Year 0: $24,000 | Year 1: $20,400 | Year 2: $17,340 | Year 3: $14,739 | Year 4: $12,528.15
Lesson 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.