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Learn to identify, model, and compare exponential functions that describe quantities increasing or decreasing by a fixed percentage over time.
Long before algebra students graphed exponential curves, merchants, bankers, and natural philosophers observed a fundamental pattern: quantities that change by a fixed proportion at each step behave very differently from those that change by a fixed amount. A population of rabbits that doubles every season grows far faster than one that adds a fixed number of offspring each year. A loan accumulating interest "on the interest" spirals in ways that simple addition cannot capture. Understanding this distinction — between additive (linear) change and multiplicative (exponential) change — is one of the most powerful ideas in all of mathematics.
The central question this lesson addresses is deceptively simple: given a function, a table, an equation, or a real-world scenario, how do you recognize whether the quantity is growing or decaying by a fixed percentage? Answering that question lets you choose the right model, make accurate predictions, and compare different exponential processes on equal footing.
Before we can recognize percent growth and decay, we need precise definitions of the building blocks. Every exponential function expressing percent change shares a common anatomy, and learning to read that anatomy is the key to classification.
The graph below is the visual centerpiece of this lesson. It shows two exponential functions on the same coordinate plane: one representing 8% growth (green curve) and one representing 8% decay (red curve). Both start at the same initial value of 100. Notice how the growth curve accelerates upward while the decay curve flattens toward zero — yet neither ever reaches infinity or zero in finite time. The dashed blue line shows a linear function growing at 8 units per period for comparison.
Several observations emerge from this graph. First, the growth curve bends upward with increasing steepness — it is concave up. The decay curve also bends upward (concave up), but it is falling toward the horizontal axis as an asymptote — it approaches zero but never reaches it. Second, the linear function grows steadily, adding 8 units per period, and it actually starts out ahead of the growth curve for the first few periods. But eventually the exponential overtakes it, because each 8% step is applied to an ever-larger base. This is the hallmark of multiplicative versus additive change.
Every percent-growth or percent-decay scenario can be captured by a single general form. Mastering this equation and its parts lets you move fluently between word problems, tables, graphs, and algebraic expressions.
The factor b encodes the percent change. When a quantity changes by r percent per period (expressed as a decimal), the relationship between b and r takes one of two forms depending on whether the quantity is increasing or decreasing.
Notice that in both cases the rate r is a positive number. The direction of change — growth or decay — is determined entirely by whether you add or subtract r from 1. This is a common source of confusion: a "12% decay rate" means r = 0.12 and b = 0.88, not b = −0.12.
This extraction formula is especially useful when you are handed an equation like y = 500(0.83)ᵗ and asked to interpret it. The factor 0.83 is less than 1, so this is decay. The rate is |0.83 − 1| = 0.17, meaning the quantity decreases by 17% per period.
In practice, you will encounter percent-change scenarios in many disguises: word problems, data tables, equations, and graphs. The diagram below provides a decision flowchart for classifying any exponential function you meet.
Let's also organize the key signatures of growth versus decay in a reference table. When you see a new function or dataset, quickly checking these features will tell you which type you are dealing with.
| Feature | Percent Growth | Percent Decay |
|---|---|---|
| Factor b | b > 1 | 0 < b < 1 |
| Rate formula | r = b − 1 | r = 1 − b |
| Graph shape | Rising, concave up | Falling, concave up |
| Table pattern | Successive ratios > 1 | Successive ratios < 1 |
| End behavior (t → ∞) | y → ∞ | y → 0⁺ (horizontal asymptote) |
| Real-world examples | Compound interest, population growth, inflation | Radioactive decay, depreciation, cooling |
A new car is purchased for $28,000. Its value depreciates by 14% each year. Write an exponential model for the car's value, determine whether it represents growth or decay, state the rate, and find the car's value after 6 years.
b = 1 − r = 1 − 0.14 = 0.86 Since 0 < 0.86 < 1, this confirms we have exponential decay.V(t) = 28000 × (0.86)ᵗ Here V(t) represents the car's value in dollars after t years.0.86² = 0.7396 → 0.86³ ≈ 0.6361 → 0.86⁶ = (0.86³)² ≈ 0.6361² ≈ 0.4046 Now multiply: V(6) = 28000 × 0.4046 ≈ $11,329The exponential model y = a · bᵗ is elegant and widely applicable, but it has important boundaries. Comparing it with the linear model highlights both its power and its limitations.
| Criterion | Linear Model (y = mx + c) | Exponential Model (y = a·bᵗ) |
|---|---|---|
| Type of change | Constant amount per period | Constant percentage per period |
| Graph shape | Straight line | Curve (concave up) |
| Table test | Constant first differences | Constant ratios of successive terms |
| Long-term behavior | Grows/shrinks without bound at a steady pace | Growth accelerates; decay approaches zero asymptotically |
| Realistic for | Short-term, controlled processes | Populations, finance, radioactive decay — but may need logistic cap for growth |
| Key parameter | Slope m | Base b (growth/decay factor) |
A critical limitation of the exponential growth model is that it assumes unlimited resources. Real populations, for instance, eventually encounter carrying-capacity limits and transition to logistic growth. Similarly, exponential decay models a quantity that approaches zero but mathematically never reaches it, which is physically accurate for processes like radioactive half-life but less so for, say, the number of cookies on a plate.
The percent-change model y = a(1 ± r)ᵗ is actually a special case of the broader continuous exponential model that uses Euler's number e ≈ 2.71828 as the base. In more advanced courses (Precalculus and Calculus), you will see that any exponential function can be rewritten in terms of e:
The connection is straightforward: since b = ek, we can always convert between the two forms. The continuous model is preferred in calculus because its derivative is proportional to itself — the rate of change at any instant is proportional to the current value, which is the mathematical definition of exponential behavior.
| Feature | Discrete: y = a·bᵗ | Continuous: y = a·eᵏᵗ |
|---|---|---|
| When to use | Percent change per fixed period | Continuous compounding or calculus |
| Growth indicator | b > 1 | k > 0 |
| Decay indicator | 0 < b < 1 | k < 0 |
| Conversion | b = eᵏ | k = ln(b) |
| Half-life / doubling time | t = ln(2) / |ln(b)| | t = ln(2) / |k| |
Another important extension is the logistic model, y = L / (1 + Ce−kt), which starts out looking exponential but levels off at a carrying capacity L. If you continue in mathematical biology or economics, you will see this model emerge naturally when resources are limited. For now, recognizing that the simple percent-change model is the first building block toward these richer models provides valuable perspective.
Recognizing percent growth and decay begins with understanding that any function of the form y = a · bᵗ describes a quantity changing by a constant percentage over equal time intervals. The growth/decay factor b is the single most important number to examine: if b > 1, the function models exponential growth at a rate of r = b − 1; if 0 < b < 1, it models exponential decay at a rate of r = 1 − b. The initial value a sets the starting point, and the exponent t counts the number of periods elapsed.
To classify a scenario, look for the constant-ratio signature in tables (divide each output by the previous one) or the characteristic concave-up curve on graphs. In word problems, phrases like "increases by ___%" or "loses ___% of its value" signal exponential behavior, as opposed to linear phrases like "increases by ___ units." These percent-change models connect forward to continuous exponential functions using base e and to logistic models that incorporate carrying capacities — but the algebraic foundation you have built here, centered on reading and interpreting the factor b, remains the essential skill at every level.