Growth vs Decay Formula

Growth vs decay is exponential growth occurs when a quantity multiplies by a factor > 1 repeatedly.

The Formula

y=a⋅bxy = a \cdot b^x where b>1b > 1 is growth, 0<b<10 < b < 1 is decay

When to use: Growth compounds: each period's increase is larger than the last. Decay shrinks: each period's decrease is smaller than the last, never quite reaching zero.

Quick Example

y=2xy = 2^x grows (doubles each step). y=(12)xy = \left(\frac{1}{2}\right)^x decays (halves each step).

Notation

Growth factor b>1b > 1; decay factor 0<b<10 < b < 1. Growth rate r=b−1r = b - 1 (so b=1+rb = 1 + r).

What This Formula Means

Exponential growth occurs when a quantity multiplies by a factor >1> 1 repeatedly; exponential decay when it multiplies by a factor between 0 and 1.

Growth compounds: each period's increase is larger than the last. Decay shrinks: each period's decrease is smaller than the last, never quite reaching zero.

Formal View

f(x)=a⋅bxf(x) = a \cdot b^x: growth   ⟺  b>1\iff b > 1 (f′>0f' > 0); decay   ⟺  0<b<1\iff 0 < b < 1 (f′<0f' < 0); with lim⁡x→∞f(x)={∞b>100<b<1\lim_{x \to \infty} f(x) = \begin{cases} \infty & b > 1 \\ 0 & 0 < b < 1 \end{cases}

Worked Examples

Example 1

easy
Classify each function as growth or decay, and find its value at x=3x=3: (a) f(x)=4⋅2xf(x)=4\cdot2^x, (b) g(x)=100⋅(0.5)xg(x)=100\cdot(0.5)^x.

Answer

(a) Growth, f(3)=32f(3)=32; (b) Decay, g(3)=12.5g(3)=12.5

First step

1
(a) Base b=2>1b=2>1: exponential growth. f(3)=4⋅8=32f(3)=4\cdot8=32.

Full solution

  1. 2
    (b) Base b=0.5b=0.5, 0<0.5<10<0.5<1: exponential decay. g(3)=100⋅(0.5)3=100⋅0.125=12.5g(3)=100\cdot(0.5)^3=100\cdot0.125=12.5.
  2. 3
    Interpretation: (a) doubles with each unit increase; (b) halves with each unit increase.
For y=a⋅bxy=a\cdot b^x with a>0a>0: if b>1b>1 the function grows exponentially; if 0<b<10<b<1 it decays exponentially. The base bb determines direction; the coefficient aa sets the initial value.

Example 2

medium
A radioactive substance has a half-life of 1010 years. Starting with 200200 g, write the decay function and find the amount remaining after 3535 years.

Example 3

medium
A car's value depreciates by 15%15\% per year. If it is worth $24{,}000 today, what is its value after 66 years?

Common Mistakes

  • Reading b=0.8b=0.8 as growth because 0.8 is positive - any base below 1 (but above 0) is decay.
  • Confusing the base with the rate - convert percents: 3% decay is b=0.97b=0.97, not 0.030.03.
  • Treating a fixed-amount-per-period change as exponential - that's linear; exponential needs a fixed multiplier.

Why This Formula Matters

This is the core read on every exponential model: population, interest, radioactive half-life, and depreciation all hinge on whether bb is above or below 1. Confusing the base with a growth rate, or exponential with linear, sends a student to the wrong model entirely. Recognizing it by "Is the quantity multiplied by the same factor each period, and is that factor above or below 1?" — rather than by familiar numbers — is what lets a student tell it apart from linear growth and growth factor vs. growth rate and saturation / logistic growth in a mixed problem set.

Frequently Asked Questions

What is the Growth vs Decay formula?

Exponential growth occurs when a quantity multiplies by a factor >1> 1 repeatedly; exponential decay when it multiplies by a factor between 0 and 1.

How do you use the Growth vs Decay formula?

Growth compounds: each period's increase is larger than the last. Decay shrinks: each period's decrease is smaller than the last, never quite reaching zero.

What do the symbols mean in the Growth vs Decay formula?

Growth factor b>1b > 1; decay factor 0<b<10 < b < 1. Growth rate r=b−1r = b - 1 (so b=1+rb = 1 + r).

Why is the Growth vs Decay formula important in Math?

This is the core read on every exponential model: population, interest, radioactive half-life, and depreciation all hinge on whether bb is above or below 1. Confusing the base with a growth rate, or exponential with linear, sends a student to the wrong model entirely. Recognizing it by "Is the quantity multiplied by the same factor each period, and is that factor above or below 1?" — rather than by familiar numbers — is what lets a student tell it apart from linear growth and growth factor vs. growth rate and saturation / logistic growth in a mixed problem set.

What do students get wrong about Growth vs Decay?

The procedure for growth vs decay is the easy part; the trap is reading b=0.8b=0.8 as growth because 0.8 is positive. Asking "Is the quantity multiplied by the same factor each period, and is that factor above or below 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Growth vs Decay formula?

Before studying the Growth vs Decay formula, you should understand: exponential function.

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Exponents and Logarithms: Rules, Proofs, and Applications →
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