Growth vs Decay Formula

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

The Formula

y=a⋅bx where b>1 is growth, 0<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=2x grows (doubles each step). y=(12)x decays (halves each step).

Notation

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

What This Formula Means

Exponential growth occurs when a quantity multiplies by a factor >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⋅bx: growth   ⟺  b>1 (f′>0); decay   ⟺  0<b<1 (f′<0); with lim⁡x→∞f(x)={∞b>100<b<1

Worked Examples

Example 1

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

Answer

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

First step

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

Full solution

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

Example 2

medium
A radioactive substance has a half-life of 10 years. Starting with 200 g, write the decay function and find the amount remaining after 35 years.

Example 3

medium
A car's value depreciates by 15% per year. If it is worth $24{,}000 today, what is its value after 6 years?

Common Mistakes

  • Reading b=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=0.97, not 0.03.
  • 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 b 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 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>1; decay factor 0<b<1. Growth rate r=b−1 (so b=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 b 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=0.8 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.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Exponents and Logarithms: Rules, Proofs, and Applications →
← Back to Growth vs Decay See All Examples Practice Problems