Growth vs Decay Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Growth vs Decay.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: f(t) = A \cdot b^t with b > 1 gives growth; 0 < b < 1 gives decay. The key is whether the multiplier per time step is above or below 1.

Common stuck point: Exponential growth eventually dominates any polynomial — b^x eventually overtakes x^{100} for any b > 1, no matter how large the polynomial degree.

Sense of Study hint: Ask yourself: what happens to the output when I double the input? If it multiplies by a fixed factor greater than 1, it is growth; less than 1, it is decay.

Worked Examples

Example 1

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

Solution

  1. 1
    (a) Base b=2>1: exponential growth. f(3)=4\cdot8=32.
  2. 2
    (b) Base b=0.5, 0<0.5<1: exponential decay. g(3)=100\cdot(0.5)^3=100\cdot0.125=12.5.
  3. 3
    Interpretation: (a) doubles with each unit increase; (b) halves with each unit increase.

Answer

(a) Growth, f(3)=32; (b) Decay, g(3)=12.5
For y=a\cdot b^x 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A population grows by 8\% per year. Starting at 5000, write the growth function and find the population after 10 years.

Example 2

hard
Show algebraically that f(x)=e^{-0.3x} is an exponential decay function by rewriting it in the form a\cdot b^x and confirming 0<b<1.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential function