Exponential growth and decay

An SAT Math micro-topic under Quadratic and exponential word problems (Advanced Math). Free to read — no account needed.

Exponential growth and decay follow the model

y = a \times b^t

: a starting amount

a

multiplied by a factor

b

once per period

t

. The single number

b

decides everything about the shape.

The graph shows both cases. When

b > 1

the curve rises faster and faster (growth); when

0 < b < 1

it falls toward zero (decay). Both pass through the starting amount

a

t = 0

To get the factor from a rate, add a percent increase to

1

or subtract a percent decrease from

1

. Growth of

8\%

per year means

b = 1.08

; decay of

15\%

per year means keeping

85\%

, so

b = 0.85

When the period is not one unit of time, put the number of periods in the exponent. If a population doubles every

3

years, then after

t

years it has doubled

\frac{t}{3}

times, giving

p \times 2^{\frac{t}{3}}

Worked examples

A colony of

500

bacteria triples every hour. How many are there after

2

hours?
Multiply by

3

each hour:

500 \times 3^2 = 500 \times 9 = 4500

A $1200 laptop loses

15\%

of its value each year. What is it worth after

2

years?
Keeping

85\%

each year means

b = 0.85

1200 \times 0.85^2 = 1200 \times 0.7225 = 867

Does

g(x) = 8 \times 0.75^x

represent growth or decay?
The base is

0.75

, and since

0 < 0.75 < 1

, the quantity shrinks each step — it is exponential decay.

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