Exponential growth and decay | Precalculus

Linear growth adds the same amount each step. Exponential growth multiplies. At first the difference is small, but soon exponential leaves linear behind.

$x$	Linear: $2 + 3 x$	Exponential: $2 \cdot 3^{x}$
0	2	2
1	5	6
2	8	18
3	11	54
4	14	162

y = a \cdot b^{x}

Work through it

Rabbits: 40, doubling each year. After 3 years?

Start and multiplier. $a = 40$ , $b = 2$

Substitute. $40 \cdot 2^{3} = 320$

A $1000 deposit grows 5% a year. Write the model.

5% growth multiplies by 1.05. $y = 1000 (1.05)^{t}$

A substance halves every 8 hours. Start 80 g. After 24 hours?

Count the half-lives. $24 \div 8 = 3$

Halve three times. $80 \cdot (\frac{1}{2})^{3} = 10$ g

Watch out for this

5% growth is $1000 (0.05)^{t}$ .

That multiplies by 5%, and shrinks fast. You keep the 100% and add 5%: 1.05.

The short version

y = a b^{x}

a

is the start and

b

is the multiplier per step. Above 1 grows, and between 0 and 1 decays.

Your turn

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