Exponential functions

Say you have 100 followers. Gaining 50 a week is linear: 100, 150, 200, 250. Growing by 50% a week is exponential: 100, 150, 225, 337.5. They start the same, and then one runs away.

y = 2^{x}

in orange and

y = x + 1

in navy. Exponential growth always wins eventually.

y = a b^{x}

a

is the value at

x = 0

b > 1

means growth, and

0 < b < 1

means decay.

A percent change turns into a multiplier. Growing 8% means multiplying by 1.08. Shrinking 8% means multiplying by 0.92.

Work through it

A $500 deposit earns 4% per year. Write its value after $t$ years.

The start is 500. $a = 500$

Growing 4% means times 1.04. $b = 1.04$

Put it together. $V = 500 (1.04)^{t}$

A car worth $20,000 loses 15% of its value each year. Write its value after $t$ years.

The start is 20,000. $a = 20000$

Losing 15% leaves 85%. $b = 0.85$

Put it together. $V = 20000 (0.85)^{t}$

Watch out for this

Writing a 4% growth rate as $b = 0.04$ or $b = 4$ .

Keep the original 100% and add 4%: $b = 1.04$ .

The short version
In

y = a b^{x}

a

is the starting value and

b

is what you multiply by for each step.

Your turn

Prefer paper? Print the Radicals and exponential functions worksheet, with an answer key.

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Work through it

A $500 deposit earns 4% per year. Write its value after t years.

A car worth $20,000 loses 15% of its value each year. Write its value after t years.

Watch out for this

Your turn

A $500 deposit earns 4% per year. Write its value after $t$ years.

A car worth $20,000 loses 15% of its value each year. Write its value after $t$ years.