TwiddleLab / Math / Growth: Exponents and Logs
Growth Curves and Percent Growth
What you will learn
- An exponential function is y = a · bˣ: start at a, multiply by b every step.
- Percent growth is a growth factor: +5% a year means b = 1.05, −10% means b = 0.9.
- Adding the same amount gives a line. Multiplying by the same factor gives a curve that wins in the end.
How the lesson goes
Your savings grow 5% a year. Is that the same as +$5 a year?
Both jars start with $100. Bet how much the 5% jar holds after 50 years.
Race the two jars.
Press Play and watch 50 years go by.
Find the growth factor of the 5% jar.
Your curve is y = 100 · bˣ. Turn b until it lies on the dashed curve.
Growing by 5% means you keep the whole amount and add 5% of it: 1 + 0.05 = 1.05 times as much each year.
y = a · bˣ
An exponential function y = a · bˣ starts at a and multiplies by b every step.
The growth factor b = 1 + r, where r is the percent change: +5% gives 1.05, −10% gives 0.9.
b > 1 grows, b = 1 stays flat, b < 1 shrinks.
Fit a curve to a growing town.
Drag the two blue handles so the curve passes through the points. Then read a and b.
Two points fix an exponential curve: the value at x = 0 is a, and the value at x = 1 divided by a is the factor b.
Line or curve?
Drag each table to the bin that describes it. Its graph appears in the bin.
Game: turn $100 into $1,000 in 20 years.
Pick a yearly rate and press Try. Land in the green box. Fewer tries, more stars.
This savings program should grow 10% a year.
It prints 101.1, 102.2, 103.3. Change one sign so it prints 110, 121, 133.1.
Growing by 10% means multiplying by 1.1 each year. Adding 1.1 is linear growth of $1.10 a year.
$1,000 plus $100 a year, or $500 growing 10% a year?
Move the year lever to the first year when the 10% offer is ahead, then Check.
The +$100 offer adds the same amount every year. The 10% offer adds 10% of a growing amount, so its yearly gain keeps rising until it passes, in year 19.
Previous lesson: Half Steps: Roots as Powers · Next lesson: Half-Life: Growth in Reverse · Part of the course: Growth: Exponents and Logs
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