Exponential functions
Some things grow by adding the same amount each step. Others grow by multiplying. That difference decides whether a savings account, a viral post, or a medicine dose ends up small or enormous. Work through the five parts below and change every slider.
1. Adding versus multiplying
Pick a deal: a penny that doubles every day, or $1,000 added every day. Drag the slider to see which one wins.
Doubling is exponential growth: the amount after x days is 0.01 · 2x. Early on it looks timid. Late on it takes over.
2. The chessboard test
An old story tells of a reward: one grain of rice on the first chessboard square, two on the second, four on the third, doubling on every square after. Meanwhile someone offers you a big pile of rice right now. Choose the size of the pile, then fill the board.
Green squares hold more grains than the whole up-front pile. The orange outline marks the first square that does. Each square holds more than all the squares before it combined, so the last squares dwarf everything else.
3. The anatomy of y = a · bx
Two numbers shape every exponential function. a is the starting value (where the curve crosses the y-axis). b is the growth factor: each time x goes up by 1, y is multiplied by b.
Try b = 1. Then nothing changes, so the graph is a flat line. Push b below 1 and the curve falls toward zero without ever touching it.
4. Compound interest
Money that earns interest on its own interest follows y = P · (1 + r)t. Here a is the deposit P, and b is 1 + r.
The rule of 72 is a mental shortcut: divide 72 by the rate to estimate the years to double. Compare it with the exact answer above.
5. Check yourself
Practice
Choose a level. Every question uses new numbers, so you can keep going as long as you like. A calculator is fine. Hints are available, but a question answered after a hint does not extend your streak. Changing the level resets your score.