One Shape, Two Directions

You'll learnto read the two numbers in a · b^x, and to say what the curve does at both ends.

An exponential curve has one of the most disciplined shapes in mathematics: it climbs the whole way or falls the whole way, it never turns, and it never touches the line y = 0. This story takes the two numbers in a · b^x one at a time, swings the base from decay to growth, and zooms into the tail that gets close to zero forever without arriving. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · f(x) = a · b^x · a > 0, b > 0, b ≠ 1 · v3

a says where it starts · b says what each step multiplies by

Before you start

What you're looking at

One exponential curve with its whole-number points marked, then a family of them sharing a starting height.

The question

Which of the two numbers sets the starting height, and which one decides the direction?

Step 1 / 12