The Exponential Function and the Number e
Treat y = aˣ as a family of curves: always positive, multiplied by a fixed factor per unit step, with a horizontal asymptote. One member, y = eˣ, has a gradient equal to its own value everywhere.
What a learner can do afterwards
- Describe the shape and asymptote of y = 2ˣ and of y = 2⁻ˣ
- Say what makes e different from any other base
- Use a model of the form A e^(kt) to describe growth and decay
1 · Read
Each step multiplies by the same base. With base 2 the values run 1, 2, 4, 8. Every curve stays positive and flattens toward the x axis on the left. That floor line is the asymptote y equals 0.
Compare y equals 2 to the x with y equals 2 to the minus x. The first climbs to the right, the second decays to the right. Flipping the sign of x mirrors the graph. Both share the same asymptote.
One base is special. The number e is about 2.718, and y equals e to the x has a gradient equal to its own value everywhere. At the y axis crossing the gradient is 1. That property is what makes e different from every other base.
The model A e to the k t fits growth and decay in proportion to size. A sets the starting amount and k sets the pace. A positive k climbs, a negative k falls toward zero. Spotting A and k in a story is half the battle.
Fixed factor per step, asymptote at zero, and e as the base whose gradient matches its value.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.