Exponential Growth & Decay

From bacteria doubling to radioactive decay — explore the power of e^(kx)

Population growth, compound interest, radioactive decay, viral spread — the most dramatic changes in nature and finance all follow exponential patterns. Exponential functions describe quantities that grow or shrink by a constant percentage in each time step, rather than a constant amount. This makes them fundamentally different from linear or polynomial growth — and far more powerful (or dangerous) over time.

The base of the natural exponential function is e ≈ 2.718, a special number that arises naturally in calculus, finance, and physics. The function y = ekx models growth when k > 0 and decay when k < 0.

In this lesson, you'll manipulate a slider to see how the growth constant k transforms the exponential curve, compare different exponential bases, and connect the math to real-world phenomena like compound interest, population growth, and radioactive decay — with an AI tutor guiding you step by step.

Related Topics

Logarithmic FunctionsQuadratic EquationsPower Functions

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What is exponential growth?
Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. The function y = e^{kx} with k > 0 models this behavior. Unlike linear growth (which adds the same amount each step), exponential growth multiplies by the same factor — so it starts slow but accelerates dramatically. A classic example: bacteria that double every hour start with 1 and reach over a million in just 20 hours.
What is the difference between exponential and polynomial growth?
Polynomial functions like x^2 or x^3 grow by increasing powers of x, but exponential functions like e^x grow by putting x in the exponent. Eventually, any exponential function with base > 1 will overtake any polynomial — no matter how high the polynomial's degree. For example, e^x eventually surpasses x^{100}.
What is the number e?
The number e ≈ 2.71828... is a mathematical constant that appears naturally when you study continuous growth. It can be defined as the limit of (1 + 1/n)^n as n approaches infinity. It is the unique base where the exponential function equals its own derivative: if f(x) = e^x, then f'(x) = e^x as well.
What are real-world examples of exponential functions?
Exponential functions model many real phenomena: compound interest (money grows exponentially when interest is reinvested), population growth (bacteria, viruses, or human populations under ideal conditions), radioactive decay (atoms decay at a rate proportional to how many remain), cooling/heating (Newton's law of cooling), and drug metabolism (medications leave the bloodstream exponentially).
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.