Exponential growth and decay

In a polynomial, x sits in the base: x², x³. An exponential function flips that — x moves up into the exponent: f(x) = bˣ. That one swap changes everything about how fast it grows.

Growth and decay

The base b (a positive number, b ≠ 1) decides the behavior:

• b > 1 → growth. Each step right multiplies the output by b. For 2ˣ, every +1 in x doubles y.
• 0 < b < 1 → decay. Each step right multiplies by a fraction, so the output shrinks. 2⁻ˣ = (½)ˣ halves each step.

Two features hold for every exponential:

• It passes through (0, 1), because b⁰ = 1 for any base.
• It has a horizontal asymptote at y = 0 — bˣ is always positive and eases toward zero, but never reaches it.

Why exponentials win the race

Compare 2ˣ to the polynomial x¹⁰⁰. For small x the polynomial is far bigger — but exponentials multiply while polynomials only add powers, so the exponential eventually overtakes any polynomial and never looks back. That runaway behavior is why "exponential growth" means fast.

The natural base e

One base shows up everywhere: e ≈ 2.718. It's the base for which the curve's steepness equals its own height at every point — the property that makes calculus with eˣ astonishingly clean. Continuous growth (compound interest, populations, radioactive decay) is written with e: A = A₀ · eᵏᵗ.

Why this matters

Anything that grows or shrinks in proportion to its current size is exponential: money at compound interest, a virus early in an outbreak, the charge draining from a capacitor. Recognizing the form bˣ tells you the rate itself is accelerating — the single most important pattern in applied math.