Exponential Function Definition

Pre-Calculus · Axiom Academy

LESSON Exponential Function Definition What makes a function exponential isn't its size — it's that the variable rides in the exponent, so every step multiplies the output by the same factor. 1. The Variable Lives in the Exponent An exponential function has the form below. The base b is a fixed number; the variable x sits up in the exponent . That is the whole distinction — in a power function like x^2 the variable is the base and the exponent is fixed, which is a completely different shape. f(x) = 2^x — the base is fixed (2), the exponent x varies. g(x) = x^2 — the exponent is fixed (2), the base x varies. 2. The Coefficient a Is Where It Starts Set x = 0 . Since b^0 = 1 for any allowed base, . So the curve always crosses the y -axis at exactly a — the coefficient is the y -intercept, the starting value before any growth or decay happens. The graph passes through — so a = 5 is the y -intercept, and from there each step multiplies by b = 3 . 3. The Base Decides: Growth or Decay With fixed, the size of b alone sets the direction. If each step multiplies by something bigger than one, so the output grows . If each step multiplies by a fraction, so the output decays toward zero. Rises as x increases. Examples: . Falls toward zero as x increases. Examples: . 4. The Heart: Multiply by b Every Step

This is the written version of the interactive lesson above. See the full Pre-Calculus course.