Exponential function

An exponential function has the form f(x) = a · bˣ, where a is the initial value, b is the base (b > 0 and b ≠ 1), and the input variable x sits in the exponent. That placement is the defining feature: in a linear function the output changes by a constant *amount* each step, but in an exponential function it changes by a constant *factor*. If b > 1 the function models exponential growth; if 0 < b < 1 it models exponential decay.

Consider f(x) = 100 · 2ˣ. Each time x increases by 1, the output doubles: 100, 200, 400, 800. The graph passes through (0, a) — here (0, 100) — rises steeply to the right, and flattens toward the x-axis on the left without ever touching it. That line, y = 0, is the graph's horizontal asymptote (shifting the function vertically moves the asymptote with it). Decay functions like f(x) = 100 · (0.5)ˣ mirror this shape, falling toward the asymptote as x grows.

Exponential functions model real-world processes that compound: population growth, compound interest, and radioactive decay. Percent-change problems translate directly into the form a(1 + r)ˣ for growth or a(1 − r)ˣ for decay. The most important base in mathematics is e ≈ 2.718; in calculus, eˣ is the function that is its own derivative, which makes exponentials central to AP Calculus limits and derivative rules.

The SAT's Advanced Math domain tests recognizing, evaluating, and interpreting exponential functions — especially telling exponential from linear growth — while AP Calculus AB tests the special limits and derivatives of exponential functions, and competition math like the AMC leans on exponent rules and exponential equations.