Exponential Functions Summary

Pre-Calculus · Axiom Academy

A recap of growth and decay, the natural base e , graph behavior, transformations, and the interest and modeling formulas. An exponential function (with ) multiplies by the constant base each step — so it eventually outgrows any linear function. The base sets the behavior: gives growth , gives decay . The graph hugs a horizontal asymptote it never reaches. The natural base arises from continuous compounding and is the most natural base for growth and decay. More frequent compounding earns more, but there is a ceiling — continuous compounding, A = Pe^ rt , is the limit. Transformations are predictable: shifts the graph and moves the asymptote to y = k . Core Concept The Exponential Function The variable lives in the exponent , not the base. The output is multiplied by the same factor b each time x increases by 1 — a constant ratio , in contrast to a line's constant difference. a : the initial value, the output when x = 0 . b : the growth or decay factor per unit step. The base alone decides direction. When the function increases without bound; when it decreases toward zero. The initial value a scales the curve but does not change whether it grows or decays. When to use: read off b to classify a model at a glance. Watch out for: a negative exponent flips a growth base into decay, e.g. 2^ -x = (1/2)^ x . Every basic exponential passes through (0, a) and approaches the horizontal asymptote y = 0 on one side. It is defined for all inputs but only ever produces positive outputs.

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