Graph of y = 2^x
Interactive graph of y = 2^x (exponential base 2). Explore exponential growth with base 2, doubling behavior, and comparison to e^x.
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Understanding the Function
The function y = 2^x is the exponential function with base 2. Like e^x, it grows rapidly for positive x and approaches zero for negative x, but it doubles every time x increases by 1. This doubling behavior makes it the natural choice for modeling binary systems, computer science, and information theory.
Base-2 exponentials appear everywhere in computing: memory sizes (2^10 = 1024), algorithm complexity (binary search is O(log2 n)), and binary data representation. In biology, bacterial populations that divide in two follow 2^x growth. Moore's Law, which describes the doubling of transistor counts roughly every two years, is fundamentally a base-2 exponential trend.
Key properties: domain is all real numbers, range is (0, infinity), y-intercept at (0, 1), horizontal asymptote at y = 0, passes through (1, 2), (2, 4), (3, 8), (-1, 0.5). The derivative is 2^x * ln(2), approximately 0.693 * 2^x. It grows slower than e^x but faster than any polynomial.