Power functions reveal how small changes in inputs explode or dampen outputs depending on the exponent.
Graph plots y = xᵖ for p = 1/8, 1/4, 1/2, 1, 2, 4, 8 over [0,1]. For x ∈ [0,1], exponents <1 curve concave down (boosting low values), =1 is linear, and >1 curve concave up (suppressing low values before rising sharply). All pass through (0,0) and (1,1).
It is used in models diminishing returns in economics, probability weighting in behavioral finance, signal compression in audio engineering, and growth curves in biology/medicine.