Polynomials/rational/exponential/log functions - Mathematics: Analysis & Approaches IB Study Notes

Polynomial Functions are expressions of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where n is a non-negative integer and aₙ ≠ 0. The degree is the highest power (n), and the leading coefficient is aₙ. Key properties: continuous everywhere, smooth curves, end behavior determined by leading term.

Rational Functions have the form f(x) = P(x)/Q(x) where P and Q are polynomials. Critical features: vertical asymptotes occur where Q(x) = 0 (denominator zero), horizontal asymptotes depend on degree comparison (if deg P < deg Q: y = 0; if deg P = deg Q: y = aₙ/bₙ; if deg P > deg Q: no horizontal asymptote but possible oblique asymptote). Holes occur when common factors cancel.

Exponential Functions follow f(x) = abˣ or f(x) = aeᵏˣ where b > 0, b ≠ 1. Properties: always positive, horizontal asymptote at y = 0 (for standard form), growth (b > 1) or decay (0 < b < 1), domain ℝ, range (0, ∞).

Logarithmic Functions are inverses of exponentials: f(x) = logₐ(x) where a > 0, a ≠ 1. Essential laws: log(xy) = log x + log y, log(x/y) = log x - log y, log(xⁿ) = n log x, logₐ(a) = 1, logₐ(1) = 0. Domain (0, ∞), range ℝ, vertical asymptote at x = 0.

Mnemonic for log laws: "Multiplication → Addition, Division → Subtraction, Power → Multiplication" (MAD-PM).

The change of base formula: logₐ(x) = log_b(x)/log_b(a) = ln(x)/ln(a), essential for calculator use.

Polynomials in Action: The trajectory of a basketball follows a quadratic (degree 2 polynomial). If h(t) = -5t² + 10t + 2 models height in meters, the parabolic path shows the ball rising then falling. The coefficient -5 represents gravitational acceleration's effect, demonstrating how degree and leading coefficient control shape.

Rational Functions Model Reality: Enzyme reaction rates in biochemistry follow the Michaelis-Menten equation v = (Vₘₐₓ[S])/(Kₘ + [S]), a rational function. As substrate concentration [S] increases, velocity approaches the horizontal asymptote Vₘₐₓ (maximum rate), never exceeding it—perfectly modeling biological saturation.

Exponential Growth Everywhere: Population growth follows P(t) = P₀e^(rt) where P₀ is initial population, r is growth rate, and e ensures continuous compounding. If bacteria double every hour, we use P(t) = P₀(2)^t. Moore's Law (computing power doubles every 18 months) is exponential: C(t) = C₀(2)^(t/1.5). The key insight: exponential functions grow by constant percentages per unit time, not constant amounts.

Logarithms Measure Scale: The Richter scale for earthquakes uses M = log(I/I₀), where I is intensity. A magnitude 7 earthquake is 10 times stronger than magnitude 6 (log properties!). pH in chemistry equals -log[H⁺], compressing huge concentration ranges (0.0000001 M to 1 M) into manageable numbers (7 to 0).

Visual Analogy: Think of exponentials as rockets accelerating faster and faster, while logarithms are speedometers that slow down, rising quickly initially then leveling off. They're inverse operations, like acceleration vs. deceleration.