📈 Precalculus

What you’ll learn

1. What a Function IsDefine functions, domain, and range, and move between formula, table, and graph.A function assigns exactly one output to each input; its domain is the allowed inputs and its range the produced outputs. Descartes's coordinate plane turns every function into a curve, checkable by the vertical line test.
2. Linear Functions: Constant ChangeInterpret slope and intercept as rate and starting value, and model steady change.Linear functions y = mx + b change by a constant amount m per unit input from a starting value b. Slope is rise over run — the language of road grades, ramps, fares, and any steady rate.
3. Quadratics and ParabolasGraph parabolas from vertex form, solve quadratics with the formula, and connect parabolas to projectiles and dishes.Quadratics graph as parabolas — the path of projectiles under constant gravity and the shape that focuses parallel rays to a point in dishes and headlights. Vertex form exposes the turning point, and the quadratic formula with its discriminant solves every quadratic equation.
4. Polynomials and End BehaviorClassify polynomials by degree and predict end behavior and roots.Degree caps the number of roots and turning points, and the leading term alone dictates end behavior: odd degrees run the ends opposite ways, even degrees together. Factored form hands over roots directly since a product is zero when any factor is.
5. Rational Functions and AsymptotesLocate vertical and horizontal asymptotes and connect rational functions to inverse-square laws.Ratios of polynomials blow up where the denominator vanishes (vertical asymptotes) and settle toward leading-term ratios in the long run (horizontal asymptotes). Inverse-square intensity laws and average-cost curves are rational functions at work.
6. Transforming GraphsApply shifts, stretches, and reflections to parent graphs, in the right order.Four moves generate whole function families: inside changes act sideways, outside changes act vertically, and stretches conventionally precede shifts. Reading y = −2(x + 1)² + 5 as "flip, stretch 2, left 1, up 5" — and writing the reverse — is the core skill.
7. Composition and InversesCompose functions, determine when inverses exist, and verify inverses by composition.Composition chains functions in order-sensitive fashion, and an inverse runs a one-to-one function backward, mirroring its graph across y = x. The verification test is f⁻¹(f(x)) = x — previewing the exponential–logarithm pair.
8. Exponential Growth and DecayModel repeated multiplication with y = a·bˣ and meet the natural base e.Exponential functions grow (or decay) by a constant factor per step, which starts slowly and then explodes past any linear rule — chessboard rice and paper-folding make the point viscerally. Compounding ever more often leads to e ≈ 2.71828, Euler's natural base for continuous growth.
9. Logarithms: Taming Huge NumbersDefine logarithms as inverse exponentials and read log scales fluently.log_b x answers "b to what power gives x," turning multiplication into addition — the trick behind Napier's 1614 invention and centuries of slide rules. Earthquake magnitudes, decibels, pH, and musical octaves are all logarithmic scales compressing vast ranges.
10. Solving Exponential and Log EquationsSolve equations with unknowns in exponents and apply half-life and doubling-time reasoning.Taking a logarithm of both sides pulls the unknown down from the exponent, answering every "how long until" question for growth and decay. Half-life powers radiocarbon dating (C-14, 5,730 years) and the rule of 70 estimates doubling times mentally.
11. Trigonometry I: The Unit CircleDefine sine and cosine on the unit circle, use radians, and memorize special angle values.On the unit circle, any angle's point is (cos θ, sin θ), extending trigonometry to all angles and revealing its periodicity. Radians measure angles by arc length (180° = π), and the special values at 0°, 30°, 45°, 60°, 90° underpin everything that follows.
12. Trigonometry II: Sinusoidal ModelsControl amplitude, period, phase, and midline in y = A sin(Bx + C) + D and fit sinusoids to cycles.Four parameters tune a sine wave to any oscillation: A the swing, 2π/B the period, C the timing, D the midline. Fundy tides (12 h 25 min period, up to 16 m range) and annual daylight are textbook sinusoidal models used for real prediction.
13. Trigonometry III: Identities and TrianglesUse core identities and the laws of sines and cosines to solve any triangle.sin²θ + cos²θ = 1 falls straight out of the unit circle, and the angle-sum identities generate the rest of the standard toolkit. The laws of sines and cosines solve arbitrary triangles — the mathematics behind triangulation surveys, including the 1850s determination of Everest's height.
14. Sequences and SeriesDistinguish arithmetic, geometric, and recursive sequences and evaluate convergent geometric series.Arithmetic sequences add a constant difference; geometric ones multiply by a constant ratio; Fibonacci's recursion appears in sunflower spiral counts with successive ratios approaching the golden ratio 1.618. Infinite geometric series with |r| < 1 converge to a/(1 − r) — a first rigorous limit.
15. To the Edge of CalculusFrame the two central questions of calculus — instantaneous change and accumulation — as limits.Instantaneous rate is the limit of secant slopes as the second point slides in — the derivative; accumulation of vanishingly small changes is the integral. Newton and Leibniz built this machinery independently in the late 1600s, and every function family from this course is its raw material.

Questions this course answers

Grounded in trusted sources

• OpenStax Precalculus (Rice University) — primary reference for definitions and conventions
• René Descartes, La Géométrie (1637)
• John Napier, Mirifici Logarithmorum Canonis Descriptio (1614)
• Leonardo of Pisa (Fibonacci), Liber Abaci (1202)
• MacTutor History of Mathematics Archive (University of St Andrews) — Euler, Napier, Newton, Leibniz
• USGS — earthquake magnitude scales
• NOAA / Canadian Hydrographic Service — Bay of Fundy tides and harmonic prediction
• NIST — radiocarbon dating and physical constants

Every Wunder lesson is built from real, reputable sources — never invented.

Related Math courses