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Precalculus: Foundations for Calculus

Mathematics

Functions are reorganized as the unifying framework for polynomial, rational, exponential, logarithmic, and trigonometric families, with fluency across multiple representations. New tools—radian measure, the unit circle, logarithms as inverses, sequences and series, vectors, parametric equations, and limits—build toward calculus and instantaneous rates of change.

What you'll learn

Domain and RangeThe domain is the complete set of allowable input values for a function, and the range is the complete set of output values the function produces.
Function Definition and NotationA relation where each input is paired with exactly one output, identifiable visually using the vertical line test.
The Unit Circle and Trigonometric ValuesThe unit circle defines sine, cosine, and tangent as coordinates and coordinate ratios of points on a circle of radius 1, extending trig functions from right triangles to all real number inputs.
Polynomial Function Behavior and GraphingMethods for predicting and sketching the global shape of a polynomial graph using its degree, leading coefficient, zeros, and their multiplicities.
Inverse Trigonometric FunctionsInverse trigonometric functions take a ratio as input and return the angle that produces that ratio, defined on restricted domains where the original trig functions are one-to-one.
Even, Odd, and Symmetry Properties of FunctionsFunctions whose graphs display mirror or rotational symmetry—even functions are symmetric about the y-axis, and odd functions are symmetric about the origin.
Function TransformationsShifts, stretches, compressions, and reflections applied to a parent function's graph, controlled by constants in a transformation equation.
Conceptual Introduction to LimitsA limit is the value that a function's output approaches as the input approaches a specific number, independent of whether the function actually reaches that value.
Parametric EquationsA pair of functions x = f(t) and y = g(t) that trace out a curve as a parameter t varies, capturing both the shape of the curve and the direction and speed of motion along it.
Fundamental Trigonometric IdentitiesEquations involving trigonometric functions that remain true for every value of the variable in their domain, derived from the unit-circle definitions of sine and cosine.
Polar Coordinates and GraphsA coordinate system that locates points by specifying a distance r from the origin and an angle θ measured counterclockwise from the positive x-axis, instead of using horizontal and vertical distances.
Exponential Functions and Growth ModelsFunctions where the variable appears in the exponent, producing output that changes by a constant multiplicative factor for equal input increments.
Rational Function Behavior and AsymptotesRational functions are ratios of polynomials where asymptotes and holes reveal what happens near undefined points and as inputs grow very large.
Graphs of Trigonometric FunctionsVisual representations of sine and cosine functions plotted on the coordinate plane, showing their periodic wave behavior and key features like zeros, maxima, and minima.
Angle Measure and RadiansA radian is a unit of angle measurement defined as the angle swept at a circle's center when the arc length equals the radius.
Solving Exponential and Logarithmic EquationsSolving these equations means finding unknown variables that appear as exponents or inside logarithms by using the inverse relationship between exponential and logarithmic operations.
Inverse FunctionsAn inverse function reverses the action of the original function, taking an output back to its corresponding input and satisfying the property that f^{-1}(f(x)) = x.
Rates of Change and Average Rate of ChangeAverage rate of change measures how much a function's output changes per unit change in input across an interval, computed as the slope of the secant line connecting two points on the graph.
Conic SectionsConic sections are the four curves—circle, ellipse, parabola, and hyperbola—formed when a plane intersects a double cone.
Geometric Sequences and SeriesA sequence in which each term is obtained by multiplying the previous term by a fixed nonzero number called the common ratio.
Solving Trigonometric EquationsA trigonometric equation uses periodic functions, which produce infinitely many solutions that are typically expressed with a general formula or found within a specified interval.
Transformations of Trigonometric FunctionsA general sine or cosine equation with parameters for amplitude, period, phase shift, and vertical shift models real periodic phenomena by stretching, compressing, and translating the basic trigonometric curve.
Increasing, Decreasing, and Extrema BehaviorThe vocabulary and intervals used to describe where a function rises, falls, and reaches its highest or lowest points across its domain.
Sequences as Discrete FunctionsA sequence is a function whose domain is restricted to positive integers, producing isolated points on a graph rather than a continuous curve.
Logarithmic Functions and Their PropertiesA logarithmic function is the inverse of an exponential function that answers the question of what exponent must be applied to a base to yield a given number.
Solving Polynomial and Rational Equations and InequalitiesSolving polynomial equations involves finding zeros using factoring, the Rational Root Theorem, synthetic division, and the quadratic formula to systematically reduce the problem to simpler cases.
Instantaneous Rate of Change and the Bridge to CalculusThe rate at which a function is changing at a single point, computed as the limit of average rates of change over progressively smaller intervals, motivating the calculus concept of the derivative.
Law of Sines and Law of CosinesThese are formulas that extend trigonometry beyond right triangles, enabling you to solve oblique triangles by relating their sides and angles using sine and cosine ratios.
Arithmetic Sequences and SeriesAn arithmetic sequence is a list of numbers where consecutive terms have a constant difference.
Introduction to VectorsA vector is a quantity that has both magnitude and direction, represented geometrically as an arrow whose length and orientation encode size and direction of travel, force, or displacement.
Function CompositionThe operation of using the output of one function as the input to another function, producing a chain of operations evaluated from inside out.
Polynomial Division and the Remainder/Factor TheoremsTechniques for dividing one polynomial by another to break down complicated polynomials into simpler factors, following a process parallel to long division of integers.
Sum, Difference, and Double-Angle FormulasThese formulas express the sine, cosine, and tangent of a sum or difference of angles in terms of the individual angles' trig values, allowing exact calculations for non-standard angles.

About this course

Why This Course Exists

You've spent Algebra I, Algebra II, and Geometry building a toolkit: solving equations, factoring, graphing lines and parabolas, working with right triangles. Precalculus takes that toolkit and reorganizes it around a single unifying idea — the function — while adding several genuinely new tools you'll need the moment you walk into a calculus classroom: radian measure, the unit circle, logarithms treated as inverses, sequences and series, vectors, parametric equations, and a first conceptual look at limits.

Think of precalculus as the last stop before calculus, and its job is to make sure two things are true when you arrive there. First, that you can fluently manipulate the algebraic families of functions — polynomial, rational, exponential, logarithmic, trigonometric — in multiple representations (equation, graph, table, words). Second, that you have enough exposure to the ideas calculus will formalize (rates of change, accumulation of small changes, behavior of a function as its input grows without bound or approaches a specific value) that when calculus introduces the derivative and the integral, you already have intuition to hang them on.

The Map of the Course

The course is organized into four broad regions, each building on the last.

1. Functions and Their Properties. You'll formalize what you already know about functions — domain, range, notation — and add precision: transformations (how shifting, stretching, and reflecting a graph relates to changes in its equation), composition (feeding one function's output into another), and inverses (undoing a function). This groundwork applies uniformly across polynomial, rational, exponential, and logarithmic functions, and later, trigonometric functions.

2. Trigonometric Functions via the Unit Circle. In geometry, sine and cosine were ratios inside a right triangle, defined only for angles between $0°$ and $90°$ . Precalculus breaks that restriction open: using the unit circle, you'll redefine sine and cosine as functions of any real number, measured in a new unit called the radian. This lets trigonometric functions model periodic phenomena — anything that repeats, like tides, sound waves, or the height of a point on a rotating wheel. You'll learn to graph these functions, transform them, prove and use trigonometric identities, and solve trigonometric equations.

3. Sequences and Series. You'll study ordered lists of numbers generated by a pattern — arithmetic sequences (constant difference) and geometric sequences (constant ratio) — and learn to add up their terms, including, in the geometric case, infinitely many terms. This is your first encounter with the idea of an infinite process producing a finite, well-defined answer, a theme that returns constantly in calculus.

4. Vectors, Parametric Equations, and Limits. You'll represent quantities that have both size and direction (vectors), describe curves using a parameter like time rather than a direct $x$ - $y$ relationship (parametric equations), and finally, conceptually approach the idea of a limit — what value a function gets arbitrarily close to — and use it to define instantaneous rate of change, the seed from which the entire derivative concept in calculus grows.

Here is how these pieces connect and build toward calculus:

Four regions of precalculus, each extending function concepts and each feeding into the concluding study of limits and rates of change.

What's Genuinely New Here

Since you're coming in comfortable with algebra and basic trigonometric ratios, it's worth being explicit about what you have not seen before, so you know where to slow down:

Radian measure and the unit circle — a new way of measuring angles and a new, more general definition of sine, cosine, and tangent that works for any real number input, not just triangle angles between $0°$ and $90°$ .
Formal function transformation and composition language — you've graphed shifted parabolas before, but not necessarily with the systematic notation $f(x-h)+k$ or composition notation $f(g(x))$ .
Logarithms as inverses of exponential functions — a new function family with its own graph, domain restrictions, and algebraic rules.
Sequences and series, especially the notion of summing infinitely many numbers to get a finite total.
Vectors and parametric equations — representing motion and direction rather than just static curves.
Limits — the conceptual tool that finally lets you talk rigorously about instantaneous rate of change, rather than only average rate of change over an interval.

How to Use This Course

Every new idea in this course will be paired with worked examples and practice, because precalculus is best learned by doing — graphing, solving, manipulating — not just reading definitions. Expect a rigorous but problem-solving-centered treatment: you will occasionally see why a rule is true (for example, why the Pythagorean identity holds), but the emphasis throughout is on being able to use each idea fluently to solve problems, communicate your reasoning clearly, and move comfortably between equations, graphs, tables, and words. By the end, you should walk into a calculus course — AP Calculus or its college equivalent — with the algebraic fluency and conceptual footing to focus on genuinely new calculus ideas, rather than getting stuck relearning precalculus along the way.