United States · College BoardSyllabus
Precalculus syllabus, dot point by dot point
Every dot point in the United States Precalculussyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Unit 1: Polynomial and Rational Functions
Module overview →- How do two quantities change together, and how do we describe whether one increases or decreases as the other does?Topic 1.1 Change in Tandem: describe how the output values of a function change as the input values change, using increasing or decreasing behavior, concavity, and the relationship shown in a graph, table or context.9 min answer →
- How can the same polynomial or rational expression be rewritten so that each form reveals a different feature?Topic 1.11 Equivalent Representations of Polynomial and Rational Expressions: rewrite polynomial and rational expressions in equivalent forms using factoring, the binomial theorem and polynomial long division to reveal zeros, asymptotes and end behavior.10 min answer →
- How do you build an explicit polynomial or rational function from a context or data, and use it to answer questions?Topic 1.14 Function Model Construction and Application: construct a polynomial or rational function model from a context, restricted domain or data set, and apply it to make predictions and solve problems.10 min answer →
- How do you choose an appropriate function model for a situation, and what assumptions does that choice make?Topic 1.13 Function Model Selection and Assumption Articulation: select an appropriate type of function to model a data set or context, and articulate the assumptions and limitations of the chosen model.9 min answer →
- How do the real and complex zeros of a polynomial, and their multiplicities, determine its factored form and graph?Topic 1.5 Polynomial Functions and Complex Zeros: relate the real and non-real complex zeros of a polynomial to its factored form, degree and graph, including the effect of multiplicity.10 min answer →
- How do the degree and leading coefficient of a polynomial determine what its graph does at the far left and far right?Topic 1.6 Polynomial Functions and End Behavior: determine the end behavior of a polynomial from its degree and leading term, and express that behavior using limit notation.9 min answer →
- How do the degree, local extrema and points of inflection of a polynomial describe the way it changes?Topic 1.4 Polynomial Functions and Rates of Change: relate the degree of a polynomial to its number of local extrema and points of inflection, and analyze where its rate of change is positive, negative or zero.9 min answer →
- What distinguishes the rate-of-change behavior of linear functions from that of quadratic functions?Topic 1.3 Rates of Change in Linear and Quadratic Functions: characterize linear functions by their constant rate of change and quadratic functions by their constant rate of change of the rate of change.9 min answer →
- How do we measure how fast a function changes, both over an interval and at a single point?Topic 1.2 Rates of Change: compute and interpret the average rate of change of a function over an interval, and estimate the rate of change at a point.9 min answer →
- What does the graph of a rational function do at its far ends, and when is there a horizontal or slant asymptote?Topic 1.7 Rational Functions and End Behavior: determine the end behavior of a rational function by comparing the degrees of its numerator and denominator, identifying horizontal or slant asymptotes.9 min answer →
- What creates a hole in the graph of a rational function, and how do you find its location?Topic 1.10 Rational Functions and Holes: identify removable discontinuities (holes) of a rational function from factors common to numerator and denominator, and find the coordinates of each hole.8 min answer →
- Where does a rational function shoot off to infinity, and how do you describe that behavior with limits?Topic 1.9 Rational Functions and Vertical Asymptotes: locate the vertical asymptotes of a rational function from the zeros of the denominator that do not cancel, and describe the behavior with one-sided limits.9 min answer →
- Where does a rational function equal zero, and how do the numerator's zeros relate to the graph?Topic 1.8 Rational Functions and Zeros: determine the real zeros of a rational function from the zeros of its numerator, accounting for values excluded by the denominator.8 min answer →
- How do shifts, stretches and reflections change the equation and graph of a function?Topic 1.12 Transformations of Functions: construct and analyze additive and multiplicative transformations (translations, dilations and reflections) of a function and their effect on the graph, domain and range.10 min answer →
Unit 2: Exponential and Logarithmic Functions
Module overview →- How do arithmetic and geometric sequences differ in the way their terms change?Topic 2.1 Change in Arithmetic and Geometric Sequences: define arithmetic sequences by a constant common difference and geometric sequences by a constant common ratio, and write their explicit and recursive forms.9 min answer →
- What is the fundamental difference between linear change and exponential change?Topic 2.2 Change in Linear and Exponential Functions: contrast linear functions, which change by a constant amount, with exponential functions, which change by a constant percentage or factor over equal-length input intervals.9 min answer →
- When two models both fit data, how do you decide which is better using residuals?Topic 2.6 Competing Function Model Validation: compare competing function models for a data set by analyzing residuals, and validate or reject a model based on the pattern and size of its residuals.9 min answer →
- How do you combine two functions by composition, and what is the domain of the result?Topic 2.7 Composition of Functions: form the composition of two functions, evaluate it, decompose a composite function, and determine the domain of a composition.9 min answer →
- How do you solve equations and inequalities that have the variable in an exponent or inside a logarithm?Topic 2.13 Exponential and Logarithmic Equations and Inequalities: solve exponential and logarithmic equations and inequalities using inverse operations, the logarithm properties, and checks for extraneous solutions.10 min answer →
- How do you build an exponential model from a context or data, and interpret its parameters?Topic 2.5 Exponential Function Context and Data Modeling: construct an exponential model from a context or data set, interpret the initial value and growth or decay factor, and use the model to make predictions.9 min answer →
- How do the exponent rules let you rewrite an exponential expression into a more useful equivalent form?Topic 2.4 Exponential Function Manipulation: rewrite exponential expressions using the product, power, negative-exponent and rational-exponent properties to reveal equivalent forms.9 min answer →
- What are the defining features of an exponential function, and how do its base and initial value shape its graph?Topic 2.3 Exponential Functions: define exponential functions, describe how the base and initial value determine growth or decay, and analyze the domain, range and horizontal asymptote of the graph.9 min answer →
- What is the inverse of a function, and how do you find and verify it?Topic 2.8 Inverse Functions: determine whether a function has an inverse, find the inverse by swapping input and output, and verify an inverse using composition and the reflection over the line y = x.9 min answer →
- Why is the logarithm the inverse of the exponential, and what does that reveal about its graph?Topic 2.10 Inverses of Exponential Functions: construct the inverse of an exponential function as a logarithmic function, and relate the graph, domain and range of each to the other.8 min answer →
- What does a logarithm mean, and how do you evaluate logarithmic expressions?Topic 2.9 Logarithmic Expressions: define a logarithm as the exponent that produces a given value, and evaluate logarithmic expressions by rewriting them in exponential form.8 min answer →
- When is a logarithmic model appropriate, and how do you build and interpret one?Topic 2.14 Logarithmic Function Context and Data Modeling: construct a logarithmic model from a context or data set, interpret its parameters, and use it to make predictions.9 min answer →
- How do the logarithm properties let you expand or condense a logarithmic expression?Topic 2.12 Logarithmic Function Manipulation: rewrite logarithmic expressions using the product, quotient, power and change-of-base properties to expand or condense them.9 min answer →
- What are the defining features of a logarithmic function and its graph?Topic 2.11 Logarithmic Functions: analyze the parent logarithmic function and its transformations, including its domain, range, vertical asymptote, and increasing or decreasing behavior.9 min answer →
- How does plotting data on a semi-log scale reveal whether it is exponential, and how do you read such a plot?Topic 2.15 Semi-log Plots: use a semi-log plot to determine whether an exponential model is appropriate, and interpret the slope and intercept of the resulting line.9 min answer →
Unit 3: Trigonometric and Polar Functions
Module overview →- How do trigonometric identities let you rewrite an expression in an equivalent form?Topic 3.12 Equivalent Representations of Trigonometric Functions: use the Pythagorean, sum and difference, and double-angle identities to rewrite trigonometric expressions in equivalent forms.9 min answer →
- How are the inverse trigonometric functions defined on restricted domains, and what are their ranges?Topic 3.9 Inverse Trigonometric Functions: define arcsine, arccosine and arctangent on restricted domains, and evaluate and interpret their outputs.9 min answer →
- What makes a relationship periodic, and how do you read its period, amplitude and midline from a graph or context?Topic 3.1 Periodic Phenomena: identify a periodic relationship, and describe its period, amplitude and key features from a graph, table or context.9 min answer →
- What are polar coordinates, and how do you convert between polar and rectangular form?Topic 3.13 Trigonometry and Polar Coordinates: locate points using polar coordinates and convert between polar and rectangular coordinates.9 min answer →
- How do you graph a polar function r = f(theta), and what shapes do the standard polar functions produce?Topic 3.14 Polar Function Graphs: construct and interpret the graph of a polar function r = f(theta), including circles, roses, limacons and spirals.9 min answer →
- How does the radius of a polar function change as the angle increases, and what does the average rate of change tell you?Topic 3.15 Rates of Change in Polar Functions: analyze how r changes as theta increases, using the average rate of change to describe whether the curve moves toward or away from the pole.9 min answer →
- What do the graphs of sine and cosine look like, and how do their period, amplitude, midline and concavity arise from the unit circle?Topic 3.4 Sine and Cosine Function Graphs: construct and interpret the graphs of sine and cosine, identifying period, amplitude, midline, zeros and concavity.9 min answer →
- How do sine and cosine values move around the unit circle, and how do symmetry and the Pythagorean identity connect them?Topic 3.3 Sine and Cosine Function Values: determine sine and cosine values using the unit circle, reference angles, symmetry and the Pythagorean identity.9 min answer →
- How are sine, cosine and tangent defined on the unit circle, and how do they relate to right-triangle ratios?Topic 3.2 Sine, Cosine, and Tangent: define sine, cosine and tangent using the unit circle and right-triangle ratios, and evaluate them at key angles.9 min answer →
- How do you build a sinusoidal model from a periodic context or data set, and how do you interpret it?Topic 3.7 Sinusoidal Function Context and Data Modeling: construct a sinusoidal model from a periodic context or data, and use it to make and interpret predictions.9 min answer →
- How does each transformation of a sinusoid change its graph, and how do you combine them?Topic 3.6 Sinusoidal Function Transformations: describe how changing each parameter transforms a sinusoid, and combine vertical and horizontal stretches, reflections and shifts.9 min answer →
- What is the general form of a sinusoidal function, and how do its four parameters control the graph?Topic 3.5 Sinusoidal Functions: write a sinusoidal function in the form a*sin(b(x - c)) + d (or with cosine) and relate amplitude, period, phase shift and vertical shift to the parameters.9 min answer →
- How are secant, cosecant and cotangent defined as reciprocals, and where are their asymptotes and ranges?Topic 3.11 The Secant, Cosecant, and Cotangent Functions: define the reciprocal trigonometric functions and identify their periods, asymptotes, ranges and graphs.9 min answer →
- What does the tangent function look like, and where are its period, asymptotes and zeros?Topic 3.8 The Tangent Function: define the tangent function, graph it, and identify its period, vertical asymptotes, zeros and behavior between asymptotes.9 min answer →
- How do you solve a trigonometric equation or inequality and find all of its solutions?Topic 3.10 Trigonometric Equations and Inequalities: solve trigonometric equations and inequalities, using inverse functions, symmetry and periodicity to find all solutions.9 min answer →
Unit 4: Functions Involving Parameters, Vectors, and Matrices
Module overview →- What are the conic sections, and how do their standard equations encode their shape and key features?Topic 4.6 Conic Sections: identify and analyze parabolas, ellipses, circles and hyperbolas from their equations, and describe their key features.9 min answer →
- What is an implicitly defined relation, and how does it differ from a function written y = f(x)?Topic 4.5 Implicitly Defined Functions: interpret a relation given by an equation in x and y, and analyze its graph even when it is not a function of x.9 min answer →
- How does a matrix represent a linear transformation of the plane, such as a rotation, reflection or scaling?Topic 4.12 Linear Transformations and Matrices: represent a linear transformation of the plane by a matrix, and identify the matrices for scalings, reflections and rotations.9 min answer →
- How is a matrix a function that takes a vector to a vector, and what do composition and inverse mean for it?Topic 4.13 Matrices as Functions: interpret a matrix as a function from vectors to vectors, and relate matrix multiplication to composition and the inverse matrix to the inverse function.9 min answer →
- How do matrices model real situations such as transitions between states over time?Topic 4.14 Matrices Modeling Contexts: use matrices to model transitions between states, and apply repeated multiplication to project the state forward in time.9 min answer →
- What is a matrix, and how do you add, scale and multiply matrices?Topic 4.10 Matrices: represent data with a matrix, and add, subtract, scale and multiply matrices, including multiplying a matrix by a vector.9 min answer →
- How fast do the coordinates of a parametric curve change, and how do those rates describe the motion?Topic 4.3 Parametric Functions and Rates of Change: compute the average rates of change of x and y with respect to t, and use them to describe the direction and relative speed of motion.9 min answer →
- How do parametric functions describe the motion of a point in the plane over time?Topic 4.2 Parametric Functions Modeling Planar Motion: use a parametric function to model the position of a moving point over time, and describe its path, direction and position at a given time.9 min answer →
- What is a parametric function, and how do x and y depend on a third variable?Topic 4.1 Parametric Functions: define a parametric function giving x and y as functions of a parameter t, and graph and interpret the curve it traces.9 min answer →
- How do you write parametric equations for a circle or a line, and how do the parameters control them?Topic 4.4 Parametrically Defined Circles and Lines: write and interpret parametric equations for circles and lines, controlling radius, center, direction and starting point.9 min answer →
- How do you find a parametrization for an implicitly defined curve such as a circle or ellipse?Topic 4.7 Parametrization of Implicitly Defined Functions: find parametric equations that trace an implicitly defined curve, and verify the parametrization satisfies the implicit equation.9 min answer →
- What are the determinant and inverse of a matrix, and what do they tell you?Topic 4.11 The Inverse and Determinant of a Matrix: compute the determinant and inverse of a 2x2 matrix, and use them to determine invertibility and solve matrix equations.9 min answer →
- What is a vector-valued function, and how does it describe position and motion in the plane?Topic 4.9 Vector-Valued Functions: interpret a vector-valued function whose output is a position vector, and relate it to parametric motion and velocity.9 min answer →
- What is a vector, and how do you add, scale and find the magnitude and direction of one?Topic 4.8 Vectors: represent a vector by components, compute its magnitude and direction, and add, subtract and scale vectors.9 min answer →