This page maps the topics of AP Precalculus to the Brilliant lessons that teach them. College Board organizes the course into four units of numbered topics, and those topics are the rows below. Search by topic number or skill to find the closest match. A topic is listed only where Brilliant content fully or in part addresses it directly.
Units 1, 2, and 3 are the units assessed on the AP Exam. Unit 4 is part of the published course framework but is never assessed on the exam; College Board describes its topics as additional material schools may include, so the Unit 4 section below is labeled accordingly.
Unit 1 — Polynomial and Rational Functions (30–40% of the multiple-choice section)
| AP topic | What the topic asks | Brilliant targeted skill |
|---|---|---|
| 1.1 | Change in Tandem. Establish the function concept — a relation mapping each input value to exactly one output value, with domain, range, independent and dependent variables, the image of an input and the preimage of an output value, and equality of two functions (same domain, same output at every input) — then describe how the input and output values of a function vary together by comparing function values, including where a function is increasing (inputs increase, outputs always increase) or decreasing, and construct a graph representing two quantities that vary with respect to each other in a contextual scenario, reading concavity from whether the rate of change is increasing or decreasing and identifying zeros where the graph meets the x-axis. | Introduction to Functions — Applying the vertical line test, Evaluating floor and ceiling functions, Finding function domains, Finding function ranges, Following conditional rules, Following function rules, Identifying conditional rules, Identifying function intervals, Identifying function rules, Matching input-output pairs, Plotting conditional functions, Plotting numeric functions, Reading continuous function graphs, Reading discontinuous function graphs, Recognizing non-functions, Writing numeric function rules; Linear Relationships — Graphing real-world relationships, Reading linear patterns; Equations and Curves — Reading values from graphs |
| 1.3 | Rates of Change in Linear and Quadratic Functions. Determine average rates of change for sequences and functions — including linear, quadratic, and other function types, understood as the slope of the secant line from (a, f(a)) to (b, f(b)) — and determine the change in those average rates of change: constant for a linear function (so its rates change at a rate of zero) and given by a linear function for a quadratic (so its rates change at a constant rate). | Linear Relationships — Calculating rates of change, Computing slope, Finding constant change |
| 1.4 | Polynomial Functions and Rates of Change. Identify key characteristics of polynomial functions related to rates of change: the analytical form of a nonconstant polynomial with its degree, leading term, and leading coefficient; local (relative) and global (absolute) maxima and minima, which occur where the function switches between increasing and decreasing or at an included endpoint of a restricted domain; the fact that between every two distinct real zeros there must be at least one input value corresponding to a local maximum or minimum; that an even-degree polynomial has either a global maximum or a global minimum (at the vertex, for a quadratic); and points of inflection, where the rate of change switches between increasing and decreasing and the graph changes concavity. | Quadratics — Finding maximum and minimum values; Polynomials — Identifying transformed features, Matching equations to graphs |
| 1.5 | Polynomial Functions and Complex Zeros. Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology — the zero/root correspondence with linear factors (x − a), multiplicity, the fact that a degree-n polynomial has exactly n complex zeros counting multiplicities, non-real zeros occurring in conjugate pairs, real zeros as x-intercepts and as endpoints for intervals satisfying polynomial inequalities, even-multiplicity zeros where the sign of the output does not change and the graph is tangent to the x-axis, and finding the degree from the least n for which successive nth differences over equal-interval inputs are constant — and determine whether a polynomial function is even or odd. | Polynomials — Applying parity properties, Classifying even and odd powers, Connecting roots and factors, Deducing roots from symmetry, Determining sign between roots, Factoring biquadratic polynomials, Finding roots from graphs, Identifying sign regions, Matching graphs to factored form, Reading polynomial graphs, Recognizing polynomial symmetry, Relating roots and degree, Solving by factoring, Using power function symmetry; Quadratics — Classifying solution types, Connecting intercepts to factors, Determining number of solutions, Graphing by factoring, Reading solutions from graphs, Using the discriminant; Complex Numbers — Solving equations with complex solutions |
| 1.6 | Polynomial Functions and End Behavior. Describe end behaviors of polynomial functions: as input values increase or decrease without bound the output values of a nonconstant polynomial increase or decrease without bound, expressed in the notation lim(x→∞) p(x) = ±∞ and lim(x→−∞) p(x) = ±∞, with the degree and sign of the leading term determining the end behavior because the leading term's values dominate those of all lower-degree terms for inputs of large magnitude. | Polynomials — Identifying dominant terms |
| 1.7 | Rational Functions and End Behavior. Describe end behaviors of rational functions, understanding a rational function as a quotient of two polynomial functions that measures the relative size of numerator to denominator on its domain, and determining end behavior by examining the quotient of the leading terms: a dominant numerator gives the end behavior of that nonconstant polynomial quotient (and, if it is linear, a slant asymptote parallel to that line); neither polynomial dominating gives a constant quotient locating a horizontal asymptote; a dominant denominator gives a horizontal asymptote at y = 0. A horizontal asymptote y = b means the output values get and stay arbitrarily close to b as inputs increase or decrease without bound, written lim(x→±∞) r(x) = b. | Equations and Curves — Matching rational functions to graphs |
| 1.8 | Rational Functions and Zeros. Determine the zeros of rational functions: the real zeros of a rational function correspond to the real zeros of the numerator for such values in its domain, and the real zeros of both polynomial functions of a rational function r are endpoints or asymptotes for intervals satisfying the inequalities r(x) ≥ 0 or r(x) ≤ 0. | Equations and Curves — Finding intercepts and sign regions |
| 1.9 | Rational Functions and Vertical Asymptotes. Determine vertical asymptotes of graphs of rational functions: if a is a real zero of the denominator polynomial and is not also a real zero of the numerator, the graph has a vertical asymptote at x = a; a vertical asymptote also occurs at x = a when the multiplicity of a as a real zero in the denominator is greater than its multiplicity in the numerator. Near a vertical asymptote the denominator values are arbitrarily close to zero, so the function increases or decreases without bound, expressed with one-sided limit notation lim(x→a⁺) r(x) = ±∞ and lim(x→a⁻) r(x) = ±∞. | Equations and Curves — Finding asymptotes and holes, Finding hyperbola asymptotes, Finding vertical asymptotes, Locating asymptotes |
| 1.10 | Rational Functions and Holes. Determine holes in graphs of rational functions: if the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, the graph has a hole at that input value; if the graph of r has a hole at x = c, its location is found by examining output values for input values sufficiently close to c — if those outputs are arbitrarily close to L, the hole is at (c, L), written lim(x→c) r(x) = L, with the two one-sided limits both equal to L. | Equations and Curves — Identifying discontinuities and sign regions |
| 1.11 | Equivalent Representations of Polynomial and Rational Expressions. Rewrite polynomial and rational expressions in equivalent forms, recognizing that factored form readily reveals real zeros and therefore information about x-intercepts, asymptotes, holes, domain, and range, while standard form reveals end behavior, and that information extracted from different analytic representations of the same function can answer questions in context; determine the quotient of two polynomial functions using long division, writing f(x) = g(x)q(x) + r(x) with the degree of the remainder less than the degree of the divisor, which helps in finding equations of slant asymptotes; and rewrite the repeated product of binomials using the binomial theorem, which uses the entries in a single row of Pascal's Triangle to expand expressions of the form (a + b)ⁿ, including polynomial functions p(x) = (x + c)ⁿ. | Equations and Curves — Adding rational expressions, Simplifying by factoring, Simplifying rational expressions; Polynomials — Factoring power differences |
| 1.12 | Transformations of Functions. Construct a function that is an additive and/or multiplicative transformation of another function: g(x) = f(x) + k is an additive transformation giving a vertical translation by k units; g(x) = f(x + h) is an additive transformation giving a horizontal translation by −h units; g(x) = a·f(x) with a ≠ 0 is a multiplicative transformation giving a vertical dilation by a factor of |a|, involving a reflection over the x-axis when a < 0; g(x) = f(bx) with b ≠ 0 is a multiplicative transformation giving a horizontal dilation by a factor of 1/|b|, involving a reflection over the y-axis when b < 0. These may be combined, taking a preimage to an image, and the domain and range of a transformed function may differ from those of the parent function. | Equations and Curves — Applying reflections and equivalent stretches, Applying shifts and stretches, Combining multiple transformations, Completing hyperbola equations, Connecting transformations to domain and range, Determining domain and range, Finding domain and range after transformations, Finding equation parameters, Finding hyperbola domain and range, Identifying transformed graphs, Shifting the square root graph, Stretching and reflecting root graphs, Transforming hyperbola graphs, Transforming the square root graph, Writing transformations from constraints; Polynomials — Applying single transformations, Combining transformations; Introduction to Functions — Combining reflections and stretches, Reflecting functions across axes, Sliding Function Graphs, Stretching Function Graphs |
| 1.14 | Function Model Construction and Application. Construct a linear, quadratic, cubic, quartic, polynomial of degree n, or related piecewise-defined function model — based on restrictions identified in a mathematical or contextual scenario, using transformations of the parent function, or using technology and regressions (linear, quadratic, cubic, and quartic) — construct a rational function model based on a context, as for quantities that are inversely proportional (for example, gravitational and electromagnetic force being inversely proportional to squared distance), and apply a function model to answer questions about a data set or contextual scenario, drawing conclusions and predicting values, rates of change, average rates of change, and changing rates of change, with appropriate units extracted or inferred from the context. | Linear Relationships — Evaluating linear equations |
Unit 2 — Exponential and Logarithmic Functions (25–40% of the multiple-choice section)
| AP topic | What the topic asks | Brilliant targeted skill |
|---|---|---|
| 2.1 | Change in Arithmetic and Geometric Sequences. Express arithmetic and geometric sequences found in mathematical and contextual scenarios as functions of the whole numbers, using the common difference or common ratio and either an initial value or a known kth term. | Exponential Functions — Identify exponential decay from a dot pattern, Identify exponential growth from a dot pattern |
| 2.2 | Change in Linear and Exponential Functions. Construct functions of the real numbers comparable to arithmetic and geometric sequences, and describe similarities and differences between linear and exponential functions. | Exponential Functions — Compare exponential growth to linear growth |
| 2.4 | Exponential Function Manipulation. Rewrite exponential expressions in equivalent forms using the product, power, and negative-exponent properties and exponential unit fractions, and connect each property to its graphical consequence. | Exponents and Radicals — Connecting fractional exponents to radicals, Evaluating general fractional exponents, Evaluating unit fraction exponents; Exponential Functions — Express decay with a negative exponent |
| 2.5 | Exponential Function Context and Data Modeling. Construct an exponential model for situations whose output values are proportional over equal-length input-value intervals, and apply exponential models to answer questions about a data set or contextual scenario. | Exponential Functions — Build a decay function with a fractional base, Build an exponential function from non-integer points, Build an exponential function from plotted points, Build an exponential function over negative domain, Compare two exponential functions, Deduce an exponential function from two points, Find the base of an exponential function, Graph and compare compound interest functions, Set the coefficient of an exponential function, Write a compound growth or decay expression |
| 2.6 | Competing Function Model Validation. Construct linear, quadratic, and exponential models based on a data set and validate a model constructed from a data set, using residuals and residual plots. | Exponential Functions — Compare exponential growth to quadratic growth |
| 2.7 | Composition of Functions. Evaluate the composition of two or more functions for given values, construct a representation of a composition, and rewrite a given function as a composition of two or more functions. | Introduction to Functions — Evaluating composite functions, Writing composite function expressions |
| 2.8 | Inverse Functions. Determine the input-output pairs of the inverse of a function and determine the inverse of a function on an invertible domain. | Introduction to Functions — Determining invertibility, Finding inverse functions; Equations and Curves — Evaluating roots and inverse compositions, Evaluating square roots of squares, Evaluating squaring and square roots, Graphing the square root function |
| 2.9 | Logarithmic Expressions. Evaluate logarithmic expressions, using the definition log_b c = a if and only if bᵃ = c, and interpret a logarithmic scale. | Logarithms — Convert between exponential and logarithmic form, Estimate a logarithm, Evaluate a logarithm exactly, Find the exponent in an exponential equation |
| 2.11 | Logarithmic Functions. Identify key characteristics of logarithmic functions — domain and range, monotonicity, concavity, extrema and inflection, vertical asymptote, and end behavior. | Logarithms — Find the domain of a logarithmic function, Graph a logarithmic function |
| 2.12 | Logarithmic Function Manipulation. Rewrite logarithmic expressions in equivalent forms using the product, power, and change-of-base properties, and connect each property to its graphical consequence. | Logarithms — Change the base of a logarithm, Convert between log bases graphically, Rewrite logarithms with the power rule, Rewrite logarithms with the product and quotient rules, Transform a log graph using identities |
| 2.13 | Exponential and Logarithmic Equations and Inequalities. Solve exponential and logarithmic equations and inequalities, and construct the inverse function for exponential and logarithmic functions. | Logarithms — Check solutions of logarithmic equations, Solve exponential equations with logarithms, Solve logarithmic equations |
Unit 3 — Trigonometric and Polar Functions (30–35% of the multiple-choice section)
| AP topic | What the topic asks | Brilliant targeted skill |
|---|---|---|
| 3.1 | Periodic Phenomena. Identify a periodic relationship between two aspects of a context — output values that repeat over successive equal-length intervals — construct the graph of the relationship from the graph of a single cycle, and describe key characteristics of a periodic function from a verbal representation, including the period as the smallest positive k with f(x+k) = f(x). | Trigonometric Functions — Predicting periodic values; Introduction to Functions — Reading periodic function graphs |
| 3.2 | Sine, Cosine, and Tangent. Determine the sine, cosine, and tangent of an angle using the unit circle: angles in standard position with a terminal ray, signed rotation and coterminal angles differing by an integer number of revolutions, radian measure as the ratio of subtended arc length to radius, sine as the vertical displacement of the intersection point P over its distance from the origin (the y-coordinate on a unit circle), cosine as the corresponding horizontal displacement ratio (the x-coordinate), and tangent as the slope of the terminal ray, equivalently y/x or sine over cosine. | Trigonometric Functions — Computing arc length, Constructing angles in radians, Constructing unit circle angles, Constructing unit circle arcs, Evaluating sine and cosine, Evaluating tangent at an angle, Finding angles beyond one revolution, Finding coterminal angles |
| 3.3 | Sine and Cosine Function Values. Determine coordinates of points on a circle centered at the origin: for an angle of measure θ in standard position and a circle of radius r, the terminal ray meets the circle at P = (r cos θ, r sin θ), and the geometry of isosceles right and equilateral triangles — while attending to the signs of the values based on the angle's quadrant — gives exact values for the cosine and sine of angles that are multiples of π/4 and π/6 radians whose terminal rays do not lie on an axis. | Trigonometric Functions — Evaluating trig functions in radians, Evaluating trig values at key angles |
| 3.4 | Sine and Cosine Function Graphs. Construct representations of the sine and cosine functions using the unit circle: f(θ) = sin θ gives the y-coordinate (vertical displacement of P from the x-axis) and f(θ) = cos θ gives the x-coordinate (horizontal displacement of P from the y-axis), each with domain all real numbers, and as the input angle measures increase the output values oscillate between −1 and 1, taking every value in between, tracking the displacement of points on the unit circle. | Trigonometric Functions — Reading cosine from circular motion, Reading sine from circular motion |
| 3.5 | Sinusoidal Functions. Identify key characteristics of the sine and cosine functions: a sinusoidal function is any function involving additive and multiplicative transformations of f(θ) = sin θ, and both sine and cosine are sinusoidal since cos θ = sin(θ + π/2); period and frequency are reciprocals (period 2π, frequency 1/(2π) for sin θ and cos θ); amplitude is half the difference between maximum and minimum values (1 for both); the midline is the average of the maximum and minimum values (y = 0 for both); the graphs oscillate between concave down and concave up as input values increase; y = sin θ has rotational symmetry about the origin and is odd, while y = cos θ has reflective symmetry over the y-axis and is even. | Trigonometric Functions — Applying symmetry and periodicity rules, Identifying periodic function features, Identifying phase shift relationships |
| 3.6 | Sinusoidal Function Transformations. Identify the amplitude, vertical shift, period, and phase shift of a sinusoidal function: functions writable as f(θ) = a·sin(b(θ+c)) + d or g(θ) = a·cos(b(θ+c)) + d with a ≠ 0 and b ≠ 0 are transformations of sine and cosine, and the transformations act identically on both because the cosine function is a phase shift of the sine function by π/2 units — adding d translates the graph and its midline vertically by d units, replacing θ by θ+c is a horizontal translation (phase shift) by −c units, multiplying by a is a vertical dilation changing amplitude by a factor of |a|, and replacing θ by bθ is a horizontal dilation changing the period by a factor of 1/|b|, so that y = a·sin(b(θ+c)) + d has amplitude |a|, period (1/|b|)·2π, midline shifted d from y = 0, and phase shift −c. | Trigonometric Functions — Computing period from rotation speed, Finding transformed function periods, Identifying amplitude and midline, Matching functions with transformations, Transforming periodic functions |
| 3.8 | The Tangent Function. Construct representations of the tangent function using the unit circle, describe its key characteristics, and describe additive and multiplicative transformations involving it: f(θ) = tan θ gives the slope of the terminal ray and equals sin θ / cos θ where cos θ ≠ 0; its period is π because the slope values repeat every half revolution; it has periodic asymptotic behavior at θ = π/2 + kπ for integer k, where cos θ = 0; it increases and changes from concave down to concave up between consecutive asymptotes; and for y = a·tan(b(θ+c)) + d, adding d translates the graph and the line containing its inflection points by d units, θ+c is a phase shift of −c units, a is a vertical dilation by a factor of |a| (with a reflection over the x-axis if a < 0), and bθ is a horizontal dilation changing the period by a factor of 1/|b| (with a reflection over the y-axis if b < 0), giving period (1/|b|)·π. | Trigonometric Functions — Determining tangent sign by quadrant |
| 3.9 | Inverse Trigonometric Functions. Construct analytical and graphical representations of the inverse of the sine, cosine, and tangent functions over a restricted domain: input and output values switch, so an inverse trigonometric function's output is often interpreted as an angle measure while its input is a value in the range of the corresponding trigonometric function; the inverses are arcsine, arccosine, and arctangent (also written sin⁻¹x, cos⁻¹x, tan⁻¹x), and because the trigonometric functions are periodic they are invertible only on restricted domains — sine restricted to [−π/2, π/2], cosine to [0, π], and tangent to (−π/2, π/2). | Trigonometric Functions — Evaluating inverse trig functions |
| 3.10 | Trigonometric Equations and Inequalities. Solve equations and inequalities involving trigonometric functions: inverse trigonometric functions are useful for solving them but solutions may need to be modified due to domain restrictions; because trigonometric functions are periodic there are often infinitely many solutions; and in equations and inequalities arising from a contextual scenario there is often a domain restriction implied by the context that limits the number of solutions. | Trigonometric Functions — Describing all solutions with periodicity, Factoring non-linear trig equations, Finding angle pairs for trig values, Solving equations from the tangent graph, Solving equations from trig graphs, Solving equations with inverse cosine, Solving equations with inverse sine, Solving trig equations on an interval |
| 3.12 | Equivalent Representations of Trigonometric Functions. Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity and with the sine and cosine sum identities, and solve equations using equivalent analytic representations: applying the Pythagorean Theorem to right triangles with unit-circle points (cos θ, sin θ) yields sin²θ + cos²θ = 1, which can be manipulated into other forms such as tan²θ = sec²θ − 1 and used to establish relationships such as arcsin x = arccos(√(1 − x²)) with appropriate domain restrictions; the sum identities are sin(α+β) = sin α cos β + cos α sin β and cos(α+β) = cos α cos β − sin α sin β, and these can also be used as difference and double-angle identities; known identities together with algebraic properties can verify additional identities, and a specific equivalent form can make information more accessible or make a trigonometric equation or inequality easier to solve. | Trigonometric Functions — Using the Pythagorean identity |
| 3.13 | Trigonometry and Polar Coordinates. Determine the location of a point in the plane using both rectangular and polar coordinates: the polar system is built on circles centered at the origin and lines through the origin with the positive x-axis as the polar axis, a point is the ordered pair (r, θ) where θ measures an angle in standard position whose terminal ray lies on a line through the point and r is the signed radial displacement (negative r meaning displacement along that line in the direction opposite the terminal ray), so the same point has many (r, θ) representations; conversions are x = r cos θ and y = r sin θ from polar to rectangular, and r = √(x² + y²) with θ = arctan(y/x) for x > 0 or θ = arctan(y/x) + π for x < 0 from rectangular to polar; and a complex number is a point in the complex plane, expressible as a + bi from rectangular coordinates (a, b) or as (r cos θ) + (r sin θ)i from polar coordinates (r, θ). | Polar Coordinate Plane — Computing angles with inverse tangent, Computing x and y from polar, Converting to polar coordinates, Finding equivalent angles, Finding quadrants from angles, Locating points from coordinates, Plotting points using radians, Plotting polar points, Plotting polar points on x-y grids, Reading polar coordinates; Complex Numbers — Finding absolute value, Finding magnitude and argument, Plotting complex numbers, Writing in polar form |
| 3.14 | Polar Function Graphs. Construct graphs of polar functions: the graph of r = f(θ) in polar coordinates consists of input-output pairs in which the inputs are angle measures and the outputs are radii; the domain of a polar function given graphically can be restricted to a desired portion by selecting endpoints corresponding to the desired angle and radius; and when graphing r = f(θ), changes in input values relate to changes in angle measure from the polar axis while changes in output values relate to changes in signed radial displacement from the origin. | Polar Coordinate Plane — Matching equations to polar graphs, Writing polar equations |
Unit 4 — Functions Involving Parameters, Vectors, and Matrices (not assessed on the AP Exam)
| AP topic | What the topic asks | Brilliant targeted skill |
|---|---|---|
| 4.5 | Implicitly Defined Functions. Construct a graph of an equation involving two variables, recognizing that such an equation can implicitly describe one or more functions, that it can be graphed by finding solutions to the equation, and that solving for one of the variables defines a function whose graph is part or all of the graph of the equation; and determine how the two related quantities vary together — for nearby ordered pairs on the graph, a positive ratio of the changes in the two variables means they simultaneously increase or both decrease while a negative ratio means one increases as the other decreases, and a rate of change of x with respect to y or of y with respect to x equal to zero indicates vertical or horizontal intervals respectively. | Equations and Curves — Analyzing implicit equations, Evaluating equations at points, Finding points on a curve, Matching equations to graphs, Matching equations to graphs, Rewriting equivalent equations, Solving systems and unions of curves, Verifying points on curves |
| 4.6 | Conic Sections. Represent conic sections with horizontal or vertical symmetry analytically: a parabola with vertex (h, k) as x − h = a(y − k)² if it opens left or right or y − k = a(x − h)² if it opens up or down (a ≠ 0); an ellipse centered at (h, k) with horizontal radius a and vertical radius b as (x − h)²/a² + (y − k)²/b² = 1, with a circle the special case a = b; and a hyperbola centered at (h, k) as (x − h)²/a² − (y − k)²/b² = 1 when it opens left and right or (y − k)²/b² − (x − h)²/a² = 1 when it opens up and down, with asymptotes y − k = ±(b/a)(x − h). | Equations and Curves — Writing circle and ellipse equations |
| 4.8 | Vectors. Identify characteristics of a vector: a vector is a directed line segment with a tail, a head, and a magnitude equal to the length of the segment; a vector from P₁ = (x₁, y₁) to P₂ = (x₂, y₂) is identified by components a = x₂ − x₁ and b = y₂ − y₁, written ⟨a, b⟩, with the zero vector ⟨0, 0⟩ the trivial case P₁ = P₂; its direction is parallel to the segment from the origin to (a, b) and its magnitude is the square root of the sum of the squares of the components; and for a vector represented geometrically the components can be found using trigonometry. Determine sums and products involving vectors — scalar multiplication multiplies each component and yields a parallel vector, vector addition adds corresponding components (represented graphically tail-to-head), and the dot product is the sum of the products of corresponding components, ⟨a₁, b₁⟩ · ⟨a₂, b₂⟩ = a₁a₂ + b₁b₂. Determine a unit vector for a given vector — a vector of magnitude 1, obtained by scalar multiplying a nonzero vector by the reciprocal of its magnitude, with ⟨a, b⟩ expressible as ai + bj where i = ⟨1, 0⟩ and j = ⟨0, 1⟩. Determine angle measures between vectors and magnitudes of vectors involved in vector addition — the dot product equals the product of the two magnitudes and the cosine of the angle between them, so a zero dot product of nonzero vectors means the vectors are perpendicular, and the Law of Sines and Law of Cosines can determine side lengths and angle measures of triangles formed by vector addition. | Vectors and Matrices — Adding vectors by components, Adding vectors from angles, Adding velocity vectors, Combining velocities from bearings, Comparing unit vectors, Computing dot products, Computing relative velocity, Computing unit vector dot products, Computing vector magnitude, Decomposing into linear combinations, Decomposing vectors with trigonometry, Drawing linear combinations, Drawing vector differences, Drawing vector sums, Drawing vectors from polar form, Evaluating linear combinations, Finding opposite vectors, Finding polar angles, Identifying scalar multiples, Representing shifts as vectors, Scaling dot products, Scaling to a target magnitude, Scaling vectors, Subtracting vectors by components, Using non-polar angle decomposition, Using standard basis vectors, Using the dot product formula |
| 4.10 | Matrices. Determine the product of two matrices: an n × m matrix is an array consisting of n rows and m columns; two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second; and the product is a new matrix whose component in the ith row and jth column is the dot product of the ith row of the first matrix and the jth column of the second. | Vectors and Matrices — Determining matrix product dimensions, Multiplying matrices, Multiplying matrices by vectors |
| 4.11 | The Inverse and Determinant of a Matrix. Determine the inverse of a 2×2 matrix: the identity matrix I is a square matrix with 1s on the diagonal from top left to bottom right and 0s everywhere else, multiplying a square matrix by its corresponding identity matrix returns the original matrix, the product of a square matrix and its inverse (when it exists) is the identity matrix of the same size, and the inverse of a 2×2 matrix, when it exists, can be calculated with or without technology. Apply the value of the determinant to invertibility and vectors: for A = [[a, b], [c, d]] the determinant is ad − bc, denoted det(A) and computable with or without technology; if a 2×2 matrix consists of two column or row vectors from R², the nonzero absolute value of its determinant is the area of the parallelogram spanned by those vectors, and a determinant of 0 means the vectors are parallel; and A has an inverse if and only if det(A) ≠ 0. | Vectors and Matrices — Computing determinants, Computing inverse matrices, Identifying non-invertible matrices, Inverting simple matrices, Solving determinant constraints, Solving systems with inverses |
| 4.12 | Linear Transformations and Matrices. Determine the output vectors of a linear transformation using a 2×2 matrix: a linear transformation is a function mapping an input vector to an output vector such that each component of the output is the sum of constant multiples of the input components; it maps the zero vector to the zero vector; a single vector in R² can be expressed as a 2×1 matrix and a set of n vectors in R² as a 2×n matrix; for a linear transformation L from R² to R² there is a unique 2×2 matrix A such that L(v) = Av, and conversely for a given 2×2 matrix A the function L(v) = Av is a linear transformation from R² to R²; and multiplying a 2×2 transformation matrix A by a 2×n matrix of n input vectors gives a 2×n matrix of the n output vectors. | Vectors and Matrices — Applying matrix transformations, Repeating matrix transformations, Transforming vectors with matrices |
| 4.13 | Matrices as Functions. Determine the association between a linear transformation and a matrix: the transformation mapping ⟨x, y⟩ to ⟨a₁₁x + a₁₂y, a₂₁x + a₂₂y⟩ is associated with the matrix [[a₁₁, a₁₂], [a₂₁, a₂₂]]; the mapping of the unit vectors provides valuable information for determining the associated matrix; the matrix [[cos θ, −sin θ], [sin θ, cos θ]] is associated with a transformation that rotates every vector by an angle θ counterclockwise about the origin; and the absolute value of the determinant of a 2×2 transformation matrix gives the magnitude of the dilation of regions in R² under the transformation. Determine the composition of two linear transformations — the composition is itself a linear transformation, and its associated matrix is the product of the matrices associated with each transformation. Determine the inverse of a linear transformation — two linear transformations are inverses if their composition maps any vector to itself, and if L is given by L(v) = Av then its inverse is L⁻¹(v) = A⁻¹v. | Vectors and Matrices — Applying sequential transformations, Composing geometric transformations, Composing transformations by multiplying, Encoding transformations as matrices, Identifying shape transformations, Reading transformation matrices |
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