United Grade 12 AP Precalculus
Chapters: 4
1. Polynomial and Rational Functions
Change in Tandem · Rates of Change · Rates of Change in Linear and Quadratic Functions · Polynomial Functions and Rates of Change · Polynomial Functions and Complex Zeros · Polynomial Functions and End Behavior · Rational Functions and End Behavior · Rational Functions and Zeros · Rational Functions and Vertical Asymptotes · Rational Functions and Holes · Equivalent Representations of Polynomial and Rational Expressions · Transformations of Functions · Function Model Selection and Assumption Articulation · Function Model Construction and Application
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
- Polynomial Functions and Their Graphs – A polynomial function is y = aₙxⁿ + … + a₁x + a₀ with whole-number powers. The degree n and the leading coefficient aₙ fix the end behaviour. Each real zero r gives a factor (x − r); the graph crosses the x-axis at a zero of odd multiplicity and touches (bounces) at a zero of even multiplicity. A degree-n polynomial has at most n real zeros and at most n − 1 turning points, and exactly n zeros when complex ones are counted. The average rate of change between two points is the slope of the secant line.
- Rational Functions: Graphs, Asymptotes and Sketching – A rational function is one polynomial divided by another, like y = (2x + 1)/(x − 1). It is not defined where the bottom is zero. Near those x-values the graph shoots up or down along a vertical asymptote. Far out, it settles near a horizontal (or slant) asymptote. Find domain, asymptotes, intercepts and holes, then sketch.
- Transformations of Functions – A transformation changes the graph of a parent function y = f(x) without changing its basic shape. Adding outside the function moves it vertically: f(x) + k moves up k. Changes inside the brackets act horizontally and the opposite way: f(x − h) moves right h. Multiplying outside, a·f(x), stretches vertically by a (and flips in the x-axis if a < 0). Multiplying inside, f(bx), squeezes horizontally by factor 1/b (and flips in the y-axis if b < 0). All together: y = a·f(b(x − h)) + k, and the point (0, 0) of the parent moves to (h, k).
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
2. Exponential and Logarithmic Functions
Change in Arithmetic and Geometric Sequences · Change in Linear and Exponential Functions · Exponential Functions · Exponential Function Manipulation · Exponential Function Context and Data Modeling · Competing Function Model Validation · Composition of Functions · Inverse Functions · Logarithmic Expressions · Inverses of Exponential Functions · Logarithmic Functions · Logarithmic Function Manipulation · Exponential and Logarithmic Equations and Inequalities · Logarithmic Function Context and Data Modeling · Semi-log Plots
- Sequences and Series: AP, GP, AM and GM – A sequence is a list of numbers in a definite order: a₁, a₂, a₃, … A series is what we get when we add the terms. In an arithmetic progression (AP) we add the same number d each time; aₙ = a + (n − 1)d and Sₙ = n/2[2a + (n − 1)d]. The arithmetic mean (AM) of a and b is (a + b)/2; n AMs between a and b use d = (b − a)/(n + 1). In a geometric progression (GP) we multiply by the same number r each time; aₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If |r| < 1 the terms shrink and the infinite sum is S∞ = a/(1 − r). The geometric mean (GM) of two positive numbers a and b is √(ab), and n GMs between them use r = (b/a)^(1/(n+1)). For positive a and b, AM ≥ GM, with equality only when a = b.
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
- Mathematical Modelling: Using Maths to Describe the Real World – A mathematical model is an equation, graph or table that describes a real situation in a simple way. The modelling cycle: understand the real problem → choose variables and make assumptions → build a model (for example linear, quadratic or exponential) → solve and predict → check the answer against real data → improve the model or state its limits. No model is perfect; a good one is simple and close enough to be useful.
- Relations and Functions (Class 12) – A relation R on a set A is any set of pairs (a, b) taken from A × A. R is reflexive if every element is related to itself, symmetric if (a, b) in R always brings (b, a), and transitive if (a, b) and (b, c) always bring (a, c). A relation with all three is an equivalence relation; it cuts A into separate equivalence classes. A function f: A → B sends every element of A to exactly one element of B. It is one-one (injective) if different inputs give different outputs, onto (surjective) if every element of B is hit, and bijective if it is both.
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
3. Trigonometric and Polar Functions
Periodic Phenomena · Sine, Cosine, and Tangent · Sine and Cosine Function Values · Sine and Cosine Function Graphs · Sinusoidal Functions · Sinusoidal Function Transformations · Sinusoidal Function Context and Data Modeling · The Tangent Function · Inverse Trigonometric Functions · Trigonometric Equations and Inequalities · The Secant, Cosecant, and Cotangent Functions · Equivalent Representations of Trigonometric Functions · Trigonometry and Polar Coordinates · Polar Function Graphs · Rates of Change in Polar Functions
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
- Inverse Trigonometric Functions (Class 12) – sin x, cos x and the other trig functions repeat, so they are not one-one and have no inverse on all of R. We cut each one to a piece (the principal value branch) where it is one-one and onto. On that piece it has an inverse: y = sin⁻¹x means sin y = x with y in [−π/2, π/2]. The graph of an inverse is the mirror image of the branch in the line y = x. Principal ranges: sin⁻¹ [−π/2, π/2], cos⁻¹ [0, π], tan⁻¹ (−π/2, π/2), cot⁻¹ (0, π), sec⁻¹ [0, π] − {π/2}, cosec⁻¹ [−π/2, π/2] − {0}.
- Polar Coordinates – Polar coordinates give the position of a point by its distance r from a fixed point (the pole) and the angle θ turned anticlockwise from a fixed ray (the initial line). Convert with x = r cosθ, y = r sinθ, and back with r² = x² + y², tanθ = y/x (check the quadrant). Equations r = f(θ) draw circles, cardioids, limaçons, roses and spirals. Calculus: slope dy/dx = (dy/dθ)/(dx/dθ), area = ½∫r²dθ.
4. Functions Involving Parameters, Vectors, and Matrices
Parametric Functions · Parametric Functions Modeling Planar Motion · Parametric Functions and Rates of Change · Parametrically Defined Circles and Lines · Implicitly Defined Functions · Conic Sections · Parametrization of Implicitly Defined Functions · Vectors · Vector-Valued Functions · Matrices · The Inverse and Determinant of a Matrix · Linear Transformations and Matrices · Matrices as Functions · Matrices Modeling Contexts
- Parametric Equations: Curves, Slopes and Arc Length – In parametric form, x and y are each written using a third variable t, called the parameter. Each value of t gives one point, and as t changes the point traces a curve. You can remove t to get a Cartesian equation, find slopes with dy/dx = (dy/dt)/(dx/dt), and find speed and arc length when t is time.
- Conic Sections (Class 11): Circle, Parabola, Ellipse and Hyperbola – Cutting a double cone with a plane gives a circle, an ellipse, a parabola or a hyperbola; a cut through the vertex gives a point, a line or a pair of lines (degenerate conics). Circle: (x − h)² + (y − k)² = r². Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, e = 1. Ellipse x²/a² + y²/b² = 1 (a > b): c² = a² − b², foci (±c, 0), e = c/a < 1, latus rectum 2b²/a, PF₁ + PF₂ = 2a. Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a > 1, latus rectum 2b²/a, |PF₁ − PF₂| = 2a.
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- Vector-Valued Functions – A vector has a length and a direction and is written in components, like ⟨3, 2⟩. A vector-valued function p(t) = ⟨x(t), y(t)⟩ gives a position vector for every value of t, so its tip traces a path. The velocity vector ⟨x′(t), y′(t)⟩ points along the path, and its length is the speed.
- Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
- Determinants: Minors, Cofactors, Adjoint, Inverse and Linear Systems – A determinant is one number made from a square matrix, written |A|. For a 2 × 2 matrix it is ad − bc, and it equals the (signed) area made by the columns. For a 3 × 3 matrix we expand along a row using minors and cofactors. |A| = 0 means A is singular and has no inverse. Half of a determinant gives the area of a triangle. The adjoint (transpose of the cofactor matrix) gives A⁻¹ = (adj A)/|A|, and then a system AX = B is solved by X = A⁻¹B. The value of |A| and (adj A)B tell us if a system is consistent.
- Matrices as Functions: Linear Transformations – A 2×2 matrix is a function that takes a vector in and gives a vector out. Its first column is where ⟨1, 0⟩ lands and its second column is where ⟨0, 1⟩ lands. The determinant ad − bc tells how areas scale; when it is not zero, the inverse matrix undoes the change. Transition matrices use the same rule to model how shares change step by step.