United Grade 12 Precalculus
Chapters: 9
1. Functions and their inverses
Composition and inverses · Logarithm and exponent inverses
- 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.
2. Polynomial and rational functions
Rational functions with asymptotes and end behavior · Rational expressions as a field (+)
- 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.
3. Trigonometry
Unit circle symmetry and periodicity (+) · Inverse trigonometric functions (+) · Sum and difference identities (+) · Trigonometry of general triangles
- The Unit Circle: Symmetry and Periodicity – The unit circle has radius 1 and centre at the origin. A point at angle θ has coordinates (cos θ, sin θ). Mirror images of this point give the rules for π − θ, π + θ and −θ, and going once round (2π) brings you back, so sine and cosine repeat every 2π and tangent every π.
- 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}.
- 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.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
4. Complex numbers
Complex plane · Polynomial identities in complex numbers (+)
- Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.
- De Moivre's Theorem and Roots of Unity – A complex number can be written by its length r and angle θ: z = r(cos θ + i sin θ) = re^{iθ}. When you multiply, the lengths multiply and the angles add. So zⁿ = rⁿ(cos nθ + i sin nθ): this is De Moivre's theorem. It gives quick powers, formulas for cos nθ and sin nθ, and the n roots of any number. The n roots of 1 sit evenly on the unit circle like the corners of a regular polygon.
5. Vectors and matrices
Vector quantities · Vector operations · Matrices and their applications · Matrices for systems of equations (+)
- 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.
- 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.
6. Conic sections
Ellipses and hyperbolas from foci (+)
- 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.
7. Polar and parametric
Polar coordinates and graphs · Parametric equations
- 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.
- 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.
8. Probability for decisions
Expected value · Using probability for decisions
- Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
9. Sequences, series and limits
Series and sigma notation · Intro to limits
- 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.
- Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.