Every lesson in the Precalculus slide course, in full text: 73 decks, 4745 slides.
1.1 Functions and Function NotationTurns the loose word 'function' into a test you can run: every input has exactly one output. Builds function notation as a name for an output rather than a multiplication, separates evaluating f(3) from solving f(x) = 3, and introduces the vertical and horizontal line tests as the same definition seen from two directions.
1.2 Domain and RangeFinds the domain of a rule by hunting for what it forbids — a zero denominator and a negative even radicand cover nearly every case — then writes the survivors in interval and set-builder notation. Reads both domain and range off a graph as its two shadows, and handles piecewise definitions, whose conditions must partition the domain without overlapping or leaving a gap.
1.3 Rates of Change and Behavior of GraphsIntroduces average rate of change as the slope of a secant line, computed from a table, a graph or a formula, with units read off the fraction. Builds the difference quotient as the same slope with the gap named h, then supplies the vocabulary for describing a graph precisely: increasing and decreasing intervals, local extrema, and absolute extrema, including why the endpoints of a closed window matter.
1.4 Composition of FunctionsBuilds new functions from old ones. Covers the pointwise arithmetic of sum, difference, product and quotient briefly, then spends the lesson on composition: evaluating it, why it is not commutative, decomposing a complicated rule into inner and outer parts, and the domain question — where the inner function's restriction survives even after the algebra hides it.
1.5 Transformation of FunctionsOrganises every graph transformation under one rule: changes written outside a function act on the output, vertically, and do what they say; changes written inside act on the input, horizontally, and do the opposite. Covers shifts, stretches, compressions and reflections, the order in which combined transformations are applied, and the even and odd symmetry tests.
1.6 Absolute Value FunctionsReads absolute value as distance rather than as sign removal, which turns equations into 'which points are this far from the centre' and inequalities into 'inside' or 'outside'. Graphs the V-shaped parent and every transformation of it from the previous section, and connects the graph to the solution sets of the corresponding equations and inequalities.
1.7 Inverse FunctionsAsks when a function can be run backwards. Defines the inverse by the two-sided composition condition, shows that only one-to-one functions have one, and settles the notation trap where the superscript minus one is not a reciprocal. Finds inverse formulas by swapping and solving, reads the graph as a reflection in the line y equals x, and uses domain restriction to rescue functions that fail the horizontal line test.
2.1 Linear FunctionsDefines a linear function by the property that its rate of change is constant, which is exactly what makes its graph straight. Computes slope from two points, from a table and from a description, interprets its sign and its units, and introduces the slope-intercept and point-slope forms along with the special cases of horizontal and vertical lines.
2.2 Graphs of Linear FunctionsGraphs lines quickly from the intercept and the slope rather than from a table, finds both intercepts, and establishes the two relationships between pairs of lines: parallel means equal slopes, and perpendicular means slopes whose product is negative one. Closes by finding where two lines cross, which is a two-variable system solved before the machinery for it arrives in Chapter 9.
2.3 Modeling with Linear FunctionsTurns a described situation into a linear rule: choosing which quantity is the input, reading the rate and the initial value out of the words, and — the step most often missed — stating the domain the situation allows rather than the one the formula would accept. Distinguishes interpolation from extrapolation and names model breakdown as the reason the distinction matters.
2.4 Fitting Linear Models to DataHandles data that trends linearly without lying on a line. Builds the scatter plot, fits a line by eye and then by least-squares regression, and introduces the correlation coefficient — including its two standard misreadings, that its sign measures quality and that a strong value implies causation, neither of which any calculation will catch.
3.1 Complex NumbersEnlarges the number system so that every quadratic has roots. Defines the imaginary unit, plots complex numbers on a plane, and works out the arithmetic — addition and multiplication behave like ordinary algebra plus the single rule that i squared is negative one, and division is handled by multiplying through by the conjugate.
3.2 Quadratic FunctionsReads a parabola entirely from its formula. Establishes vertex form as the transformation form of the squaring rule, converts to it by completing the square and back by expanding, locates the vertex from standard form directly, and uses the discriminant to predict how many real roots exist. Closes with optimisation, where the vertex answers maximum and minimum questions without calculus.
3.3 Power Functions and Polynomial FunctionsGeneralises beyond quadratics. Classifies power functions by the parity of their exponent, defines polynomials by degree and leading coefficient, and establishes that end behaviour depends on those two numbers alone because every lower-degree term becomes negligible far from the origin. Bounds the number of roots and turning points by the degree.
3.4 Graphs of Polynomial FunctionsHandles the middle of a polynomial graph. Finds roots by factoring, and uses each root's multiplicity to decide whether the curve crosses the axis, bounces off it, or flattens through. Combines roots, multiplicities and end behaviour into a complete sketch, recovers a formula from a graph, and uses sign changes to guarantee roots exist.
3.5 Dividing PolynomialsSupplies the machinery §3.6 needs. Establishes the division algorithm for polynomials, performs long division and its stripped-down form synthetic division, and proves the Remainder Theorem — that dividing by x minus c leaves f(c) as the remainder, which turns division into a root test and makes degree reduction possible.
3.6 Zeros of Polynomial FunctionsAnswers the question the chapter has been deferring: how to find a root when none is obvious. The Rational Zero Theorem reduces an infinite search to a finite list of candidates; synthetic division tests them cheaply; and the Fundamental Theorem of Algebra guarantees the count is exact over the complex numbers. Closes with the conjugate pair theorem and Descartes' Rule of Signs.
3.7 Rational FunctionsIntroduces the first functions in the course whose graphs break. Factors numerator and denominator to separate holes from vertical asymptotes, compares degrees to find horizontal or slant asymptotes, locates intercepts, and assembles the whole into a sketch. Closes with rational inequalities, solved by sign analysis across the critical values.
3.8 Inverses and Radical FunctionsApplies §1.7's inverse machinery to the families of this chapter. Restricts a quadratic or higher power to a branch on which it is one-to-one, inverts it to produce a radical function, and inverts rational functions by collecting and factoring. Solves radical equations, where squaring both sides can create solutions the original never had.
3.9 Modeling Using VariationCloses the chapter with power functions met through the situations that produce them. Translates the phrases direct, inverse and joint variation into formulas with an unknown constant, finds that constant from a single data point, and works out how the output responds to scaling the input — where the exponent, not the constant, does all the work.
4.1 Exponential FunctionsMoves the variable from the base to the exponent, which changes growth from additive to multiplicative. Identifies exponential functions from tables and formulas, distinguishes growth from decay by the base, establishes that an exponential eventually outgrows every power function, and introduces compound interest and the number e as the limit of ever-more-frequent compounding.
4.2 Graphs of Exponential FunctionsApplies Chapter 1's transformations to the exponential parent. Establishes the three features to track — the point at height one, the horizontal asymptote, and the range — and works out which transformations move which. The vertical shift is the one that moves the asymptote, and forgetting that is what costs the range and the end behaviour together.
4.3 Logarithmic FunctionsIntroduces the inverse of the exponential and the tool for solving for an exponent. Establishes the equivalence between logarithmic and exponential form, evaluates logarithms by asking what power the base needs, derives the restricted domain from the exponential's restricted range, and names the common and natural logarithms.
4.4 Graphs of Logarithmic FunctionsTransforms the logarithm parent, where the vertical asymptote and the domain boundary are the same line and therefore move together. Establishes that a horizontal shift moves both, that a reflection in the vertical axis moves the domain to the negative inputs, and that the domain can always be found by requiring the argument to be strictly positive.
4.5 Logarithmic PropertiesEstablishes the three logarithm properties as exponent rules read through the definition: products become sums, quotients become differences, and powers become multipliers. Expands and condenses logarithmic expressions, warns against the false property for sums, and supplies the change-of-base formula that makes any logarithm computable from the two a calculator provides.
4.6 Exponential and Logarithmic EquationsPuts the chapter's machinery to work. Solves exponential equations by taking a logarithm of both sides so the power property can bring the exponent down, or by equating exponents when a common base is available. Solves logarithmic equations by condensing and converting to exponential form, and insists on checking, since condensing can produce solutions the original equation rejects.
4.7 Exponential and Logarithmic ModelsThe chapter's applications. Builds and uses four model families — unbounded exponential growth, decay described by half-life, Newton's law of cooling with its shifted asymptote, and logistic growth with a carrying capacity — and solves each for a time using §4.6's technique. Chooses between them by asking what the quantity approaches.
4.8 Fitting Exponential Models to DataCloses the chapter by fitting curves to scattered data. Uses linearisation — taking a logarithm of the outputs — to turn exponential data into linear data that Chapter 2's regression already handles, then converts the fitted line back into an exponential model. Distinguishes exponential, logarithmic and power models by which transformation straightens the plot, and carries every caution from §2.4 forward.
5.1 AnglesSets up the chapter. Places angles in standard position with a sign convention, introduces radian measure as a ratio of arc length to radius, converts between degrees and radians, and derives the arc length and sector area formulas from the fraction-of-a-circle idea. Closes with coterminal angles, reference angles and angular speed.
5.2 Unit Circle: Sine and Cosine FunctionsDefines cosine and sine as the coordinates of the point where an angle's terminal side meets the unit circle, which makes them defined for every angle. Derives the Pythagorean identity as the circle's own equation, establishes the exact values at the special angles, and reads the sign of each function in each quadrant directly off the coordinate axes.
5.3 The Other Trigonometric FunctionsDefines the remaining four trigonometric functions as ratios and reciprocals of the cosine and sine. Establishes where each is undefined by asking which denominator vanishes, reads the tangent as the slope of the terminal side, and derives the two further Pythagorean identities by dividing the original one through.
5.4 Right Triangle TrigonometryConnects the circle definition back to the right triangle ratios. Shows that the two agree because the triangles are similar, defines all six functions as side ratios, solves right triangles from one side and one angle, and applies the results to angles of elevation and depression, where the measurement convention causes most of the errors.
6.1 Graphs of the Sine and Cosine FunctionsTurns the circle's rotation into a wave. Unrolls the unit circle to produce the sine and cosine graphs, identifies the amplitude, period and midline, and applies Chapter 1's transformations — with the added complication that an inside factor changes the period, so the phase shift must be read after factoring.
6.2 Graphs of the Other Trigonometric FunctionsGraphs the four derived trigonometric functions by reading consequences off the sine and cosine. Places the vertical asymptotes at the undefined points, explains the tangent's shorter period through its slope reading, sketches the secant and cosecant as reciprocals hugging the curves they invert, and applies the standard transformations to all four.
6.3 Inverse Trigonometric FunctionsBuilds the inverse trigonometric functions by restricting each domain until the horizontal line test passes. Explains why each standard restriction was chosen, evaluates inverses exactly and with a calculator, composes them with the original functions, and shows how to recover the other angle a problem may want.
7.1 Solving Trigonometric Equations with IdentitiesSeparates identities from equations, then assembles the fundamental identities into four small families. Derives the Pythagorean identities from the unit circle, uses reciprocal, quotient and even-odd relations to rewrite expressions, and establishes the discipline of verifying by transforming a single side.
7.2 Sum and Difference IdentitiesStates the sum and difference identities for the sine, cosine and tangent, and puts them to work finding exact values for angles beyond the special ones. Explains the cosine's reversed sign pattern, derives the cofunction identities as a consequence, and uses the identities to verify and to simplify.
7.3 Double-Angle, Half-Angle, and Reduction FormulasDerives the double-angle identities by setting the two angles equal in the sum formulas, gives the cosine's three equivalent forms and says when to use each, rearranges them into the reduction formulas, and runs them backwards to reach the half-angle formulas — where the sign is decided by the halved angle's quadrant.
7.4 Sum-to-Product and Product-to-Sum FormulasDerives the product-to-sum formulas by adding and subtracting the sum and difference identities, then inverts them to get the sum-to-product formulas with their half-sum and half-difference arguments. Applies both directions to verification, exact values, and the acoustic phenomenon of beats.
7.5 Solving Trigonometric EquationsSolves trigonometric equations by isolating a function, by factoring, and by using identities to reduce to a single function. Handles the infinitely many solutions periodicity produces, the extra solutions an inside coefficient creates, and the extraneous ones that squaring introduces.
7.6 Modeling with Trigonometric FunctionsBuilds sinusoidal models from measured data, computing amplitude, midline, period and phase shift from a maximum and a minimum. Extends to damped oscillation, where a decaying exponential multiplies the wave, and uses the models to answer questions by solving the equations of the previous section.
8.1 Non-right Triangles: Law of SinesExtends trigonometry to triangles with no right angle. States the law of sines, applies it to the AAS and ASA cases, works carefully through the ambiguous SSA case where two triangles may satisfy the data, and computes areas from two sides and their included angle.
8.2 Non-right Triangles: Law of CosinesSupplies the tool for the two cases the law of sines cannot start. Presents the law of cosines as the Pythagorean theorem with a correction term, uses it for SAS and SSS triangles, shows why solving for an angle with it is never ambiguous, and reaches areas from three sides with Heron's formula.
8.3 Polar CoordinatesIntroduces a coordinate system built on a distance and a direction rather than two perpendicular displacements. Plots points, converts in both directions using the right-triangle relations, handles the quadrant question the inverse tangent leaves open, and converts equations between the two systems.
8.4 Polar Coordinates: GraphsCatalogues the curves polar equations describe most naturally. Uses symmetry tests to halve the plotting work, classifies limacons by comparing the two coefficients, derives the rose curves' petal-count parity rule, and identifies circles and lemniscates from the form of their equations.
8.5 Polar Form of Complex NumbersWrites a complex number as a modulus and an argument, turning multiplication into a stretch and a rotation. Derives De Moivre's theorem for powers from the multiplication rule, and finds all n distinct nth roots as equally spaced points on a circle.
8.6 Parametric EquationsDescribes a curve by giving both coordinates as functions of a third variable, which adds direction, timing and repetition that an equation cannot record. Eliminates the parameter to identify a shape, preserves the domain restriction that elimination would otherwise lose, and parametrises projectile motion.
8.7 Parametric Equations: GraphsGraphs parametric equations with their orientation marked, handles curves that loop or revisit points, and distinguishes paths that cross from objects that collide. Uses the parametric form to answer simultaneous-motion questions an equation cannot state.
8.8 VectorsIntroduces quantities with both magnitude and direction, and an algebra for combining them. Adds vectors geometrically and by components, resolves a vector into components using the right-triangle relations, computes magnitudes and direction angles, and applies the whole apparatus to forces and to navigation.
9.1 Systems of Linear Equations: Two VariablesSolves two linear equations in two unknowns by graphing, substitution and elimination, and classifies every system into the three possible outcomes by matching its geometric arrangement to the algebraic signal that identifies it. Applies the machinery to mixture and rate problems.
9.2 Systems of Linear Equations: Three VariablesExtends elimination to three equations in three unknowns, reducing to a triangular system and back-substituting. Interprets each equation as a plane, classifies the possible outcomes, and distinguishes the two geometrically different ways a system can have infinitely many solutions.
9.3 Systems of Nonlinear Equations and Inequalities: Two VariablesSolves systems in which at least one equation is not linear, predicting the number of solutions from the geometry and checking every candidate against the originals. Covers substitution and elimination for nonlinear systems, extraneous solutions, and the shaded regions that nonlinear inequalities describe.
9.4 Partial FractionsReverses the addition of fractions, breaking a rational expression into a sum of simpler ones. Chooses the correct decomposition form from the denominator's factorisation, then finds the unknown numerators either by substituting convenient values or by matching coefficients.
9.5 Matrices and Matrix OperationsIntroduces the matrix as an object with its own arithmetic. Covers dimensions and notation, addition and scalar multiplication entry by entry, and the row-times-column rule for products — including why the dimension condition exists and why matrix multiplication is not commutative.
9.6 Solving Systems with Gaussian EliminationWrites a system as an augmented matrix and solves it by row operations. Establishes why the three operations preserve the solution set, drives the matrix to row echelon form by a mechanical column-by-column procedure, and reads the inconsistent and dependent cases off recognisable rows.
9.7 Solving Systems with InversesIntroduces the identity matrix and the multiplicative inverse, computes inverses for two-by-two and larger matrices, and solves a whole system in one multiplication. Connects a vanishing determinant to the absence of an inverse and to the special cases of the earlier sections.
9.8 Solving Systems with Cramer's RuleExpresses each unknown of a square system as a ratio of determinants, with the coefficient determinant as the common denominator and a column replaced by the constants in each numerator. Extends to three variables, and reads a vanishing denominator as the singular case.
10.1 The EllipseDefines the ellipse by a constant sum of distances to two foci, derives the standard equation from that definition, and identifies the centre, axes, vertices and foci from an equation. Covers both orientations, translated ellipses, and the relation among the three constants.
10.2 The HyperbolaChanges the ellipse's constant sum to a constant difference and follows the consequences: two unbounded branches, foci outside the curve, a rearranged relation among the constants, and asymptotes found from a central rectangle. Covers both orientations and translated hyperbolas.
10.3 The ParabolaDefines the parabola by equal distances to a focus and a directrix, connects that definition to the quadratic graphs of chapter 3, identifies the vertex, focus, directrix and axis from a standard equation in all four orientations, and derives the reflection property.
10.4 Rotation of AxesExplains the cross term as a sign that a conic is tilted, gives the rotation formulas and the angle that removes it, and introduces the discriminant — a quantity unchanged by rotation that classifies any second-degree equation without rotating it at all.
10.5 Conic Sections in Polar CoordinatesDescribes all three conics with a single polar equation by placing the pole at a focus. Identifies the conic from its eccentricity, reads the directrix's position from the form, converts between polar and rectangular equations, and applies the result to orbital motion.
11.1 Sequences and Their NotationsPresents a sequence as a function whose domain is the counting numbers, introduces subscript notation, and distinguishes explicit formulas from recursive ones — including why a recursive definition needs a starting value. Covers factorial notation and the conventions attached to it.
11.2 Arithmetic SequencesDevelops the sequences whose consecutive terms differ by a constant, connecting them to the linear functions of chapter 2. Derives the term formula and explains its off-by-one, derives the sum formula by pairing terms from the ends, and applies both to modelling.
11.3 Geometric SequencesDevelops the sequences whose consecutive terms have a constant ratio, presenting them as the multiplicative counterpart of arithmetic sequences. Covers the exponential term formula, the sum formula and its derivation, and the distinction between growth and decay.
11.4 Series and Their NotationsIntroduces summation notation and the distinction between a sequence and a series, then develops finite arithmetic and geometric sums before showing that an infinite geometric series has a finite total exactly when its ratio is smaller than one in size.
11.5 Counting PrinciplesCounts the ways something can happen. Distinguishes the addition principle for alternatives from the multiplication principle for stages, then separates permutations from combinations by whether order matters — the single question that decides nearly every problem in the section.
11.6 Binomial TheoremExpands a binomial raised to a whole power, showing that the coefficients are the combination counts of the previous section and explaining why. Covers Pascal's triangle as a generating device and the general term formula for extracting a single term without expanding.
11.7 ProbabilityDefines probability as a ratio of counts under the equally-likely assumption, then develops the rules for combining events: subtraction for overlapping unions, multiplication for independent events with the independence condition made explicit, and the complement rule for at-least-one questions.
12.1 Finding Limits: Numerical and Graphical ApproachesIntroduces the limit as what a function's outputs approach near a point, distinct from its value there. Estimates limits from tables and graphs, distinguishes one-sided from two-sided limits, and identifies the situations in which a limit fails to exist.
12.2 Finding Limits: Properties of LimitsReplaces numerical estimation with exact computation. Gives the properties that let a limit pass through sums, products, quotients and powers, establishes substitution as the first move, and develops the algebraic techniques for resolving indeterminate forms.
12.3 ContinuityNames the property that makes substitution valid and states it as three conditions. Classifies discontinuities by which condition fails, distinguishes removable failures from the rest, and states the intermediate value property that continuity guarantees.
12.4 DerivativesDefines the derivative as the limit of a difference quotient, showing that the slope of a tangent and an instantaneous rate of change are the same question. Computes derivatives from the definition, interprets their sign and size, and closes the course by naming what calculus does with them.
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