Every lesson in the Algebra 2 slide course, in full text: 100 decks, 6497 slides.
Factoring Higher-Degree PolynomialsOne of the three gaps from the Algebra 2 pre-class diagnostic. Why factored form hands you the zeros, the GCF that has to come first, factoring four terms by grouping, the sum and difference of cubes with their three signs, and quartics that are quadratics in disguise - finishing with the complete-factoring procedure and solving by the zero-product property.
Radicals and Rational ExponentsThe second gap from the Algebra 2 pre-class diagnostic. What an index really controls, simplifying by extracting the largest perfect power, the bridge between radical and fractional-exponent notation, adding and multiplying and rationalising, and solving radical equations with the extraneous-solution check that decides the answer.
Conic SectionsThe third gap from the Algebra 2 pre-class diagnostic. Where all four conics come from, identifying any of them from the two squared coefficients, completing the square through a factored-out coefficient, and reading centre, radius, vertices, foci, directrix and asymptotes off standard form - with the ellipse and hyperbola relations kept firmly apart.
1.1 Properties of Real NumbersThe real number line and its subsets, ordering negatives, the five properties of addition and multiplication, subtraction and division as definitions, and unit analysis as a self-check.
1.2 Evaluating and Simplifying ExpressionsPowers and their bases, the order of operations, substituting into an algebraic expression, identifying terms and coefficients, combining like terms, and building an expression from a verbal model.
1.3 Solving Linear EquationsEquations, solutions and equivalent equations; the four properties of equality; isolating a variable on one side and then on both; clearing brackets and fractions; and writing an equation from a verbal model.
1.4 Rewriting Formulas and EquationsSolving a formula or an equation for one of its variables: one-step and two-step rearrangements, rewriting a linear equation for y, and factoring out the target variable when it appears in more than one term.
1.5 Problem Solving Strategies and ModelsTurning a described situation into an equation: the verbal model with units, and the three strategies the lesson names — use a formula, look for a pattern, and draw a diagram — plus the two-category problems that need only one letter.
1.6 Solving Linear InequalitiesSimple and compound inequalities, their graphs with open and solid endpoints, the transformations that preserve them, the rule that reverses the symbol when you multiply or divide by a negative, and modelling a range of acceptable values.
1.7 Absolute Value Equations and InequalitiesAbsolute value as distance on a number line, solving an absolute value equation by splitting it into two, rejecting extraneous solutions, the and/or rule for absolute value inequalities, and writing a tolerance as one.
2.1 Relations and FunctionsRelations and their four representations, domain and range, the definition of a function and the vertical line test, equations in two variables and their graphs, and function notation for linear and non-linear functions.
2.2 Slope and Rate of ChangeSlope as the ratio of rise to run, the two-point formula and the consistent-order rule, what the sign of a slope says about a line, the slope conditions for parallel and perpendicular lines, and slope as an average rate of change with units.
2.3 Graphing Equations of LinesThe parent function for linear functions, slope-intercept form and the four-step graphing procedure, reading slope and intercept as a rate and an initial value, standard form and the intercept method, and the equations of horizontal and vertical lines.
2.4 Writing Equations of LinesWriting a linear equation from slope and intercept, from slope and a point using point-slope form, and from two points; finding lines parallel or perpendicular to a given line through a given point; and building a linear model from two data values.
2.5 Direct VariationThe direct variation equation y equals a x, the constant of variation, the family of lines through the origin, the constant-ratio test for deciding whether data shows direct variation, and building a direct variation model from one data pair.
2.6 Scatter Plots and Best-Fitting LinesScatter plots and the three kinds of correlation, the correlation coefficient r, approximating a best-fitting line by hand in four steps, using a line of fit to predict, and finding the regression line with a graphing calculator.
2.7 Absolute Value Functions and TransformationsThe absolute value parent function and its vertex, translating a graph horizontally and vertically, stretching, shrinking and reflecting it, the combined general form, and writing an equation from a graph.
2.8 Linear Inequalities in Two VariablesSolutions of a linear inequality in two variables, boundary lines and half-planes, dashed against solid boundaries, the two-step graphing procedure with a test point, one-variable inequalities graphed in the plane, and modelling a real constraint as a shaded region.
3.1 Solving Linear Systems by GraphingSystems of two linear equations and what a solution means, solving by graphing and checking in both equations, systems with infinitely many solutions or none, the consistent and independent vocabulary, and modelling a comparison with a system.
3.2 Solving Linear Systems AlgebraicallyThe substitution method and the elimination method for solving a linear system exactly, how to choose between them, what the no-solution and infinitely-many cases look like algebraically, and modelling with a system solved by algebra.
3.3 Systems of Linear InequalitiesSystems of linear inequalities, the graph as the overlap of half-planes, systems whose regions do not meet, systems of three or more inequalities and their bounded regions, absolute value inequalities inside a system, and modelling a pair of constraints.
3.4 Systems in Three VariablesLinear equations in three variables and their graphs as planes, ordered triples as solutions, solving a three-equation system by eliminating one variable twice, the degenerate cases, and modelling a situation with three unknowns.
3.5 Basic Matrix OperationsWhat a matrix is, its dimensions and elements, when two matrices are equal, adding and subtracting element by element, scalar multiplication, combining the operations, and organising real data in matrix form.
3.6 Multiplying MatricesWhen a matrix product is defined and what its dimensions are, the row-by-column recipe for each element, computing a full product, why matrix multiplication is not commutative, and modelling a cost calculation as a matrix product.
3.7 Determinants and Cramer's RuleThe determinant of a 2 by 2 and a 3 by 3 matrix, using a determinant to find the area of a triangle, the coefficient matrix, Cramer's rule for solving a system, and what a zero determinant tells you.
3.8 Inverse Matrices and SystemsThe identity matrix, inverse matrices and the determinant condition for their existence, the formula for the inverse of a 2 by 2, writing a linear system as a matrix equation, and solving it with a single multiplication.
4.1 Quadratic Functions in Standard FormThe quadratic parent function and its parabola, vertex and axis of symmetry, how a controls width and direction and c controls height, the vertex formula for the general standard form, and finding a minimum or maximum value in a revenue model.
4.10 Writing Quadratic Functions from Their Graphs and DataBuilding a quadratic function from a vertex and a point, from two x-intercepts and a point, and from three arbitrary points using a system of equations; choosing the right form for the information given; and finding a best-fitting quadratic model from data.
4.2 Vertex Form and Intercept FormVertex form and the vertex it names directly, graphing from it, modelling a suspension cable, intercept form and the x-intercepts it names, the axis of symmetry halfway between them, and converting between the three forms.
4.3 Factoring x^2 + bx + c and the Zero Product PropertyFactoring a monic trinomial by finding two numbers with the right product and sum, the difference of two squares and perfect square trinomial patterns, the zero product property that turns factors into roots, a doubling-the-area model, and the zeros of a quadratic function.
4.4 Factoring When the Leading Coefficient Is Not OneThe four-integer search for factoring ax squared plus bx plus c, the special patterns when a and c are perfect squares, pulling out a common monomial first, solving equations that must be rewritten in standard form, and maximising revenue through the average of the zeros.
4.5 Square Roots and Equations of the Form x Squared Equals sThe product and quotient properties of square roots, the two conditions for a radical expression to be simplified, rationalising denominators using conjugates, solving quadratic equations by taking square roots, and the dropped-object height model.
4.6 The Imaginary Unit and Complex ArithmeticThe imaginary unit i and the square root of a negative number, standard form and the families of complex numbers, adding and subtracting, multiplying with FOIL, dividing by a complex conjugate, and plotting in the complex plane with absolute value.
4.7 Completing the SquareSolving when one side is already a perfect square, the number that completes x squared plus bx, solving any quadratic equation by completing the square with any leading coefficient, an area model, and rewriting a function in vertex form to read off its maximum.
4.8 The Quadratic Formula and the DiscriminantThe quadratic formula obtained by completing the square once in general, the three kinds of solution, the discriminant as a predictor of how many and what type, the link between the discriminant and the number of x-intercepts, and the vertical-motion model with an initial velocity.
4.9 Quadratic Inequalities in One and Two VariablesGraphing a quadratic inequality in two variables, a rope-strength model and systems of quadratic inequalities, solving a one-variable inequality by table and by graph, a robotics growth model, and the algebraic critical-value method.
5.1 Properties of Exponents and Scientific NotationWhy exponents add when powers multiply, the seven properties of exponents, evaluating numerical expressions, scientific notation in a large counting problem, simplifying algebraic expressions to positive exponents, and comparing volumes by scaling.
5.2 Polynomial Functions, Synthetic Substitution and End BehaviourWhat counts as a polynomial function and how degree names it, evaluating by direct and by synthetic substitution, reading end behaviour from the degree and the leading coefficient, and graphing polynomial functions including a wave-energy model.
5.3 Adding, Subtracting and Multiplying PolynomialsAdding polynomials in vertical and horizontal formats, subtracting by adding the opposite, multiplying two and three polynomials, the sum-and-difference, square and cube patterns, and multiplying two models to build a third.
5.4 Factoring and Solving Higher-Degree Polynomial EquationsCommon monomial factors, the sum and difference of two cubes, factoring by grouping, recognising quadratic form, and solving higher-degree polynomial equations with the zero product property including a basin volume model.
5.5 Polynomial Division and the Remainder and Factor TheoremsPolynomial long division, synthetic division by a linear divisor, the remainder theorem linking division to evaluation, the factor theorem and its four equivalent statements, and using one known solution to finish a polynomial model.
5.6 The Rational Zero Theorem and Finding All Real ZerosThe rational zero theorem and the finite candidate list it produces, testing candidates with synthetic division, narrowing a long list with a graph, repeating on the quotient until it is quadratic, and solving a pyramid-volume model.
5.7 The Fundamental Theorem of Algebra and Classifying ZerosThe fundamental theorem of algebra and its corollary counting zeros with multiplicity, finding every zero including imaginary ones, the complex and irrational conjugate theorems, building a polynomial from given zeros, Descartes' rule of signs, and approximating real zeros in a model.
5.8 Graphing Polynomial Functions from Their InterceptsThe equivalence of zeros, factors, solutions and x-intercepts, graphing a polynomial from its intercepts and end behaviour, turning points and the rules that count them, local maxima and minima, and maximising a volume model.
5.9 Writing Polynomial Functions from Graphs and DataWriting a cubic from its graph and a fourth point, computing finite differences, the two properties linking constant nth-order differences to degree n, building a function from a system of four equations, and polynomial regression on measured data.
6.1 nth Roots and Rational Exponentsnth roots and how many real ones a number has, why a to the power one over n is the nth root, evaluating expressions with rational exponents in both forms, solving equations by taking nth roots, and a cube-root model for a fish's length.
6.2 Properties of Rational Exponents and RadicalsThe six exponent properties restated for rational exponents, a mammal surface-area model, the product and quotient properties of radicals, the two conditions for simplest form, combining like radicals, and simplifying variable expressions including the absolute-value rule.
6.3 Operations on Functions and CompositionAdding, subtracting, multiplying and dividing functions and finding the domains, power functions and a rhino heartbeat model, evaluating a composition at a number, composing functions symbolically with domain care, and a discount model in which the order of composition changes the answer.
6.4 Inverse Relations and Inverse FunctionsInverse relations found by switching x and y, verifying a pair of inverses with both compositions, the horizontal line test and restricting a domain, inverses of cubic and higher power functions, and inverting a model to solve for the other variable.
6.5 Graphing Square Root and Cube Root FunctionsThe square root and cube root parent functions with their domains and ranges, vertical stretches, shrinks and reflections, translating a square root function and reading its new domain and range, translating a cube root function, and a pendulum period model.
6.6 Solving Radical Equations and Equations with Rational ExponentsIsolating a radical and raising both sides to the index, a hurricane wind-speed model, solving equations with rational exponents using reciprocal exponents, extraneous solutions introduced by squaring, and equations with two radicals that require squaring twice.
7.1 Exponential Growth Functions and Compound InterestThe exponential parent function with a base greater than one, the effect of the coefficient a on the y-intercept, translations and horizontal asymptotes, exponential growth models built from a percent increase, and the compound interest formula.
7.2 Exponential Decay Functions and DepreciationThe exponential parent function for a base between zero and one, the coefficient a and the y-intercept, translations and asymptotes, decay models built from a percent decrease, and the reflection relationship linking growth to decay.
7.3 The Natural Base e and Continuous GrowthThe number e as the limit of one plus one over n raised to the n, simplifying and evaluating natural base expressions, graphing y equals a times e to the rx, translating and modelling with natural base functions, and continuously compounded interest.
7.4 Logarithms and Logarithmic FunctionsThe definition of a logarithm and the equivalence of logarithmic and exponential form, evaluating logarithms including common and natural ones, the inverse properties and finding inverses of exponential and logarithmic functions, and graphing and translating logarithmic functions.
7.5 Properties of LogarithmsThe product, quotient and power properties of logarithms, expanding a single logarithm into several and condensing several into one, the change-of-base formula for evaluating any logarithm on a calculator, and applying the properties to a decibel model.
7.6 Solving Exponential and Logarithmic EquationsThe property of equality for exponential equations and solving by equating exponents, taking a logarithm of each side, Newton's law of cooling as an exponential model, the property of equality for logarithmic equations, exponentiating each side, and checking every apparent solution for extraneousness.
7.7 Writing Exponential and Power ModelsWriting an exponential function through two points, the semi-log transformation that straightens exponential data, finding an exponential model from a scatter plot and by regression, writing a power function through two points, and the log-log transformation that straightens power data.
8.1 Inverse and Joint VariationClassifying relationships as direct variation, inverse variation or neither; writing an inverse variation equation from a single data pair; building inverse variation models and reading them from tables; testing data by checking whether the products are constant; and writing joint and combined variation equations from a sentence.
8.2 Graphing Simple Rational FunctionsThe definition of a rational function, the parent hyperbola y equals 1 over x with its two asymptotes, graphing y equals a over x, translating to y equals a over x minus h plus k, finding both asymptotes of the general linear-over-linear form, and using a rational model for average cost.
8.3 Graphing General Rational FunctionsThe three characteristics of a rational graph — x-intercepts from the numerator's zeros, a vertical asymptote at each zero of the denominator, and a horizontal asymptote decided by comparing degrees — worked through all three degree cases, and an optimization problem solved with a rational model.
8.4 Multiplying and Dividing Rational ExpressionsSimplified form and the factor-then-cancel routine, the crucial difference between cancelling factors and cancelling terms, multiplying rational expressions including opposite factors and polynomial factors, dividing by multiplying by the reciprocal, and comparing package designs with surface-area-to-volume ratios.
8.5 Adding and Subtracting Rational ExpressionsAdding and subtracting rational expressions with like denominators, building the least common multiple of two polynomials, working with unlike denominators, distributing a subtraction across a whole numerator, simplifying the combined result, and simplifying complex fractions by two methods.
8.6 Solving Rational EquationsSolving a rational equation by cross multiplying when each side is a single fraction, building and using a mixture model, clearing all denominators by multiplying through by the least common denominator, handling equations that become quadratics, and checking every candidate for extraneousness.
9.1 The Distance and Midpoint FormulasThe distance formula and its derivation from the Pythagorean theorem, classifying a triangle by comparing side lengths, the midpoint formula, writing the equation of a perpendicular bisector, and locating a circle's centre and diameter from three points on it.
9.2 Parabolas as Conic SectionsThe focus-directrix definition of a parabola, the two standard equations with vertex at the origin, identifying the focus, directrix and axis of symmetry from an equation, writing an equation from a given focus or directrix, and applying the model to parabolic reflectors.
9.3 Circles as Conic SectionsThe definition of a circle and the derivation of its standard equation from the distance formula, graphing a circle from a rearranged equation, writing an equation from a point on the circle, finding tangent lines using the perpendicular radius, and using circular inequalities to model coverage regions.
9.4 Ellipses as Conic SectionsThe two-foci definition of an ellipse and its vocabulary, the two standard equations with centre at the origin, identifying the major axis and locating vertices, co-vertices and foci, writing an equation from a vertex together with a co-vertex or a focus, and finding the area of an elliptical region.
9.5 Hyperbolas as Conic SectionsThe difference-of-distances definition of a hyperbola, the two standard equations with centre at the origin, the central rectangle and the asymptotes, identifying the transverse axis and locating vertices and foci, writing an equation from foci and vertices, and modelling a hyperbolic mirror.
9.6 Translating and Classifying ConicsThe standard forms of translated conic sections, graphing a translated circle and hyperbola, writing equations of translated parabolas and ellipses from their foci and vertices, identifying lines of symmetry, and classifying a general second-degree equation using the discriminant.
9.7 Solving Quadratic SystemsQuadratic systems and the possible numbers of intersection points, solving a linear-quadratic system by graphing and by substitution, solving a system of two second-degree equations by elimination, and using two hyperbolas to locate a ship.
10.1 Counting Principle and PermutationsTree diagrams and the fundamental counting principle, counting arrangements with and without repeated characters, factorials and permutations of n objects, permutations of n objects taken r at a time, and permutations of objects that include repeats.
10.2 Combinations and the Binomial TheoremCombinations and how they differ from permutations, deciding whether to multiply or add combinations, using subtraction from a total for at-least problems, building and reading Pascal's triangle, and using the binomial theorem to expand a power or to find a single coefficient.
10.3 Defining and Using ProbabilityTheoretical probability as a ratio of counts, probabilities computed with permutations and combinations, odds in favour of and against an event, experimental probability from surveys and trials, and geometric probability computed from lengths and areas.
10.4 Disjoint and Overlapping EventsCompound events as unions and intersections, disjoint events whose probabilities add, the general addition rule for overlapping events, rearranging that rule to find the probability of an intersection, complements, and using a complement to answer an at-least question.
10.5 Independent and Dependent EventsIndependent events and the multiplication rule, several independent events together with complements for at-least questions, dependent events and conditional probability read from a two-way table, drawing with and without replacement, and probability tree diagrams.
10.6 Binomial DistributionsRandom variables and probability distributions, reading a most likely value and a range from a histogram, the conditions of a binomial experiment and the formula for exactly k successes, constructing and interpreting a binomial distribution, and describing a distribution as symmetric or skewed.
11.1 Central Tendency and DispersionMean, median and mode as measures of central tendency, range and standard deviation as measures of dispersion, the effect of an outlier on each of the five statistics, and choosing which measure best describes a data set.
11.2 Transformations of DataThe effect of adding a constant to every data value on the mean, median, mode, range and standard deviation; why the two spread measures are unchanged by a shift; the effect of multiplying every value by a constant; why every statistic scales; and the general transformation that scales and then shifts.
11.3 Normal DistributionsThe normal curve and the 68-95-99.7 rule for area under it, finding probabilities from the rule, interpreting real normally distributed data, converting values to z-scores, and reading probabilities from the standard normal table.
11.4 Sampling and Margin of ErrorPopulations and samples, the four common sampling methods, recognising a biased sample, describing a procedure for drawing a random sample, computing a margin of error from the sample size, and finding the sample size needed for a given margin of error.
11.5 Choosing a Model for Two-Variable DataThe five function families used to model paired data, reading a scatter plot's shape to choose among them, fitting linear, exponential, quadratic and cubic models with regression, and preferring the simpler model when two fit comparably well.
12.1 Sequences and SeriesSequences as functions whose domain is a set of consecutive integers, writing terms from a rule and a rule from terms, graphing a sequence as isolated points, series and summation notation, and the three special formulas for sums.
12.2 Arithmetic Sequences and SeriesThe common difference, the rule for the nth term of an arithmetic sequence, recovering that rule from a term and d or from two terms, the linear graph, and the formula for the sum of a finite arithmetic series.
12.3 Geometric Sequences and SeriesThe common ratio, the rule for the nth term of a geometric sequence, recovering that rule from a term and r or from two terms, the exponential graph, the sum of a finite geometric series, and percent growth as a geometric model.
12.4 Infinite Geometric SeriesPartial sums and the value they approach, the sum of an infinite geometric series and the condition on the common ratio, series that have no sum, infinite series as models, and converting repeating decimals to fractions.
12.5 Recursive RulesExplicit against recursive rules, generating terms from a recursive rule, writing recursive rules for arithmetic and geometric sequences and for the Fibonacci and factorial sequences, recursive models with long-run behaviour, and iterating a function.
13.1 Right Triangle TrigonometryThe six trigonometric ratios of an acute angle, finding all six from one of them, the exact values at 30, 45 and 60 degrees, solving a right triangle with a calculator, and indirect measurement using angles of elevation and depression.
13.2 General Angles and Radian MeasureAngles in standard position with any real measure, coterminal angles, the definition of a radian, conversion between degrees and radians, the degree and radian measures of the special angles, and the arc length and area of a sector.
13.3 Trigonometric Functions of Any AngleThe general definitions of the six trigonometric functions from a point on the terminal side, the unit circle and quadrantal angles, reference angles, the signs of the functions by quadrant, and the three-step procedure for evaluating any angle.
13.4 Inverse Trigonometric FunctionsWhy the trigonometric functions need restricted domains before they can be inverted, the ranges of inverse sine, cosine and tangent, evaluating inverse expressions exactly, solving a trigonometric equation in a stated quadrant, and finding angles in applied right triangles.
13.5 The Law of SinesThe law of sines, solving triangles in the AAS and ASA cases, the ambiguous SSA case and how to count the possible triangles, SSA with none, one or two solutions, and the area of a triangle from two sides and the included angle.
13.6 The Law of CosinesThe law of cosines and its relationship to the Pythagorean theorem, solving triangles in the SAS and SSS cases, why the largest angle is found first, choosing between the two laws, and Heron's formula for the area from three sides.
14.1 Graphing Trigonometric FunctionsThe parent graphs of sine and cosine, amplitude and period, the five key points for graphing a sine bx and a cosine bx, frequency and sine models of oscillation, and the tangent graph with its vertical asymptotes.
14.2 Translating Trigonometric GraphsThe general form a sine b of x minus h plus k, vertical and horizontal translations, the midline, reflections when the leading coefficient is negative, translated and reflected tangent graphs, and models built from them.
14.3 Trigonometric IdentitiesWhere the fundamental identities come from, the reciprocal, quotient, Pythagorean, cofunction and negative-angle families, finding all six function values from one, simplifying trigonometric expressions, and verifying identities.
14.4 Solving Trigonometric EquationsGeneral solutions of trigonometric equations, solutions restricted to an interval, solving by factoring and by the quadratic formula, discarding impossible roots, and recognising extraneous solutions created by squaring.
14.5 Writing Trigonometric ModelsRecovering the amplitude, period, and vertical and horizontal shifts from a sinusoid's graph, choosing between sine and cosine and deciding whether to reflect, converting a period or frequency into b, modelling circular motion, and fitting a sinusoidal regression to data.
14.6 Sum and Difference FormulasThe six sum and difference formulas for sine, cosine and tangent, finding exact values by splitting a special angle, evaluating with given function values and quadrant signs, simplifying expressions and deriving reduction identities, and solving trigonometric equations and models.
14.7 Double-Angle and Half-Angle FormulasThe double-angle formulas including the three equivalent forms for cosine, the half-angle formulas and the quadrant rule for their signs, finding exact values by halving a special angle, working from a given function value, and simplifying expressions and deriving a projectile range model.
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