College Algebra

Units

1. 1

   Algebraic expressions

   7 lessons

   1. Exponent rules, radicals and rational exponents; sums and products of rational and irrational numbersSimplify expressions with exponents and radicals, then justify whether exact results are rational or irrational.
   2. Adding, subtracting and multiplying polynomialsAdd, subtract and multiply polynomials; use special products and identify a product’s degree.
   3. Factoring out the GCF and factoring x^2 + bx + cFactor a quadratic completely by taking out the greatest common factor, then finding the pair with the right product and sum, and read the zeros from the factors.
   4. Factoring ax^2 + bx + c by groupingFactor a trinomial whose x² term has a coefficient by splitting the middle term with the product ac, grouping, and checking by multiplying back.
   5. Special products: difference of squares and perfect square trinomialsSpot a difference of squares or a perfect square trinomial, factor it without a search, and factor completely when a common factor or a repeated pattern is hiding.
   6. Rational exponents and radicalsEvaluate rational powers, combine them using exponent laws, and simplify radicals.
   7. Rewriting rational expressions: simplify, divide, add and multiplyRewrite rational expressions in four ways while keeping every original denominator restriction.
2. 2

   Equations and inequalities

   13 lessons

   1. Solving as reasoning: justifying each stepSolve linear equations, justify each line, and check solutions in the original equation.
   2. Solving multistep linear equations, including special casesSolve multistep equations and distinguish one solution, no solution, and every real number as a solution.
   3. Rearranging formulas to highlight a quantityRearrange a formula to leave a requested variable alone, including when it appears in two terms.
   4. Solving and graphing linear inequalities, including compound inequalitiesSolve and graph single and compound inequalities, including reversing signs when dividing by a negative.
   5. Absolute value equations and inequalitiesSolve absolute value equations and inequalities, graph solutions, and check candidates in the original statement.
   6. Solving quadratic equations by factoringSolve a quadratic equation by making one side zero, factoring, and checking every solution.
   7. Completing the square to reveal the vertex and solveRewrite a quadratic to find its vertex, then complete the square to solve an equation.
   8. The quadratic formula and the discriminantSolve quadratics with the formula and predict the number of real solutions from the discriminant.
   9. Choosing a method to solve a quadraticChoose an efficient method, solve exactly when possible, and check which solutions fit the question.
   10. Solving radical equations and spotting extraneous solutionsSolve square-root equations and check every candidate in the original equation.
   11. Solving rational equationsSolve rational equations by clearing denominators, then reject answers that make an original denominator zero.
   12. The number i and arithmetic with complex numbersSimplify complex-number sums, products, powers of i, and quotients into a + bi form.
   13. Complex solutions of quadratic equationsSolve quadratics with non-real roots and connect those roots to graphs and factors.
3. 3

   Functions and graphs

   11 lessons

   1. What a function is: one output per input, function notation, domain and rangeTest whether a relation is a function, read function notation, and find domain and range in context.
   2. Interpreting key features of graphs and tables in contextInterpret graph and table features with input and output units, then sketch a graph from a story.
   3. Average rate of change over an intervalFind and interpret average rates from graphs, tables, and formulas, with appropriate units.
   4. Slope-intercept form: graphing and writing linesGraph a line from y = mx + b, write y = mx + b from a graph, a point and slope, or two points, and say what m and b mean in a story.
   5. Point-slope and standard form; converting between forms; horizontal and vertical linesWrite a line in point-slope form from one point and a slope, graph from standard form using intercepts, convert between all three forms, and choose the form the information suggests.
   6. Graphing quadratic functions: vertex, axis of symmetry and interceptsSketch a parabola from its opening, axis, vertex, intercepts and mirror points, and say what its minimum or maximum value is.
   7. Vertex form and maximum/minimum problemsRead the vertex from y = a(x − h)² + k, rewrite a quadratic into that form, and use it to find the best input and the best value in a situation.
   8. Graphing square root, cube root, absolute value, piecewise and step functionsLocate familiar graph shapes, move them, and draw only the parts allowed by each input rule.
   9. Transformations across function familiesMove, stretch and reflect any parent function, track its key points and features, read a transformation from a formula, and test for even and odd symmetry.
   10. Building functions by combining: sums, products, quotients and compositionBuild and evaluate function combinations, compare composition orders, and find allowed inputs.
   11. Inverse functions: finding, graphing and restricting the domainDecide whether a function can be undone, find its inverse by swapping and solving, and restrict the domain when it cannot.
4. 4

   Polynomial and rational functions

   7 lessons

   1. Seeing structure: factoring higher-degree polynomials by patternFactor higher-degree polynomials completely by spotting chunks, choosing a pattern, and checking every factor.
   2. Polynomial arithmetic, closure and proving identitiesMultiply polynomials, prove an identity, generate Pythagorean triples, and explain closure.
   3. Polynomial division, the Remainder Theorem and the Factor TheoremDivide polynomials, find remainders by substitution, and decide when a linear expression is a factor.
   4. Zeros, multiplicity and end behavior: sketching polynomial graphsSketch a polynomial using its zeros, crossings or touches, end behavior, and y-intercept.
   5. The Fundamental Theorem of Algebra and complex zeros of polynomialsFind and count complex zeros, identify real x-intercepts, and build real polynomials from zeros.
   6. Graphing rational functions: asymptotes, holes and interceptsSketch a rational graph from its holes, asymptotes, intercepts, and behavior near a vertical asymptote.
   7. Polynomial and rational inequalities with sign chartsSolve polynomial and rational inequalities by marking boundaries, testing signs, and choosing endpoints correctly.
5. 5

   Exponential and logarithmic functions

   5 lessons

   1. Exponential growth and decay models, e and continuous growthChoose an exponential model, then calculate an amount from a given time.
   2. Logarithms as exponentsRewrite logarithms as powers, evaluate them, use log and ln, and identify invalid inputs.
   3. Properties of logarithmsExpand, condense, and evaluate logarithms using the product, quotient, power, and change-of-base properties.
   4. Solving exponential and logarithmic equationsSolve exponential and logarithmic equations, give exact log answers and calculator approximations, and check log inputs.
   5. Exponential and logarithmic functions as inverses: graphs and featuresGraph an exponential–logarithmic inverse pair, identify its features, and find an inverse after a shift.
6. 6

   Systems of equations

   6 lessons

   1. Systems of linear equations: solutions as intersectionsFind and check a shared point, then predict how many solutions two lines have.
   2. Solving systems by substitutionSolve a system by substitution, check its ordered pair, and recognize no solution or infinitely many solutions.
   3. Solving systems by elimination, and why it worksSolve systems by elimination and explain why scaling and reversible combinations preserve their solutions.
   4. Systems of a linear and a quadratic equationFind and check every point shared by a line and a parabola; use the same substitution idea with a circle.
   5. Modelling with systems and choosing a methodModel two facts with a system, choose a solution method, and check the answer in context.
   6. Graphing linear inequalities and systems of inequalities as half-planesGraph each inequality and identify the points that satisfy every rule in a system.
7. 7

   Sequences and series

   4 lessons

   1. Arithmetic and geometric sequences: recursive and explicit rules, and finite geometric seriesWrite recursive and explicit rules, find a distant term, and sum a finite geometric series.
   2. Sigma notation and finite arithmetic and geometric seriesEvaluate finite sums in sigma notation and use a geometric sum to find a loan payment.
   3. Infinite geometric series and convergenceDecide whether an infinite geometric series has a finite sum, and find that sum for decimals and rebound distances.
   4. The Binomial Theorem and Pascal's triangleUse Pascal's triangle to expand a binomial or find a specified term.