College Algebra · Modules 1 to 14

College Algebra Study Tools

See algebra move. Every module shows each idea as a picture and as symbols at the same time: graphs that draw themselves, equations that balance like a scale, tiles that complete the square. Walkthroughs include animated worked examples, and every Lab has an explorer with sliders, an unlimited practice-problem trainer with hints and worked solutions, a game and real-life cases from health, money, sports and science. Everything runs in your browser on a computer, tablet or phone. No login, nothing graded.

How to use this: open the walkthrough first and go at your own pace with the arrow keys or Next. Then open the Lab: play with the explorer, then practice in the Problem Trainer, using the hints and the animated worked solutions. Use the tools bar at the top of each page to pause the animations, slow them down, have the text read aloud, or get a written description of each picture.

Module 1 Real numbers, exponents and radicals

The real number system, exponents, radicals, scientific notation and order of operations.

Start here · Animated walkthroughReal numbers, exponents and radicals WalkthroughWatch the number sets nest inside each other, then see order of operations, exponent rules, scientific notation, radicals and fraction exponents worked out one animated line at a time as factors cancel and pairs come out of the root. Explorer + practiceModule 1 LabBuild their own exponent expressions and watch the tiles cancel, practice unlimited problems at three levels with hints and worked solutions, race the clock, and apply the ideas to real problems in health, money, sports and science.

Module 2 Polynomials and factoring

Adding, multiplying and factoring polynomials, and simplifying rational expressions.

Start here · Animated walkthroughPolynomials and factoring WalkthroughWatch polynomials come apart into terms, multiply as growing rectangles and factor back into the rectangles they came from, through GCF, grouping, trinomials, special patterns, cubes and rational expressions. Then drag algebra tiles into rectangles to factor trinomials, practise unlimited problems with worked solutions, race the clock in Factor Match, and use factoring on real-life problems. Explorer + practiceModule 2 LabExplorer, problem trainer with worked solutions, a game and real-life cases.

Module 3 Linear equations and inequalities

Solving linear equations and inequalities, absolute value equations and formulas.

Start here · Animated walkthroughLinear equations and inequalities WalkthroughWatch equations solved on a balance scale, see fractions cleared, see the inequality sign flip on a mirrored number line and see absolute value as distance, all in 20 animated steps with 11 worked examples. Explorer + practiceModule 3 LabSolve equations by choosing each move on a balance scale, practise with unlimited problems at three levels with hints and worked solutions, race the clock on interval notation, and apply linear equations to nursing, money, sports and health situations.

Module 4 Lines, slope and graphs

Slope, intercepts and the equations of lines, parallel and perpendicular.

Start here · Animated walkthroughLines, slope and graphs WalkthroughWatch lines build themselves on the coordinate plane, from plotting points and climbing the rise-over-run staircase to writing the same line in slope-intercept, point-slope and standard form, with worked examples for intercepts, perpendicular lines, distance, midpoint and real-data rates. Explorer + practiceModule 4 LabDrag two points and see the slope triangle, all three equations and the intercepts update live, then practise with unlimited worked problems, a timed graph-matching game and real-life cases from nursing, sports, money and science.

Module 5 Functions, domain and range

What a function is, function notation, domain and range, and reading graphs.

Start here · Animated walkthroughFunctions, domain and range WalkthroughWatch inputs drop through a function machine and become points on a graph, then sweep vertical lines, read values, and see the domain and range shade in as shadows on the axes, with worked examples on interval notation, piecewise functions, rate of change, the eight toolkit graphs and even or odd symmetry. Explorer + practiceModule 5 LabDrag points along toolkit or typed functions to see f(x), the domain and range and the secant slope update live, then practise with unlimited worked problems, a timed Toolkit Match game and real-life cases from nursing, money, sports and science.

Module 6 Transformations of functions

Shifting, stretching and reflecting graphs, and combining functions.

Start here · Animated walkthroughTransformations of functions WalkthroughWatch toolkit graphs slide, flip, stretch and squeeze over their ghosted parents while the equation changes beside them, with an animated explanation of why f(x − h) moves right and worked examples for combining moves in order, factoring the inside, reciprocal asymptotes, writing equations from graphs, even and odd tests, a cost function and a dose response curve. Explorer + practiceModule 6 LabDrive a, b, h and k with sliders on any toolkit graph and play the moves in order, then practise unlimited worked problems, race the clock in Transformation Match and apply shifts and stretches to coupons, dosing, sports, temperature and jet lag.

Module 7 Quadratic functions and equations

Parabolas, vertex form and four ways to solve a quadratic equation.

Start here · Animated walkthroughQuadratic functions and equations WalkthroughWatch parabolas draw themselves outward from the vertex, slide into vertex form, and cross the x-axis at their roots, then solve quadratics four ways (factoring, square roots, completing the square with algebra tiles, and the quadratic formula derived from those tiles) in 20 animated steps with 12 worked examples, including the discriminant, complex solutions and projectile, area and revenue problems. Explorer + practiceModule 7 LabDrive a parabola with a, b, c or a, h, k sliders and see the vertex, axis, roots, discriminant and all three forms update live, then practise unlimited problems at three levels with hints and worked solutions, race the clock in Parabola Match, and apply quadratics to sports, money, health, science and everyday cases.

Module 8 Polynomial functions and their graphs

Graphs of higher-degree polynomials: end behavior, zeros and multiplicity.

Start here · Animated walkthroughPolynomial functions and their graphs WalkthroughZoom out until a polynomial looks like its leading term, watch graphs cross, bounce and flatten at their zeros, and watch long and synthetic division fill in one column at a time, then use the Remainder, Factor and Rational Zero Theorems to find every zero and build polynomials from their zeros. Explorer + practiceModule 8 LabDrag zeros along the x-axis and set multiplicities and the leading coefficient while the graph, end-behavior arrows and factored and expanded forms update live, then practise ten problem types at three levels with hints and worked solutions, race the clock, and apply polynomials to packaging, medicine, sports, business and lab design.

Module 9 Rational functions and asymptotes

Rational functions, asymptotes, holes and rational equations.

Start here · Animated walkthroughRational functions and asymptotes WalkthroughWatch 1/x shoot up near zero and flatten far away, see common factors cancel into open-circle holes, zoom far out to find horizontal asymptotes, divide to find slant ones, and solve rational equations and inequalities with extraneous-solution checks and sign charts, in 18 animated steps with 9 worked examples and 3 applications. Explorer + practiceModule 9 LabBuild a function from numerator and denominator factors and watch the asymptotes, holes, intercepts and behavior near each asymptote update live, then practise unlimited problems at three levels with hints and worked solutions, race the clock in Asymptote Rush, and apply rational functions to cost, medication, dilution, sports, circuits and work-rate cases.

Module 10 Composition and inverse functions

Combining and composing functions, then undoing them: one-to-one functions, inverses and their mirror-image graphs.

Start here · Animated walkthroughComposition and inverse functions WalkthroughWatch functions add and stack, chain together as animated machines and run backwards, with worked examples for composing from formulas, tables and graphs, the domain of a composition, swap and solve, checking f(f⁻¹(x)) = x, and the graph of an inverse folding over y = x, plus restricted domains and temperature, dose and currency applications. Explorer + practiceModule 10 LabBuild a function from blocks, toggle its folded inverse, the horizontal line test and a chained second function, then practise ten kinds of unlimited worked problems, race the clock in Inverse Match and solve real-life cases from cooking, nursing, travel money, running and shoe sizes.

Module 11 Exponential functions: growth and decay

Growth and decay that multiplies: exponential models, interest and the number e.

Start here · Animated walkthroughExponential functions: growth and decay WalkthroughWatch adding race against multiplying, see a·bˣ bend as a and b change, follow a drug dose halving every half-life and bacteria doubling in a dish, and watch the number e appear as interest compounds more and more often, across 19 animated steps with 11 worked examples. Explorer + practiceModule 11 LabDrive a and b with sliders while doubling or half-life markers and a value table update live, race a line against an exponential, practise unlimited worked problems at three levels, beat the clock in Rate Match, and apply exponential models to nursing, microbiology, credit cards, sports and viral videos.

Module 12 Logarithms and log equations

Logarithms as exponents, the log rules, and solving exponential and log equations.

Start here · Animated walkthroughLogarithms and log equations WalkthroughLearn that a logarithm is an exponent by counting factors, watch y = 2ˣ flip across y = x into y = log₂ x, see the product, quotient and power rules as stacks of exponent blocks, and follow 22 animated steps with worked examples for change of base, exponential and log equations (including an extraneous solution shown on the graph), pH with the 7.35 to 7.45 blood range, decibels, earthquake magnitude, doubling time and half-life. Explorer + practiceModule 12 LabDrag a point on a log curve and watch its mirror image on the inverse exponential while sliders change the base and shifts, place real liquids and sounds on a log number line, practise unlimited problems at three levels with hints and worked solutions, race the clock in Log Match and apply logs to blood pH, medication half-life, hearing, earthquakes and savings.

Module 13 Systems of equations and matrices

Systems of equations and inequalities by graphing, substitution, elimination and matrices.

Start here · Animated walkthroughSystems of equations and matrices WalkthroughWatch two lines meet at the one point that solves both equations, see parallel and coincident lines slide into place, and follow animated worked examples of substitution, elimination with scaled equations, row reduction on an augmented matrix, line-and-parabola and line-and-circle systems, shaded inequalities with feasible regions and corner points, and applications from IV saline mixtures to break-even, tickets and linear programming. Explorer + practiceModule 13 LabDrag the slope and intercept handles of two lines to watch the solution, both forms of the system and its classification update live, switch to inequality shading to find a feasible region and test its corners, practise unlimited worked problems at three levels, race the clock in Solution Sort and solve real-life cases.

Module 14 Sequences, series and counting

Arithmetic and geometric sequences, series, counting and the binomial theorem.

Start here · Animated walkthroughSequences, series and counting WalkthroughWatch arithmetic staircases climb, geometric terms grow and shrink, Gauss's pairing trick and the subtract-and-cancel trick add whole series, and a square fill up as an infinite series converges, then apply it all to savings, a drug dose model reaching steady state, theatre seating and a bouncing ball before counting outfits, orderings and committees and building Pascal's triangle for the binomial theorem, in 21 animated steps with 16 worked examples. Explorer + practiceModule 14 LabDrive arithmetic and geometric sequences with sliders and watch the terms, running sum and limit line respond, build Pascal's triangle and expand (a + b)ⁿ, practise unlimited problems in ten types at three levels with hints and worked solutions, race four Beat the Clock rounds, and solve real-life cases in nursing, money, sports, science and everyday life.