1.1 Real Numbers
Classify, compare, and calculate with real numbers.
Explore topic ↗Textbook-aligned learning
Precalculus
Build the foundations for calculus, one textbook section at a time. Learn the ideas, follow worked examples, then practise and print a worksheet.
91 textbook sections · Each has rules, two worked examples, online practice, and a printable worksheet with a worked key. The practice questions are original and focus on the named section.
Search a lesson title or section number, such as 1.1.
91 lessons in textbook order.
No lessons found
Try a section number or a shorter lesson title.
CHAPTER 1
12 lessons
1.2 Exponents and Radicals
Focus on exponent rule and negative exponent.
Explore topic ↗1.3 Algebraic Expressions
Expand, factor, and simplify polynomials.
Explore topic ↗1.4 Rational Expressions
Combine fractions and keep domain restrictions.
Explore topic ↗1.5 Equations
Focus on solve an equation and quadratic equation.
Explore topic ↗1.6 Complex Numbers
Focus on complex addition and complex conjugate.
Explore topic ↗1.7 Modeling with Equations
Focus on translate and solve and distance model.
Explore topic ↗1.8 Inequalities
Solve and write solution intervals.
Explore topic ↗1.9 The Coordinate Plane; Graphs of Equations; Circles
Read a circle’s center and radius.
Explore topic ↗1.10 Lines
Focus on slope and parallel slopes.
Explore topic ↗1.11 Solving Equations and Inequalities Graphically
Focus on graph intersection and interpret an intersection.
Explore topic ↗1.12 Modeling Variation
Focus on direct variation and direct variation constant.
Explore topic ↗CHAPTER 2
9 lessons
2.1 Functions
Evaluate a function at a given input.
Explore topic ↗2.2 Graphs of Functions
Identify allowed inputs and possible outputs.
Explore topic ↗2.3 Getting Information from the Graph of a Function
Find intercepts, extrema, and intervals.
Explore topic ↗2.4 Average Rate of Change of a Function
Measure the change in output per unit input.
Explore topic ↗2.5 Linear Functions and Models
Focus on slope and rate from a table.
Explore topic ↗2.6 Transformations of Functions
Track shifts of familiar function graphs.
Explore topic ↗2.7 Combining Functions
Add, multiply, divide, and compose functions.
Explore topic ↗2.8 One-to-One Functions and Their Inverses
Reverse a one-to-one function.
Explore topic ↗2.M Focus on Modeling (Chapter Supplement)
Build, test, and optimize function models from geometric and real-world constraints.
Explore topic ↗CHAPTER 3
8 lessons
3.1 Quadratic Functions and Models
Focus on vertex and maximum value.
Explore topic ↗3.2 Polynomial Functions and Their Graphs
Connect factored form to zeros.
Explore topic ↗3.3 Dividing Polynomials
Use the remainder theorem.
Explore topic ↗3.4 Real Zeros of Polynomials
Focus on factor theorem and evaluate a polynomial.
Explore topic ↗3.5 Complex Zeros and the Fundamental Theorem of Algebra
Pair nonreal zeros, factor over the complex numbers, and build a polynomial from its zeros.
Explore topic ↗3.6 Rational Functions
Focus on excluded input and horizontal asymptote.
Explore topic ↗3.7 Polynomial and Rational Inequalities
Focus on sign of factored product and exclude denominator zero.
Explore topic ↗3.M Focus on Modeling (Chapter Supplement)
Fit polynomial models to data and interpret predictions with realistic constraints.
Explore topic ↗CHAPTER 4
8 lessons
4.1 Exponential Functions
Evaluate and interpret exponential growth and decay.
Explore topic ↗4.2 The Natural Exponential Function
Focus on natural exponential and natural logarithm inverse.
Explore topic ↗4.3 Logarithmic Functions
Connect logarithms to exponents.
Explore topic ↗4.4 Laws of Logarithms
Focus on product law and quotient law.
Explore topic ↗4.5 Exponential and Logarithmic Equations
Focus on solve an exponential equation and check domain.
Explore topic ↗4.6 Modeling with Exponential Functions
Focus on exponential growth and decay model.
Explore topic ↗4.7 Logarithmic Scales
Focus on logarithmic ratio and interpret equal readings.
Explore topic ↗4.M Focus on Modeling (Chapter Supplement)
Fit exponential and power models, interpret logarithmic scales, and test predictions against data.
Explore topic ↗CHAPTER 5
7 lessons
5.1 The Unit Circle
Connect angle measure to coordinates.
Explore topic ↗5.2 Trigonometric Functions of Real Numbers
Use sine, cosine, tangent, and reciprocal ratios.
Explore topic ↗5.3 Trigonometric Graphs
Read amplitude and period from sine and cosine.
Explore topic ↗5.4 More Trigonometric Graphs
Focus on tangent period and tangent asymptote.
Explore topic ↗5.5 Inverse Trigonometric Functions and Their Graphs
Focus on principal inverse sine and inverse sine domain.
Explore topic ↗5.6 Modeling Harmonic Motion
Focus on amplitude and midline and harmonic midline.
Explore topic ↗5.M Focus on Modeling (Chapter Supplement)
Fit sinusoidal models to periodic data and evaluate predictions within a meaningful domain.
Explore topic ↗CHAPTER 6
6 lessons
6.1 Angle Measure
Focus on degree conversion and arc length.
Explore topic ↗6.2 Trigonometry of Right Triangles
Focus on right triangle sine and tangent ratio.
Explore topic ↗6.3 Trigonometric Functions of Angles
Focus on trigonometry from a point and tangent from coordinates.
Explore topic ↗6.4 Inverse Trigonometric Functions and Right Triangles
Focus on inverse cosine and right triangle angle.
Explore topic ↗6.5 The Law of Sines
Focus on law of sines and ambiguous case.
Explore topic ↗6.6 The Law of Cosines
Focus on law of cosines and find cosine from sides.
Explore topic ↗CHAPTER 7
5 lessons
7.1 Trigonometric Identities
Focus on pythagorean identity and reciprocal identity.
Explore topic ↗7.2 Addition and Subtraction Formulas
Focus on sine addition and sine sum.
Explore topic ↗7.3 Double-Angle, Half-Angle, and Product-Sum Formulas
Focus on double-angle sine and half angle sign.
Explore topic ↗7.4 Basic Trigonometric Equations
Focus on sine equation and unit circle sine solution.
Explore topic ↗7.5 More Trigonometric Equations
Focus on squared trig equation and square-root branches.
Explore topic ↗CHAPTER 8
4 lessons
8.1 Polar Coordinates
Focus on polar conversion and polar equivalent.
Explore topic ↗8.2 Graphs of Polar Equations
Focus on constant polar radius and polar line.
Explore topic ↗8.3 Polar Form of Complex Numbers; De Moivre’s Theorem
Focus on de moivre modulus and multiply polar moduli.
Explore topic ↗8.4 Plane Curves and Parametric Equations
Focus on evaluate a parametric point and direction of travel.
Explore topic ↗CHAPTER 9
6 lessons
9.1 Vectors in Two Dimensions
Add and scale two-dimensional vectors.
Explore topic ↗9.2 The Dot Product
Focus on dot product and dot product calculation.
Explore topic ↗9.3 Three-Dimensional Coordinate Geometry
Focus on distance in space and sphere equation.
Explore topic ↗9.4 Vectors in Three Dimensions
Focus on vector addition and magnitude.
Explore topic ↗9.5 The Cross Product
Focus on cross product and parallel cross product.
Explore topic ↗9.6 Equations of Lines and Planes
Focus on plane normal and normal vector.
Explore topic ↗CHAPTER 10
9 lessons
10.1 Systems of Linear Equations in Two Variables
Focus on linear system and no-solution system.
Explore topic ↗10.2 Systems of Linear Equations in Several Variables
Focus on three-variable system and dependent system.
Explore topic ↗10.3 Matrices and Systems of Linear Equations
Focus on augmented matrix and identify pivot.
Explore topic ↗10.4 The Algebra of Matrices
Focus on matrix addition and matrix multiplication dimensions.
Explore topic ↗10.5 Inverses of Matrices and Matrix Equations
Focus on invertibility and solve a matrix equation.
Explore topic ↗10.6 Determinants and Cramer’s Rule
Focus on determinant and cramer rule condition.
Explore topic ↗10.7 Partial Fractions
Focus on partial fractions and proper rational expression.
Explore topic ↗10.8 Systems of Nonlinear Equations
Focus on line and parabola and count intersections.
Explore topic ↗10.9 Systems of Inequalities
Focus on system of inequalities and boundary type.
Explore topic ↗CHAPTER 11
6 lessons
11.1 Parabolas
Focus on parabola focus and parabola orientation.
Explore topic ↗11.2 Ellipses
Focus on ellipse vertex and ellipse center.
Explore topic ↗11.3 Hyperbolas
Focus on hyperbola vertex and hyperbola orientation.
Explore topic ↗11.4 Shifted Conics
Focus on shifted center and shifted circle radius.
Explore topic ↗11.5 Rotation of Axes
Focus on new-coordinate formula and conic discriminant.
Explore topic ↗11.6 Polar Equations of Conics
Focus on classify a conic and hyperbola eccentricity.
Explore topic ↗CHAPTER 12
6 lessons
12.1 Sequences and Summation Notation
Focus on summation and sequence term.
Explore topic ↗12.2 Arithmetic Sequences
Focus on arithmetic nth term and arithmetic sum.
Explore topic ↗12.3 Geometric Sequences
Focus on geometric nth term and finite geometric sum.
Explore topic ↗12.4 Mathematics of Finance
Focus on future value of an annuity and monthly loan payment.
Explore topic ↗12.5 Mathematical Induction
Verify a base case and prove an induction step.
Explore topic ↗12.6 The Binomial Theorem
Focus on binomial coefficient and expand a binomial square.
Explore topic ↗CHAPTER 13
5 lessons
13.1 Finding Limits Numerically and Graphically
Focus on continuous limit and undefined point.
Explore topic ↗13.2 Finding Limits Algebraically
Focus on removable factor and direct substitution.
Explore topic ↗13.3 Tangent Lines and Derivatives
Focus on tangent slope and tangent line.
Explore topic ↗13.4 Limits at Infinity; Limits of Sequences
Focus on sequence limit and sequence convergence.
Explore topic ↗13.5 Areas
Focus on area under constant graph and triangle under a line.
Explore topic ↗Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.