About This Resource vii

Acknowledgements xiv

1 Lines and linear relationships 1

1.1 Overview 2

1.2 Linear equations and inequations 3

1.3 Systems of simultaneous linear equations 8

1.4 Linear graphs and their equations 15

1.5 Intersections of lines and their applications 24

1.6 Straight lines and gradients 31

1.7 Bisection and lengths of line segments 38

1.8 Review 45

Answers 49

2 Algebraic foundations 55

2.1 Overview 56

2.2 Algebraic skills 57

2.3 Pascal’s triangle and binomial expansions 67

2.4 The binomial theorem 71

2.5 Sets of real numbers 80

2.6 Surds 87

2.7 Review 99

Answers 103

3 Quadratic relationships 109

3.1 Overview 110

3.2 Quadratic equations with rational roots 111

3.3 Quadratics over R 116

3.4 Applications of quadratic equations 129

3.5 Graphs of quadratic polynomials 134

3.6 Determining the rule of a quadratic polynomial from a graph 146

3.7 Quadratic inequations 152

3.8 Quadratic models and applications 160

3.9 Review 166

Answers 171

4 Cubic polynomials 183

4.1 Overview 184

4.2 Polynomials 185

4.3 The remainder and factor theorems 197

4.4 Graphs of cubic polynomials 207

4.5 Equations of cubic polynomials 218

4.6 Cubic models and applications 230

4.7 Review 237

Answers 241

5 Quartic polynomials 257

5.1 Overview 258

5.2 Quartic polynomials 259

5.3 Families of polynomials 272

5.4 Numerical approximation of roots of polynomial equations 277

5.5 Review 287

Answers 291

6 Functions and relations 301

6.1 Overview 302

6.2 Functions and relations 303

6.3 The rectangular hyperbola and the truncus 314

6.4 The square root function 332

6.5 Other functions and relations 339

6.6 Transformations of functions 353

6.7 Review 365

Answers 370

7 Probability 389

7.1 Overview 390

7.2 Probability review 391

7.3 Conditional probability 403

7.4 Independence 413

7.5 Counting techniques 420

7.6 Binomial coefficients and Pascal’s triangle 434

7.7 Review 444

Answers 449

8 Trigonometric functions 455

8.1 Overview 456

8.2 Trigonometric ratios 457

8.3 Circular measure 466

8.4 Unit circle definitions 475

8.5 Symmetry properties 486

8.6 Graphs of the sine and cosine functions 497

8.7 Review 508

Answers 512

9 Trigonometric functions and applications 521

9.1 Overview 522

9.2 Trigonometric equations 523

9.3 Transformations of sine and cosine graphs 533

9.4 Applications of sine and cosine functions 542

9.5 The tangent function 549

9.6 Trigonometric identities and properties 558

9.7 Review 567

Answers 572

10 Exponential functions and logarithms 585

10.1 Overview 586

10.2 Indices as exponents 587

10.3 Indices as logarithms 596

10.4 Graphs of exponential functions 606

10.5 Applications of exponential functions 617

10.6 Inverses of exponential functions 623

10.7 Review 636

Answers 641

11 Introduction to differential calculus 653

11.1 Overview 654

11.2 Rates of change 655

11.3 Gradients of secants 664

11.4 The derivative function 669

11.5 Differentiation of polynomials by rule 677

11.6 Review 688

Answers 692

12 Differentiation and applications 701

12.1 Overview 702

12.2 Limits, continuity and differentiability 703

12.3 Coordinate geometry applications of differentiation 714

12.4 Curve sketching 724

12.5 Optimisation problems 733

12.6 Rates of change and kinematics 739

12.7 Derivatives of power functions (extending) 750

12.8 Review 756

Answers 761

13 Anti-differentiation and introduction to integral calculus 771

13.1 Overview 772

13.2 Anti-derivatives 773

13.3 Anti-derivative functions and graphs 780

13.4 Applications of anti-differentiation 787

13.5 The definite integral 794

13.6 Review 806

Answers 811

Algorithms and Pseudocode

Glossary 819

Index 827