# Topic 2 – Functions and Equations Mathematics – International Baccalaureate

### Topic 2 – Functions and Equations Mathematics

# | TOPIC | TITLE | |
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1 | Study Plan | Study plan – Topic 2 – Functions & Equations | |

Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision. | |||

2 | Logarithms-Power of 2 | Powers of 2. | |

Objective: On completion of the lesson the student should be able to convert between logarithmic statements and index statements to the power of 2. | |||

3 | Logarithms-Equations and logs | Equations of type log x to the base 3 = 4. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the number from which the logarithm evolves. | |||

4 | Logarithms-Equations and logs | Equations of type log 32 to the base x = 5. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the base from which the number came. | |||

5 | Logarithms-Log laws | Laws of logarithms. | |

Objective: On completion of the lesson the student will be familiar with 5 logarithm laws. | |||

6 | Logarithms-Log laws expansion | Using the log laws to expand logarithmic expressions. | |

Objective: On completion of the lesson the student will be able to use the log laws to expand logarithmic expressions. | |||

7 | Logarithms-Log laws simplifying | Using the log laws to simplify expressions involving logarithms. | |

Objective: On completion of the lesson the student will be able to simplify logarithmic expressions using the log laws. | |||

8 | Logarithms-Log laws numbers | Using the log laws to find the logarithms of numbers. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the use of the log laws and be able to do more applications with numerical examples. | |||

9 | Logarithms-Equations and logs | Equations involving logarithms. | |

Objective: On completion of the lesson the student will be able to solve equations with log terms. | |||

10 | Logarithms-Logs to solve equations | Using logarithms to solve equations. | |

Objective: On completion of the lesson the student will be able to use logarithms to solve index equations with the assistance of a calculator. | |||

11 | Logarithms-Change base formula | Change of base formula | |

Objective: On completion of the lesson the student will have seen the change of base formula for logarithms and be capable of using it to change the logarithm of one base to another base. | |||

12 | Matrices | Translation in the number plane | |

Objective: On completion of the lesson the student will be able to place ordered pairs into a matrix, then perform translation by addition using a transformation matrix, then extract ordered pairs from an answer matrix. | |||

13 | Matrices | Translation by matrix multiplication | |

Objective: On completion of the lesson the student will be able to convert ordered pairs to elements of a matrix, multiply matrices together, where possible, and interpret the answer matrix on a number plane. | |||

14 | Transformations | Special transformations – reflections, rotations and enlargements. | |

Objective: On completion of the lesson the student will be able to perform transformations: to rotate, reflect and change the size of various shapes and or points where applicable. | |||

15 | Vectors | Vectors | |

Objective: On completion of the lesson the student will be able to represent a vector in matrix and diagrammatic form, as well as add two vectors using matrices and/or a diagram. | |||

16 | Simultaneous equations | Number of solutions (Stage 2) | |

Objective: On completion of the lesson of the lesson the student will identify simultaneous equations that are consistent, inconsistent or the same. | |||

17 | Vectors | 2 vector addition in 2 and 3D (stage 2) | |

Objective: On completion of the lesson the student will understand and use component forms for vector resolution. | |||

18 | Linear systems | Optimal solutions (Stage 2) – Vectors | |

Objective: On completion of the lesson the student will understand the process of linear programming to find optimal solutions. | |||

19 | Linear systems | Linear systems with matrices (Stage 2) | |

Objective: On completion of the lesson the student will process matrices formed from linear systems of equations. | |||

20 | Linear systems | Row-echelon form (Stage 2) | |

Objective: On completion of the lesson the student will process matrices formed from linear systems of equations using the row-echelon form. | |||

21 | Linear systems | Gauss Jordan elimination method (Stage 2) | |

Objective: On completion of the lesson the student will process matrices formed from linear systems of equations using the Gauss Jordan elimination method. | |||

22 | Functions | Definition, domain and range | |

Objective: On completion of this lesson the student will be able to select functions from relations by referring to the domain and range. | |||

23 | Functions | Notation and evaluations | |

Objective: On completion of the lesson the student will be understand different notations for functions. | |||

24 | Functions | More on domain and range | |

Objective: On completion of the lesson the student will be able to describe the domain and range using appropriate set notation. | |||

25 | Functions | Domain and range from graphical representations | |

Objective: On completion of the lesson the student will be able to describe the domain and range using appropriate set notation from graphical representations. | |||

26 | Functions | Evaluating and graphing piecewise functions | |

Objective: On completion of the lesson the student will be able to evaluate and graph piecewise functions. | |||

27 | Functions | Functions combinations | |

Objective: On completion of the lesson the student will be able to perform operations with functions while working with their domains. | |||

28 | Functions | Composition of functions | |

Objective: On completion of the lesson the student will understand composition of functions or a function of a function. | |||

29 | Functions | Inverse functions | |

Objective: On completion of the lesson the student will be able to find inverse functions, use the notation correctly and the horizontal line test will be used. | |||

30 | Functions | Rational functions Part 1 | |

Objective: On completion of the lesson the student will be able to work with the division of functions and to interpret this on the coordinate number plane showing vertical and horizontal asymptotes. | |||

31 | Functions | Rational functions Part 2 | |

Objective: On completion of the lesson the student will be able to use the degree of polynomials and polynomial division to assist in graphing rational functions on the coordinate number plane showing vertical, horizontal and slant asymptotes. | |||

32 | Functions | Parametric equations (Stage 2) | |

Objective: On completion of the lesson the student will be able to eliminate the parameter from a set of equations and identify appropriate restrictions on the domain and range. | |||

33 | Functions | Polynomial addition etc in combining and simplifying functions (Stage 2) | |

Objective: On completion of the lesson the student will have multiple techniques to understand and construct graphs using algebra. | |||

34 | Functions | Parametric functions (Stage 2) | |

Objective: On completion of the lesson the student will understand some standard parametric forms using trigonometric identities, appreciate the beauty of the the graphs that can be generated and an application to projectile motion. | |||

35 | Co-ordinate Geometry-Two point formula | Two point formula: equation of a line which joins a pair of points. | |

Objective: On completion of the lesson the student will be able to calculate the equation of a line given any two named points on the line. | |||

36 | Co-ordinate Geometry-Intercept form | Intercept form of a straight line: find the equation when given x and y | |

Objective: On completion of the lesson the student will have an effective and efficient method for calculating the equation of a straight line. | |||

37 | Co-ordinate Geometry-Parallel lines equations | Parallel lines: identify equation of a line parallel to another | |

Objective: On completion of the lesson the student will be able to decide if two or more lines are parallel or not and to solve problems involving parallel lines. | |||

38 | Co-ordinate Geometry-Perpendicular lines | Perpendicular lines. | |

Objective: On completion of the lesson the student will be able to derive the equation of a line, given that it is perpendicular to another stated line. | |||

39 | Co-ordinate Geometry-Inequalities | Inequalities on the number plane. | |

Objective: On completion of the lesson the student will be able to derive the expression for an inequality given its graph. The student will also be able to solve some problems using inequalities. | |||

40 | Co-ordinate Geometry-Theorems | Perpendicular distance | |

Objective: On completion of the lesson the student will be able to derive the formula to calculate the distance between a given point and a given line. The student will also be able to calculate the distance between parallel lines. | |||

41 | Co-ordinate Geometry-Theorems | Line through intersection of two given lines | |

Objective: On completion of the lesson the student will be able to calculate the equation of a line which goes through the intersection of two given lines and also through another named point or satisfies some other specified condition. | |||

42 | Co-ordinate Geometry-Theorems | Angles between two lines | |

Objective: On completion of the lesson the student will be able to calculate the angle between given lines and derive the equation of a line given its angle to another line. | |||

43 | Co-ordinate Geometry-Theorems | Internal and external division of an interval | |

Objective: On completion of the lesson the student will be able to divide an interval according to a given ratio and to calculate what point divides an interval in a given ratio for both internal and external divisions. | |||

44 | Translations | Transformations – reflections | |

Objective: On completion of the lesson the student will be able to take a pre-image and using the appropriate techniques, accurately show its image after reflection. | |||

45 | Geometric transformations | Geometry transformations without matrices: reflection (Stage 2) | |

Objective: On completion of this lesson the student will use and understand the language used in geometric transformations and perform reflections in a number plane. | |||

46 | Geometric transformations | Geometry transformations without matrices: translation (Stage 2) | |

Objective: On completion of this lesson the student will perform translations in a number plane. | |||

47 | Geometric transformations | Geometry transformations without matrices: rotation (Stage 2) | |

Objective: On completion of this lesson the student will perform and construct rotations. | |||

48 | Geometric transformations | Geometry transformations without matrices: dilation or enlargement (Stage 2) | |

Objective: On completion of this lesson the student will perform the non-congruent transformation of dilation or emlargement and calculate scale factor. | |||

49 | Geometric transformations | The definition and concept of combined transformations resulting in an equivalent single transformation. | |

Objective: On completion of this lesson the student will combine reflections and glide transformations to produce single isometric transformations. | |||

50 | Graphs, polynomials | Graphs of polynomials | |

Objective: On completion of the lesson the student will understand how to graph polynomials using the zeros of polynomials, the y intercepts and the direction of the curves. | |||

51 | Quadratic equations | Introduction to quadratic equations. | |

Objective: On completion of the lesson the student will understand simple quadratic equations. | |||

52 | Quadratic equations | Quadratic equations with factorisation. | |

Objective: On completion of the lesson the student will be able to find both roots of a quadratic equation by factorising. | |||

53 | Quadratic equations | Solving quadratic equations. | |

Objective: On completion of the lesson the student will have gained more confidence in working with quadratic equations. | |||

54 | Quadratic equations | Completing the square | |

Objective: On completion of the lesson the student will understand the process of completing the square. | |||

55 | Quadratic equations | Solving quadratic equations by completing the square | |

Objective: On completion of the lesson the student will understand the reasoning behind completing the square. | |||

56 | Quadratic equations | The quadratic formula | |

Objective: On completion of the lesson the student will be familiar with the quadratic formula. | |||

57 | Quadratic equations | Problem solving with quadratic equations | |

Objective: On completion of the lesson the student will be able to express a problem as a quadratic equation and then solve it. | |||

58 | Quadratic equations | Solving simultaneous quadratic equations graphically | |

Objective: On completion of the lesson the student will better understand why quadratic equations have two solutions and will be capable of solving quadratic equations and problems graphically.. | |||

59 | Functions and graphs | Quadratic polynomials of the form y = ax. + bx + c. | |

Objective: On completion of the lesson the student will be able to predict the general shape of a parabola and verify the predictions by sketching the parabola. The student will also be introduced to the discriminant and the axis. | |||

60 | Functions and graphs | Graphing perfect squares: y=(a-x) squared | |

Objective: On completion of the lesson the student will be able to analyse a curve and then check their work by graphing the curve. | |||

61 | Graphing roots | Graphing irrational roots | |

Objective: On completion of the lesson the student will be able to solve any polynomial which has real roots, whether they are rational or irrational. | |||

62 | Coordinate geometry | Solve by graphing | |

Objective: On completion of the lesson students will use the slope intercept form of a line to create graphs and find points of intersection. | |||

63 | Logarithms-Graph-log curve | The graph of the logarithmic curve | |

Objective: On completion of the lesson the student will be able to draw a logarithmic curve to a given base and know the general properties of log curves. | |||

64 | Logarithms-Log curves | Working with log curves. | |

Objective: On completion of the lesson the student will be able to solve problems with log curves | |||

65 | Logarithms-Complex numbers | Imaginary numbers and standard form | |

Objective: On completion of the lesson the student will use the a+bi form of complex numbers for addition and subtraction. | |||

66 | Logarithms-Complex numbers | Complex numbers – multiplication and division | |

Objective: On completion of the lesson the student will use the a+bi form of complex numbers for multiplication and division. | |||

67 | Logarithms-Complex numbers | Plotting complex number and graphical representation | |

Objective: On completion of the lesson the student will use the argand diagram to assist in the addition and subtraction of complex numbers. | |||

68 | Logarithms-Complex numbers | Absolute value | |

Objective: On completion of the lesson the student will use the absolute value or modulus of complex numbers | |||

69 | Logarithms-Complex numbers | Trigonometric form of a complex number | |

Objective: On completion of the lesson the student will write complex numbers in trigonometric or polar form. This may also be known as mod-ard form. | |||

70 | Logarithms-Complex numbers | Multiplication and division of complex numbers in trig form (Stage 2) | |

Objective: On completion of the lesson the student will use the trig form of complex numbers for multiplication and division. | |||

71 | Logarithms-Complex numbers | DeMoivre’s theorem (Stage 2) | |

Objective: On completion of the lesson the student will use DeMoivre’s theorem to find powers of complex numbers in trig form. | |||

72 | Logarithms-Complex numbers | The nth root of real and complex numbers (Stage 2) | |

Objective: On completion of the lesson the student will use DeMoivre’s theorem to find roots of complex numbers in trig form. | |||

73 | Logarithms-Complex numbers | Fundamental theorem of algebra (Stage 2) | |

Objective: On completion of the lesson the student will recognise and use the fundamental theorem of algebra to find factors for polynomials with real coefficients over the complex number field. | |||

74 | Sequences and Series-Compound interest | Compound interest | |

Objective: On completion of the G.P. lesson the student will understand the compound interest formula and how to use it and adjust the values of r and n, if required, for different compounding periods. | |||

75 | Sequences and Series-Superannuation | Superannuation. | |

Objective: On completion of the lesson the student will understand the method of finding the accumulated amount of a superannuation investment using the sum formula for a G.P. | |||

76 | Sequences and Series-Time payments | Time payments. | |

Objective: On completion of the lesson the student will have examined examples carefully and be capable of setting out the long method of calculating a regular payment for a reducible interest loan. | |||

77 | Sequences and Series | Applications of arithmetic sequences | |

Objective: On completion of the lesson the student will be capable of problems involving practical situations with arithmetic series. | |||

78 | Exam | Exam – Topic 2 – Functions & Equations | |

Objective: Exam |