Mathematics

Course name: Advanced Functions Course code: (MHF4U)

Grade: 12

Type: University Preparation

This resource extends Learners` experience with functions. Learners will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Learners will also refine their use of the mathematical processes necessary for success in senior mathematics. This resource is intended both for Learners studying Calculus and Vectors as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.

Grade: 12

Type: University Preparation

Prerequisites: None

This resource extends Learners` experience with functions. Learners will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Learners will also refine their use of the mathematical processes necessary for success in senior mathematics. This resource is intended both for Learners studying Calculus and Vectors as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.

These learning resources support both Teacher led instruction and independent study.

For independent study, Learners may engage in active notebook reflection whenever prompted to participate in Discussions, Partner Activities, or Quizzes. No assignments can be submitted to TVO for grading or credit.

Educators may refer to the For Teachers page for extra information.

These learning resources support both Teacher led instruction and independent study.

For independent study, Learners may engage in active notebook reflection whenever prompted to participate in Discussions, Partner Activities, or Quizzes. No assignments can be submitted to TVO for grading or credit.

Educators may refer to the For Teachers page for extra information.

41 Lessons

15 Lessons

1.1 Recognizing applications of polynomial expressions

1.2 Describing polynomial expressions

1.3 Identifying polynomial functions

1.4 Classifying polynomial functions

1.5 Exploring the effect of degree on polynomial functions

1.6 Determining the end behaviours of polynomial functions

1.7 Describing polynomials in factored form

1.8 Developing the factor theorem and remainder theorem

1.9 Applying long division to polynomial functions

1.10 Investigating transformations of polynomial functions

1.11 Using symmetry to characterize odd and even polynomial functions

1.12 Exploring the graphs of reciprocal functions

Course name: Advanced Functions Course code: (MHF4U)

These learning resources support both Teacher led instruction and independent study.

These learning resources support both Teacher led instruction and independent study.

Unit 1 : Polynomial and rational functions 15 Lessons .chain-link{fill:#3d4649;} .icon-circle{fill:#e6e6e6;}

1.1 Recognizing applications of polynomial expressions 1.1 Recognizing applications of polynomial expressions

1.2 Describing polynomial expressions 1.2 Describing polynomial expressions

1.3 Identifying polynomial functions 1.3 Identifying polynomial functions

1.4 Classifying polynomial functions 1.4 Classifying polynomial functions

1.5 Exploring the effect of degree on polynomial functions 1.5 Exploring the effect of degree on polynomial functions

1.6 Determining the end behaviours of polynomial functions 1.6 Determining the end behaviours of polynomial functions

1.7 Describing polynomials in factored form 1.7 Describing polynomials in factored form

1.8 Developing the factor theorem and remainder theorem 1.8 Developing the factor theorem and remainder theorem

1.9 Applying long division to polynomial functions 1.9 Applying long division to polynomial functions

1.10 Investigating transformations of polynomial functions 1.10 Investigating transformations of polynomial functions

1.11 Using symmetry to characterize odd and even polynomial functions 1.11 Using symmetry to characterize odd and even polynomial functions

1.12 Exploring the graphs of reciprocal functions 1.12 Exploring the graphs of reciprocal functions

1.13 Connecting algebraic and graphical representations of rational functions 1.13 Connecting algebraic and graphical representations of rational functions

1.14 Solving rational equations 1.14 Solving rational equations

1.15 Solving inequalities 1.15 Solving inequalities

Unit 2 : Exponential and logarithmic functions 6 Lessons .chain-link{fill:#3d4649;} .icon-circle{fill:#e6e6e6;}

2.1 Defining logarithmic functions 2.1 Defining logarithmic functions

2.2 Exploring the characteristics of logarithmic functions 2.2 Exploring the characteristics of logarithmic functions

2.3 Connecting exponential and logarithmic functions 2.3 Connecting exponential and logarithmic functions

2.4 Solving logarithmic and exponential equations: Part 1 2.4 Solving logarithmic and exponential equations: Part 1

2.5 Solving logarithmic and exponential equations: Part 2 2.5 Solving logarithmic and exponential equations: Part 2

2.6 Exploring transformations of logarithmic functions 2.6 Exploring transformations of logarithmic functions

Unit 3 : Trigonometric functions 12 Lessons .chain-link{fill:#3d4649;} .icon-circle{fill:#e6e6e6;}

3.1 Introducing trigonometric functions 3.1 Introducing trigonometric functions

3.2 Introducing radian measure 3.2 Introducing radian measure

3.3 Exploring angles expressed in radian measure 3.3 Exploring angles expressed in radian measure

3.4 Exploring characteristics of the sine, cosine and tangent functions 3.4 Exploring characteristics of the sine, cosine and tangent functions

3.5 Graphing the Reciprocal Trigonometric Functions 3.5 Graphing the Reciprocal Trigonometric Functions

3.6 Investigating transformations of sinusoidal functions 3.6 Investigating transformations of sinusoidal functions

3.7 Connecting algebraic and graphical representations of sinusoidal functions 3.7 Connecting algebraic and graphical representations of sinusoidal functions

3.8 Solving trigonometric equations 3.8 Solving trigonometric equations

3.9 Verifying and applying the compound angle formulas 3.9 Verifying and applying the compound angle formulas

3.10 Exploring equivalent trigonometric expressions 3.10 Exploring equivalent trigonometric expressions

3.11 Deriving and applying the double angle and Pythagorean formulas 3.11 Deriving and applying the double angle and Pythagorean formulas

3.12 Solving trigonometric equations 3.12 Solving trigonometric equations

Unit 4 : Characteristics of functions 8 Lessons .chain-link{fill:#3d4649;} .icon-circle{fill:#e6e6e6;}

4.1 Expressing a function as the combination of other functions 4.1 Expressing a function as the combination of other functions

4.2 Combining functions by adding, subtracting or multiplying 4.2 Combining functions by adding, subtracting or multiplying

4.3 Combining functions by dividing or applying one function to another 4.3 Combining functions by dividing or applying one function to another

4.4 Determining the domain and range of combined functions 4.4 Determining the domain and range of combined functions

4.5 Exploring the average rate of change of a function 4.5 Exploring the average rate of change of a function

4.6 Determining the instantaneous rate of change of a function 4.6 Determining the instantaneous rate of change of a function

4.7 Describing composite functions using symmetry 4.7 Describing composite functions using symmetry

4.8 Solving the equations of composite functions arising from real world applications 4.8 Solving the equations of composite functions arising from real world applications

15 Lessons

1.1 Recognizing applications of polynomial expressions

1.2 Describing polynomial expressions

1.3 Identifying polynomial functions

1.4 Classifying polynomial functions

1.5 Exploring the effect of degree on polynomial functions

1.6 Determining the end behaviours of polynomial functions

1.7 Describing polynomials in factored form

1.8 Developing the factor theorem and remainder theorem

1.9 Applying long division to polynomial functions

1.10 Investigating transformations of polynomial functions

1.11 Using symmetry to characterize odd and even polynomial functions

1.12 Exploring the graphs of reciprocal functions

1.13 Connecting algebraic and graphical representations of rational functions

1.14 Solving rational equations

1.15 Solving inequalities

6 Lessons

2.1 Defining logarithmic functions

2.2 Exploring the characteristics of logarithmic functions

2.3 Connecting exponential and logarithmic functions

2.4 Solving logarithmic and exponential equations: Part 1

2.5 Solving logarithmic and exponential equations: Part 2

2.6 Exploring transformations of logarithmic functions

12 Lessons

3.1 Introducing trigonometric functions

3.2 Introducing radian measure

3.3 Exploring angles expressed in radian measure

3.4 Exploring characteristics of the sine, cosine and tangent functions

3.5 Graphing the Reciprocal Trigonometric Functions

3.6 Investigating transformations of sinusoidal functions

3.7 Connecting algebraic and graphical representations of sinusoidal functions

3.8 Solving trigonometric equations

3.9 Verifying and applying the compound angle formulas

3.10 Exploring equivalent trigonometric expressions

3.11 Deriving and applying the double angle and Pythagorean formulas

3.12 Solving trigonometric equations

8 Lessons

4.1 Expressing a function as the combination of other functions

4.2 Combining functions by adding, subtracting or multiplying

4.3 Combining functions by dividing or applying one function to another

4.4 Determining the domain and range of combined functions

4.5 Exploring the average rate of change of a function

4.6 Determining the instantaneous rate of change of a function

4.7 Describing composite functions using symmetry

4.8 Solving the equations of composite functions arising from real world applications