MHF4U – Advanced Functions – Grade 12
This course extends students’ experience with functions. Students 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. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course 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.
$450.00
*Course outline is subject to change
Overall Curriculum Expectations
By the end of this course, students will :
Exponential and Logarithmic Functions
- A1 – Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
- A2 – Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
- A3 – Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
Trigonometric Functions
- B1 – Demonstrate an understanding of the meaning and application of radian measure;
- B2 – Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
- B3 – Solve problems involving trigonometric equations and prove trigonometric identities.
Polynomial and Rational Functions
- C1 – Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
- C2 – Identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
- C3 – Solve problems involving polynomial and simple rational equations graphically and algebraically;
- C4 – Demonstrate an understanding of solving polynomial and simple rational inequalities.
Characteristics of Functions
- D1 – Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
- D2 – Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
- D3 – Compare the characteristics of functions, and solve problems by modeling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
Outline of Course Content
In this unit, students determine properties of the graph of polynomial functions and sketch the graph them. They then describe the nature of change in polynomial functions and determine the nature of change in polynomial functions. Finally, they will understand the Remainder and Factor Theorems. Students will factor polynomial expressions, determine the roots of polynomial equations and determine the real roots of non-factorable polynomial equations. They will compare the nature of change of polynomial functions with that of linear and quadratic functions and solve problems involving the abstract extensions of algorithms as well as solve factorable and non-factorable polynomial inequalities. Finally, they will write the equation of a family of polynomial functions and describe intervals and distances.
In this unit, students will sketch the graphs of sine and cosine functions for the angle expressed in radians, and determine and describe some key properties. They will also graph the tangent and reciprocal trigonometric functions, and relate those graphs to the graphs of sine and cosine. Students will learn how to graph transformations of trigonometric functions and will learn to determine the period, amplitude and phase shift of the transformed functions. Students will also solve a variety of problems including solving equations and real-world applications. Students will study the connections between exponentials and logarithms, simplify and evaluate numerical expressions, determine the key features of the graph of a logarithmic function and its transformations, and pose and solve problems involving exponential and logarithmic functions and equations. Average and instantaneous rates of change will also be analyzed.
In this unit, students will verify the product and quotient laws of logarithms and use them to simplify and evaluate numerical expressions. They will simplify and solve exponential and logarithmic equations, as well as solve applications problems by the use of exponential and logarithmic equations.
In this unit, students will develop an understanding of the combining of functions, especially the composition of functions. They will also use function models to solve a variety of problems.
Final evaluation should be designed to provide the opportunity for students to demonstrate comprehensive learning in each of the four Achievement Chart categories. Due to the emphasis of cumulative tests and examinations in university programs, a final evaluation should play a prominent role in the final assessment of the student.