Grade 12, University Preparation (MHF4U)
CourseTitle: Advanced Functions, Grade 12, University Preparation(MHF4U)
CourseType: University Preparation
Prerequisite: Functions, Grade 11, University Preparation or Mathematics for College Technology for CollegeTechnology
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2010 (Revised)
This course extends students’ experience with functions. Students will investigate the propertiesof 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 consolidatetheir understanding of mathematics before proceeding to any one of a variety of universityprograms.
Titles and Descriptions
Time and Sequence
Basic Skills Review
Many situations can be modeled graphically. Interpreting those graphs is something that requires you to become familiar with all of the aspects of these graphs. Students will recall a few basic facts of a distance time graph. In previous math courses, students saw some transformations and studied their effects on a given graph. These will also be reviewed. Finally, the concepts of function notation, relation, range, domain, and function notation will all be reviewed.
In this unit, students learn to identify and describe characteristics of polynomial functions including key features of their graphs and the relationship between finite differences and equations. The connection between equations and graphs of polynomial functions will also beexamined.
Equation-solving skills and graphing skills are combined to solve polynomial equations and inequalities. The relationship between the Remainder Theorem and the Factor Theorem is identified. Techniques for factoring polynomial functions of degree greater than two are examined and also applied to determine the roots of polynomial equations. Families of polynomial functions are analyzed. Finally, inequalities are solved graphically using technology, and algebraic methods for solving factorable polynomial inequalities.
In this Unit, students learn to analyse properties of those rational functions created by taking the reciprocal of linear functions and quadratic functions. The equations and key features of the graphs of these rational functions are analyzed. Different forms of rational functions are explored, and solved using a variety of methods, such as algebraically and using technology. Finally, connections between real-
world situations and rational functions are explored through problem solving.
In this unit, students his unit begins by studying trigonometry concepts and then applies these concepts to analyze trigonometric functions. The graphs of the sine, cosine, and tangent functions are analyzed and their key features are identified. Transformations of these graphs will also be examined. Finally, trigonometric equations are solved by combining factoring techniques with knowledge of trigonometric ratios of special angles.
Exponential and Logarithmic Functions
In this unit students study the exponential function and its inverse, including writing equations to fit data and graphing inverse functions. Then, logarithms and transformations of logarithmic functions are explored. The power law of logarithms is examined, including solving problems, evaluating logarithms, and graphing logarithmic functions. Problems and applications connecting logarithms and the physical sciences will be solved. Techniques to solve exponential equations are investigated and applied. The Product and Quotient Laws of logarithms are studied and techniques to solve logarithmic equations are demonstrated. Finally, mathematical modeling with exponential and logarithmic equations is examined to solve problems.
Characteristics of Functions
This unit develops students understanding of average and instantaneous rate of change, both numerically and graphically, and how to 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. Students will be taught how to determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, to describe some properties of the resulting functions, and to solve related problems. The unit helps students discover how to 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.
The final assessment task is a three hour exam worth 30% of the student’s final mark.