Functions - MCR3U
ILC Course Code: MCR3U-C
Grade 11, University Preparation, 1.0 credit
Principles of Mathematics (MPM2D-C)
This course introduces the mathematical concept of the function by extending your experiences with linear and quadratic relations. You will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. You will reason mathematically and communicate your thinking as you solve multi-step problems.
Online submission of course work is required for this course using Microsoft Word © or OpenOffice Writer.
You will need a regular scientific calculator, i.e. one that can handle exponential and trigonometric functions.
Unit 1: Characteristics of Functions
In mathematics, the word “function” refers to the idea that one quantity depends on another. The concept of a function is evident in many daily situations. For example, the time it takes for a vehicle to travel 10 km depends on its speed; the length of a tree’s shadow depends on the time of day; the amount of an investment depends on the rate of interest earned.
In previous math courses, you studied linear functions and quadratic functions. You learned about the shape and features of the graphs of these functions in relation to their corresponding equations. In this course, you will extend your knowledge of linear and quadratic functions, as well as investigate and learn about other types of functions. Before you do that, you will need to develop a set of tools that can be used to explore and identify key characteristics of any type of function.
In the Key Questions for this unit you’ll practise simplifying polynomial expressions by using inverse operations. You’ll distinguish between relations and functions, and you’ll determine if the inverse of a function is also a function. Finally, you’ll identify the parameters of both linear and quadratic functions and use them to sketch the functions and solve problems.
Unit 2: Exponential Functions
In this unit, you will learn about a new type of function called an exponential function. This type of function arises from a variety of real-world applications, such as the radioactive decay of a chemical substance, population growth, and compound interest earned on investments and annuities.
Exponential functions are represented by powers that have a numerical base and a variable exponent. Before you can solve problems involving exponential functions, you will have to review and extend skills with powers, numerical bases, and exponents.
In the Key Questions for this unit you’ll identify exponential functions based on their characteristics, graph them, and solve problems involving them.
Unit 3: Trigonometric Functions
This unit is different from the other units in this course because it deals specifically with functions that are connected to the properties of triangles. Trigonometry, which is the study of the relationships between the angles and sides of triangles, has many real-world applications, as you will see. This unit begins by reviewing familiar concepts from Grade 10 and then extending them. Real-world situations that are periodic, such as tides and temperature changes, can be modelled by trigonometric functions and their transformations. Connections to the transformations investigated in Units 1 and 2 will be made.
In the Key Questions for this unit you’ll solve problems involving the six trigonometric ratios, the sine law, and the cosine law. You’ll also prove trigonometric identities, and pose and solve problems regarding the characteristics of sinusoidal functions.
Unit 4: Discrete Functions
In Unit 4, you will explore the concepts of sequences and series. You will investigate the patterns found in the Fibonacci sequence and Pascal’s triangle. This will lay the groundwork for your study of geometric and arithmetic sequences. In Lesson 18, you will examine series and solve real-world problems that relate to series. The remainder of the unit covers topics that someday could provide you with a big payoff: annuities, simple interest, and compound interest.
In the Key Questions for this unit you’ll represent sequences and series in a variety of ways and make connections between them. You’ll find the sums of arithmetic and geometric series, and apply what you’ve learned to practical examples involving compound interest and annuities.
This is the English version of the course.
Click here to view the French version of the course.