Advanced Functions
Course Description
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.
Overall Curriculum Expectations
By the end of this course, students will:
A. CHARACTERISTICS OF FUNCTIONS
| A1 | 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; |
| A2 | 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; |
| A3 | compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques. |
B. POLYNOMIAL AND RATIONAL FUNCTIONS
| B1 | identify and describe some key features of polynomial functions, and make connections between numeric, graphical, and algebraic representations of polynomial functions; |
| B2 | identify and describe some key features of the graphs of rational functions, and represent functions graphically; |
| B3 | solve problems involving polynomial and simple rational equations graphically and algebraically; |
| B4 | demonstrate an understanding of solving polynomial and simple rational inequalities. |
C. TRIGONOMETRIC FUNCTIONS
| C1 | demonstrate an understanding of the meaning and application of radian measure; |
| C2 | make connections between trigonometric ratios and the graphical and algebraic representations the corresponding trigonometric functions and between trigonometric functions and their and use these connections to solve problems; |
| C3 | solve problems involving trigonometric equations and prove trigonometric identities. |
D. PROBABILITY DISTRIBUTIONS
| D1 | demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions; |
| D2 | 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; |
| D3 | solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications. |
Course Content
| Unit | Title | Total Unit Time |
|---|---|---|
| UNIT 1 | Functions Exploring Absolute Value Properties of Graphs of Functions Sketching Graphs of Functions Inverse Relations Piecewise Functions Exploring Operations with Functions Modelling with FunctionsDetermining Average Rate of Change Estimating Instantaneous Rates of Change from Tables of Values and Equations Exploring Instantaneous Rates of Change Using Graphs Using Rates of Change to Create a Graphical Model Solving Problems Involving Rates of ChangeConsolidate your understanding of the characteristics of functions Create new functions by adding, subtracting, multiplying, and dividing functions Investigate the creation of composite functions numerically, graphically, and algebraically Determine key characteristics of these new functions Solve problems using a variety of function models |
28 hours |
| UNIT 2 | Identify and describe key characteristics of polynomial functions Divide one polynomial by another polynomial Factor polynomial expressions Solve problems that involve polynomial equations and inequalities graphically and algebraicallyDetermine the roots of polynomial equations, with and without technology Solve polynomial inequalities, with and without technology Solve problems involving polynomial function models Graph the reciprocal functions of linear and quadratic functions Identify the key characteristics of rational functions from their equations and use these characteristics to sketch their graphs Solve rational equations and inequalities with and without graphing technology Determine average and instantaneous rates of change in situations that are modelled by rational functions |
34 hours |
| UNIT 3 | Understand radian measure and its relationship to degree measure Use radian measure with trigonometric functions Make connections between trigonometric ratios and the graphs of the primary and reciprocal trigonometric functions Pose, model, and solve problems that involve trigonometric functions Solve problems that involve rates of change in trigonometric functionsRecognize equivalent trigonometric relationships Use compound angle formulas to determine the exact values of trigonometric ratios that involve sums, differences, and products of special angles Prove trigonometric identities using a variety of strategies Solve trigonometric equations using a variety of strategies |
30 hours |
| UNIT 4 | Relate logarithmic functions to exponential functions Describe the characteristics of logarithmic functions and their graphs Evaluate logarithms and simplify logarithmic expressions Solve exponential and logarithmic equations Use exponential and logarithmic functions to solve problems involving exponential growth and decay, and applications of logarithmic scales |
16 hours |
| Final evaluation (s) | Final Exam | 2 hours |
| Total : 110 hrs |
Evaluation in this course will be continuous throughout the year and will include a variety of evaluation methods. The tools highlighted in yellow will be used for the three different types of assessments:
Assessment of Learning |
Assessment of Learning |
Assessment of Learning |
Student Product
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Student Product
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Student Product
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Observation
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Observation
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Observation
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Conversation
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Conversation
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There are four levels of achievement for students who are passing the course:
- Level 1 (50-59%)
- Level 2 (60-69%)
- Level 3 (70-79%)
- Level 4 (80-100%)
Level 3 is the provincial standard for student achievement.
The percentage grade represents the quality of the students’ overall achievement of the expectations for the course and reflects the corresponding achievement as described in the achievement chart for mathematics. Term work will be 70% of the overall grade for the course; the summative evaluations will be 30% of the overall grade, incorporating summative evaluation and a final written examination.
| Percentage of Final Mark | Categories of Mark Breakdown |
|---|---|
| 70% | Term Work student product (tests) Observations(performance tasks) |
| 30% | Final Written Exam 30% |
Within the 70% term mark and the 30% summative mark, the breakdown of the achievement chart categories will be approximately 25% Knowledge/Understanding, Application 25%, Communication 25%, and Thinking/Inquiry 25%.
The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning. The final percentage grade represents the quality of the student’s overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline.
Proctoring
The tests are typically a paper-pen evaluation written at a mutually agreed time, date, and location. The tests will be proctored, meaning a suitable adult with a dedicated identifiable and authentic email address will supervise you writing the tests. This process ensures the security and integrity of the test. Any person related or affiliated to the student in a personal way cannot serve as a test supervisor.
Resources required by the student
- A non-programmable, non-graphing, scientific calculator
- A scanner, smart phone camera, or similar device to upload handwritten or hand-drawn work
- A front-facing camera on a desktop, laptop, or mobile device to allow for proctoring over the internet
- Internet access and a modern standards-compliant web browser
The tuition for this course is $800 for Canadian students and $2000 for international students.
Refunds
Maple Leaf School does not issue refunds. When a student enrolls in our course, MLS administration team undertakes many tasks including establishing electronic/physical files, assigning teachers and tracking the enrolment for Ministry purposes, etc. The work is completed by our school the moment you register online.
Course Curriculum
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| U2L5 | 00:00:00 | ||
| U2L6 | 00:00:00 | ||
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| U3L1 B | 00:00:00 | ||
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| U4L1 B | 00:00:00 | ||
| U4L2 A | 00:00:00 | ||
| U4L2 B | 00:00:00 | ||
| Final Exam Request | |||
| How to request? | 00:00:00 | ||
I learned a lot.
I will apply what I have learned to my daily life.
Perfect course
The perfect course! It has benefited me so much! Thanks to the teacher! And thanks to maths!