North Toronto Christian School MATH DEPARTMENT

Transcription

1 North Toronto Christian School MATH DEPARTMENT Course: Advanced Functions, Grade 12 MHF4U Teacher: D. Brooks 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. Text: Erdman, et al. Advanced Functions 12. McGraw-Hill Ryerson, Required Technology: Students must have their own TI-84 graphing calculator and bring it to all classes. The graphing calculator will be needed for class work, homework/practice exercises. The graphing calculator is also needed for evaluations including the final exam. Students will also need access to Excel or Google Sheets. Extra Help: Students are encouraged to come after classes from 3:00-4:00 pm to practice and get feedback on current work. Bring specific questions or come with a friend to work on homework. Bulletin Board: Posted information includes daily homework assignments and information regarding expectations for tests and assignments. Assessment and Evaluation: The assessments for the course include a test for each unit of the course and at least one real-world application assignment for each major strand of the course. The assignments will involve using the TI84 to model functions for real data. Tests are scheduled for the dates shown and will not change unless there are major conflicts with other school events. There will be numerous other assessments including observations of class work (e.g. investigations, work on black board), conversations, self /peer/teacher assessed quizzes on current work. The purpose is to help everyone be aware of current levels of progress on specific concepts and not all assessments will be included in the term grade.

2 North Toronto Christian School MATH DEPARTMENT Tests Real-world Application Asgn Other Assessments Term 1 test 1 18% assignment 1 5% Observations 3% Sept 28 TI84 quiz (F Sept 15) test % Observations 1% Oct 27 test % assignment % Observations and/or 1% Nov 27 Conversations test 4 Jan 18 20% Quizzes (special angles) 2% Term 2 quiz 4.5 Feb 5 test 5 Mar 7 test 6 Apr 17 test 6-7 May 15 test 8 Jun 1 4% assignment % Observations and/or 1% Conversations 20% Observations 1% 15% Observations 1% 20% assignment % Conversation: reading assignment 2% 15% (asgn 8, depending on TBD Observations 1% work load) Percentages refer to the percentage of the report card grade for the term. Each term is worth 35% of the final grade and the final exam is worth 30%. (i.e., 35% + 35% + 30% = 100%) The school office will calculate the official grade for each report card. You can use the stated weights to estimate your own grade during the term. (Please note that your teacher is not supposed to do this for you.) Final Exam: The final exam evaluates all overall expectations for the course according to all achievement categories. It will include a section on modelling functions with the TI84 that is similar to the real-world application assignments. Late Policy: Assignments are due by 3:30 p.m. on the previously announced date. Late submissions will be penalized 5% per day. In the event of significant extenuating circumstances, individual exceptions may be considered, but these should be discussed with the teacher in advance of the deadline. (Please note that work load is not a special circumstance; everyone is busy.) Work must be submitted in the assigned format paper or electronic or both to be considered on time. Academic Honesty: All submitted work is expected to be the original work of the person who hands it in. Discussion may be allowed in or out of class, but there should be substantial difference between the work of different students. Plagiarized work will receive a grade of zero. Course : The learning goals for the course are listed in detail in the course schedule on the following pages. Use the schedule to keep track of timing and your progress.

3 Polynomial & Rational Functions Unit 1 - Polynomial Functions 1 Review and develop TI84 skills TI84 quiz at start of 2 nd week of class Master common procedures, ZOOM, WINDOW, STAT, LIST, etc Investigate the key features of a polynomial function Investigate and indentify key features of power functions and their graphs Identify domain, range, end behavior, degree using correct terminology Identify even and odd functions Investigate and identify the degree of a polynomial functions Understand and use finite difference tables to determine degree of a polynomial Use a regression to determine the equation of a polynomial function Identify key features of polynomial functions (e.g. leading coefficient) Investigate and understand key features of polynomials Investigate the x-intercepts of polynomial graphs and relate them to equations Identify the order of polynomial roots Understand transformations of polynomial functions Identify and describe transformations Graph transformed functions and identify key features Use transformations in various Understand and use the secant of a function Calculate average rates of change using a secant Calculate average rate of change for real-world situations Understand and use the tangent of a function Calculate instantaneous rate of change using a table Calculate instantaneous rate of change for real-world situations Review and solve problems Test Unit 1 test Apply average and instantaneous rates of change in real world situations Calculate rates of change and explain their significance for real-world Real-world application assignment

4 Unit 2 Polynomial Equations and Inequalities Divide a polynomial function by a binomial Use long division to divide a polynomial by a binomial Use synthetic division to divide a polynomial Find the division statement for a polynomial quotient Investigate and use the Remainder and Factor theorems Use the remainder theorem to find remainders of polynomial division Use the remainder theorem to explain the factor theorem Use the factor theorem to find the factors of a polynomial without dividing Use the factor theorem Use the factor theorem to find polynomial factors and solve polynomial equations Understand smart guesses when using the factor theorem Use inspection and the features of polynomial roots to determine and justify the factorization of a polynomial Solve polynomial equations Solve polynomial equations using factoring: grouping, sum or difference of cubes, remainder theorem and division Solve polynomial equations numerically using the TI84 Identify most appropriate solution methods for solving a particular polynomial equation Investigate characteristics of families of polynomials Identify and describe the characteristics of a family of polynomials Solve polynomial inequalities numerically Understand the concept of an inequality as compared to an equation Solve polynomial inequalities graphically using TI Solve polynomial inequalities algebraically Solve polynomial inequalities by factoring and using a graph Solve polynomial inequalities using a table or number line Review and solve problems 29 Test Unit 1&2 test

5 Unit 3 Rational Functions Investigate and understand linear reciprocal functions Identify the general form of a reciprocal function (as compared to an inverse function) Sketch the graph of a linear reciprocal function Identify the key features of a linear reciprocal function Investigate and understand quadratic reciprocal functions Identify the key features and graphs of the reciprocal of a quadratic function Relate the key features of the graph of a quadratic reciprocal function to the degree of the quadratic polynomial Extend the concept of a reciprocal function to degree polynomials az b Investigate and understand rational functions, f x cx d Identify the key features of a linear rational function Graph a linear rational function given the equation Determine the equation of a linear rational function given the graph or key features of the function Explain concept of a horizontal asymptote and relate it to polynomial rational functions Solve Rational Equations & Inequalities Solve rational equations by factoring Solve rational equations using a hand drawn graph or the TI84 Solve rational inequalities using a hand drawn graph or the TI84 Solve rational inequalities by factoring and using a table or number line Solve rational inequalities by using cases Canadian Senior Math Contest, W November 23 (optional) Review and solve problems Test Unit 2&3 test

6 , 1-3 Solve real-world problems using rational functions and equations Model real-world using rational functions Use the TI84 (STATPLOT, regression, etc.) to analyze and calculate rates for realworld rational functions Real life application assignment Trigonometric Functions Unit 4 Trigonometry Understand radian measure Investigate radian measure and define an angle of 1 radian Express angles in terms of fractions of radians Evaluate trigonometric ratios of special angles Express special angles as fractions of radians Master (thoroughly memorize) trig ratios of special angles Evaluate trig ratios of special angles exactly using special triangles, the quadrant diagram (CAST rule) and graphs Investigate equivalent trigonometric expressions Investigate trigonometric expressions and identify identities using the TI84 Find and explain trigonometric identities such as the co-function identities using a variety of methods (graph, special triangle, quadrant diagram) Use trigonometric identities to express trig ratios in alternate forms Investigate and use compound angle identities Understand the purpose of the compound angle identities Identify and use the sum and difference identities for cosine and sine in a variety of situations (e.g. to express a trig function in a simpler or more useful form) Use the sum identities to determine the double angle identities Use the double angle identities in a variety of situations Christmas Break

7 Proving trig identities Use compound angle and other identities to prove trig identities algebraically Justify that a give trig equation is not an identity using a variety of methods (finding a counter example, using the TI84) Review and solve problems Test Unit 4 test Unit 5 Trigonometric Functions Identify and describe the basic trigonometric functions Plot the sine, cosine and tangent graphs and identify their key features Explore and explain various relationship between the trig functions (e.g. tan has the same x-intercepts as sin) Identify and describe reciprocal trigonometric functions Identify the reciprocal trig functions, cosecant, secant and cotangent, and distinguish them from the inverse trig functions (sin- 1 x) Identify the graphs of the reciprocal trig functions and state their key features Use the reciprocal trig functions to solve a variety of problems involving realworld geometry Understand transformations of sinusoidal functions Identify transformations of the sine and cosine functions Use transformed sinusoidal functions to model real-world situations Solve trigonometric equations using a variety of methods Solve trig equations using special angles Solve trig equations numerically using the TI84 Solve quadratic trig equations by factoring Solve trig equations algebraically using a variety of trig identities Review and solve problems

8 Test Unit 4&5 test Solve real-worlds problems using trig functions and graphs Model real-world using trig functions and real-world data Use the TI84 (STATPLOT, regression, etc.) to analyze and calculate rates for realworld rational functions Analyze and describe real-world situations and justify conclusions using graphs and rates. Real-world application assignment Exponential & Logarithmic Functions and Equations Unit 6 Exponential and Logarithmic Functions Understand the exponential function and investigate its inverse Master exponent laws Investigate the exponential function using a finite difference table Investigate the inverse of the exponential function using graphical methods and describe its key features Define and investigate the logarithmic function Understand the logarithmic function as the inverse of the exponential function Graph the logarithmic function and identify its key features Evaluate logarithmic expressions by relating them to power and exponent rules Investigate the use of semi-log graphs March Break Review & continue logarithms Investigate transformations of the logarithmic function Identify the key features of transformed log functions Plot transformed log functions State the equation of a transformed log functions given its graph

9 Investigate the power law of logarithms Investigate equivalent log expressions and state the power law Relate the power law to transformations of log functions Use the power law for logs in a variety of situations (e.g. simplifying log expressions, proving simple identities, etc.) Use logarithmic scales to describe quantities in the physical sciences Understand the purpose of a logarithmic scale Use the Richter, decibel and ph and other logarithmic scales to solve simple problems Euclid Contest W April 11 (optional/recommended) Review and solve problems Test Unit 6 test Unit 7 Exponential and Logarithmic Equations Identify and use equivalent powers Identify various growth patterns using powers Solve exponential equations by considering equivalent powers Solve exponential equations Identify and solve a variety of types of exponential equations including equivalent powers, quadratic and numerical methods Investigate the product & quotient laws of logarithms Investigate equivalent log expressions and state the product and quotient rules for logarithms Apply the product and quotient rules for logarithms in a variety of problems (e.g. simplifying log expressions, proving simple identities)

10 Solve logarithmic equations Use the laws of logarithms to simplify functions and solve logarithmic equations involving polynomial expressions Use inverse functions to solve exponential and logarithmic equations Use a variety of algebraic methods to solve equations involving combinations of logarithmic and other types of functions Solve real-worlds problems using exponential functions and graphs Model real-world using exponential functions Use the exponential regression to find an exponential function for a data set Justify the choice of an exponential (or other) function to model real-world data Understand the criteria for a good mathematical model and asses the suitability of various models Identify the key features of the logistic function and its purpose for modelling realworld data Review and solve problems Test Real-world application assignment Unit 6&7 test Combining Functions Unit 8 Combining functions Describe and identify various combined functions Investigate the results of basic operations with various functions Use the superposition principle to graph the sum of two functions Identify the key features of combined functions (review of unit 1,2,3 functions) Investigate and describe the composition of functions Understand the concept of composite functions in terms of mapping diagrams and real-world Use the notation for composite functions Solve problems involving composite functions Investigate and explain the composite of inverse functions

11 Solve equations & inequalities with non-algebraic solutions Investigate various equations and identify appropriate solution methods Identify equations which cannot be solved algebraically Solve equations and inequalities numerically using the TI Review equations from units 2,3,5 & 7 Review presentations 125 Review test equations Equations test 2.3, 3.4, 5.4, 7.4, 8.4 Specific content to be determined Model real-world situations with combined functions Use combined functions to model involving a variety of types of growth rate Review for final exam Review presentations