Advanced_Precalculus_Curriculum__Aug_07_

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					Course Title:          Advanced Precalculus, Level 5
Grade:                        11
Length of Course:      One Year (5 credits)
Prerequisites:         Algebra, Geometry, Algebra 2

Description:
       For the serious cyclist, keeping focus on the road ahead is essential. Precalculus instruction
       should use a similar approach. In keeping with this thought we concentrate on those topics
       which are essential for success in calculus, emphasizing depth rather than breadth. A
       central theme of this course is functions as models of change. Functions express the way
       one variable quantity is related to another quantity. This course emphasizes that functions
       can be grouped into families and that functions can be models for real world data. Once
       introduced a family of functions is compared and contrasted with other families of
       functions. Linear, exponential, power, logarithmic, trigonometric and rational functions are
       covered in depth.

        Recognizing that for some, precalculus can be a capstone course, inclusion of quantitative
        literacy topics such as combinatorics, probability, sequences and series has been provided.

        This class we will be guided by two principles. The first is The Rule of Three which
        requires that every topic be presented geometrically, numerically and algebraically. The
        second guiding principle is The Way of Archimedes which states that formal definitions
        and procedures evolve from the investigation of practical problems. The problems we
        consider come from the both the Natural and Social Sciences, as well as Business arenas
        and are generally understood to be important.

        The graphing calculator is a marvelous tool which this course employs both as a problem solver
        and an exploratory tool to anticipate upcoming concepts. The National Council of Teachers of
        Mathematics states: “Some mathematics becomes more important because technology requires it;
        some mathematics becomes less important because technology replaces it. Some mathematics
        becomes possible because technology allows it.” Throughout our studies appropriate use of
        technology is incorporated.

        This course strives to give students a proper balance between the mastery of skills and the
        comprehension of key concepts. With that in mind, this curriculum guide clearly defines
        the learning objectives for each unit in terms of the key skills and key concepts that must be
        mastered within each unit.

Evaluation:
      Student performance will be measured using a variety of instructor-specific quizzes and
      chapter tests as well as a common departmental Midterm and Final Exam. Assessments
      will equally emphasize measurement of the degree to which required skills have been
      mastered as well as how well key concepts have been understood.

Text:
        Precalculus with Trigonometry, Concepts and Applications, Paul A. Foerster, Key Curriculum
        Press 2003

Reference Texts:
      Advanced Mathematics, Richard G. Brown, Houghton-Mifflin 1992
                                                        Advanced Precalculus Grade 11
                                                 Unit 1: Functions & Mathematical Models
        Topic                                     Learning Objectives: Key Definitions, Skills and Concepts

1.1 Functions:           What is a function? What are the four ways a function can be represented? What are the domain and range of a
Algebraically,           function?
Numerically,
Graphically and          Skills check, ability to:
Verbally                     Given the graph of a function, be able to answer questions regarding function values at specific inputs, find its
                             domain and range
                             Given a table of values for a function, be able to graph and find domain and range
                             Given an equation of a function, be able to graph, find function values at specific inputs and find domain and
                             range
                             Given anecdotes, be able to sketch appropriate graphs

                         Concept check:
                            Given that altitude is a function of time, is it always, sometimes or never true that time is a function of
                             altitude?
                            What is the difference between interpolation and extrapolation? Why is it important to recognize these
                             differences?
                              Does all real life data form a function?


1.2 Kinds of Functions   What is the mathematical definition of a function? How do you use Euler notation to describe a function? What
                         are some of the functions families that have already been studied? How do we use Boolean logic to restrict the
                         domain of a function on a graphing calculator? What is the vertical line test?

                         Skills check, ability to:
                             Given the equation of a function and a restricted domain, create a graph on the calculator and state its range
                             and intercepts
                             Given the sketch of a function, identify the family to which it most likely belongs
                             Given two quantities that are related, create a reasonable sketch
                             Use the Vertical Line Test to determine if a sketch is that of a function

                         Concept check:
                            Explain how the vertical line test is a graphical interpretation of the Golden Rule of Functions.


                                                         Grade 11 Advanced Precalculus, page 2
                                                    Advanced Precalculus Grade 11
        Topic                                   Learning Objectives: Key Definitions, Skills and Concepts

1.3 Dilations and    What are dilations and translations? How are these accomplished graphically and algebraically?
Translations of
Function Graphs      Skills check, ability to:
                         Draw the graph of f(x  c), f(cx), cf(x), and f(x)  c, given the graph of f
                         Given the equation of f(x), write the equations for g(x) if g(x) is a dilation and/or translation of f(x)

                     Concept check:
                        What is the benefit of understanding about transformations of functions?
                        What are the real world applications of composition of functions?
                        If c > 0, why does f ( x  c) shift the graph of f (x) to the right and not to the left as one might expect?


1.4 Composition of   What is composition of functions? What is the domain of a composed function? How can the graphing calculator
Functions             be used to display composed functions?

                     Skills check, ability to:
                         Evaluate f(g(c)) given rules, graphs or tables for f and g
                         Find f(g(x)) given rules for f and g
                         Find the domain of f(g(x)) given domains for f and g
                         Express a complicated function as the composition of easier functions

                     Concept check:
                        Is the function f(g(x)) the same as g(f(x))? Why or why not?
                        If f(x) is linear and g(x) is linear, must f(g(x)) also be linear? Why or why not?




                                                      Grade 11 Advanced Precalculus, page 3
                                                      Advanced Precalculus Grade 11
         Topic                                    Learning Objectives: Key Definitions, Skills and Concepts

1.5 Inverse of          What is the definition of an inverse function? Do all functions have inverse functions? How can you tell if a
Functions               function will have an inverse that is also a function? What is the horizontal like test? How can you graphically
                        construct inverses? What is a one-to-one function? How can you algebraically construct inverses? What are
                        parametric equations?

                        Skills check, ability to:
                            Test for whether a given function is one-to-one and the existence of an inverse function (horizontal line test)
                            Verify whether two functions are inverses
                            Find the inverse of a one-to-one function (algebraic, graphical, and numerical method) and its corresponding
                            domain and range
                            Use the parametric mode on a graphing calculator to plot a function and its inverse

                        Concept check:
                           How can a function that is not one-to-one, have an inverse function?
                           In the expression f 1 (x), is -1 an exponent?
                           Can a discrete function that is not always monotonic have an inverse? Why or why not?
                            Identify real world examples where change should most accurately be described using a parameter


1.6 Reflections,        How are reflections over the coordinate axes accomplished using transformations? What are even and odd
Absolute Values and     functions? What is the Greatest Integer Function? What is a piecewise function?
Other Transformations
                        Skills check, ability to:
                           Given the graph of f(x), sketch f(-x). –f(x), f ( x) and f( x )
                           Sketch the Greatest Integer Function and identify the places where it has a step discontinuity
                           Sketch piecewise functions

                        Concept check:
                           Is every function even or odd?
                           What is the visual impact of a function that is even/odd?
                           Create an anecdote that would result in a step function




                                                        Grade 11 Advanced Precalculus, page 4
                                                     Advanced Precalculus Grade 11
                                         Unit 2: Periodic Functions and Right Triangle Problems
        Topic                                   Learning Objectives: Key Definitions, Skills and Concepts

2.2 Measurement of    How are angles drawn on the Cartesian Plane? What is standard position of an angle? What does a negative
Rotation              angle measure mean? What are quadrantal angles? What are coterminal angles? What is a reference angle?

                      Skills check, ability to:
                          Given an angle in degrees, both positive and negative, correctly sketch it in standard form, identify its
                          quadrant, and its reference angle
                          Correctly interpret subdivisions of a degree (minutes/seconds)

                      Concept check:
                        True/False: The reference angle of 90º is 0º
                        True/False: The position of the terminal ray of an angle drawn in standard position is unique to the angle
                        measure
                        Find a formula for all angles coterminal with a given angle 

2.3 Sine and Cosine   What is a periodic function? What are the definitions of the Sine and Cosine Functions on the Unit Circle?
Functions
                      Skills check, ability to:
                          Correctly find the sine & cosine of an angle using a calculator
                          Correctly find the sine and cosine of an angle given a point on its terminal ray
                          Correctly sketch the graph of the sine and cosine function and identify its domain and range
                          Correctly sketch transformations of the sine and cosine functions
                          Give the exact values of the sine and cosine functions for the quadrantal angles
                          Give the exact values of the sine and cosine functions for the special angles (30º, 60º, 45º, 120º, etc)

                      Concept check:
                         Explain why the function vales for the sine function change sign where they do
                         Where is the sine function increasing/decreasing? Why is this happening?
                         State the quadrants where one would expect to see a positive/negative sine. Explain how you arrived at your
                         answer.
                         Why is range of both the sine and cosine functions [-1, 1]?
                         Why doesn’t the radius of the circle influence the values of the sine and cosine functions?
                         Draw the BOX diagram for the sine and cosine functions showing input, output and rule
                         True/False: If A and B are angles of a triangle such that A > B, then cos A > cos B. Justify your answer.

                                                       Grade 11 Advanced Precalculus, page 5
                                                        Advanced Precalculus Grade 11
         Topic                                      Learning Objectives: Key Definitions, Skills and Concepts

2.4 Values of the Six     What are the definitions of the six trig functions?
Trigonometric
Functions                 Skills check, ability to:
                              Correctly state the exact values of the six trig functions of special and quadrantal angles
                              Correctly state the values of the six trig functions of an angle given a point on its terminal ray
                              Correctly identify the domain and range of the six trig functions
                              Correctly state the values of five trig functions of an angle given the quadrant in which its terminal side lies
                              and the value of one of its trig functions

                          Concept check:
                             Explain where and why each of the six trig functions are increasing/decreasing
                             State the quadrants in which one would expect to find positive/negative functions. Justify your answer.
                             Which of the following does not represent a real number:
                             A) sin 30º
                             B) tan 45º
                             C) cos 90º
                             D) csc 90º
                             E) sec 90º

                              Before calculators became common classroom tools, students used trig tables to find trigonometric ratios.
                              Below is a simplified trig table for angles between 40º and 50º. Without using a calculator, can you
                              determine which column gives sine values, which gives cosine values and which gives tangent values?
                              Justify your answers.
                        Angle                         ?                            ?                           ?
                        40º                           0.8391                       0.6428                      0.7660
                        42º                           0.9004                       0.6691                      0.7431
                        44º                           0.9657                       0.6947                      0.7193
                        46º                           1.0355                       0.7193                      0.6947
                        48º                           1.1106                       0.7431                      0.6691
                        50º                           1.1917                       0.7660                      0.6428




                                                           Grade 11 Advanced Precalculus, page 6
                                                       Advanced Precalculus Grade 11
        Topic                                      Learning Objectives: Key Definitions, Skills and Concepts

2.5 Inverse Trig         What is the principal branch of the sine and cosine functions? What does the notation sin   1
                                                                                                                          (x) =  mean?
Functions and Triangle   What is an angle of elevation/depression?
Problems
                         Skills check, ability to:
                            Correctly use a calculator to find sin 1 (c)
                            Correctly solve right triangle problems that require finding sides and/or angles

                         Concept Check:
                            Why must we use the principal branch of the periodic functions to develop an inverse function?
                            Why isn’t the entire sine function invertible?
                            State the domain and range of the inverse sine and cosine functions. Justify your answer
                            Draw the BOX diagram for the inverse sine and cosine functions showing input, output and rule
                            Why is sin 1 (2) undefined? What inputs to the inverse sine BOX result in an undefined output? Why?
                            Why isn’t cos 1 (-.6) a negative number?

                            To get a rough idea of the height of a building, John paces off 50 feet from the base of the building, and then
                            measures the angle of elevation to the top of the building to be 58º. About how tall is the building? Justify
                            your choice.
                            A) 31 feet
                            B) 42 feet
                            C) 59 feet
                            D) 80 feet
                            E) 417 feet




                                                         Grade 11 Advanced Precalculus, page 7
                                                       Advanced Precalculus Grade 11
                                         Unit 3: Applications of Trigonometric and Circular Functions
         Topic                                    Learning Objectives: Key Definitions, Skills and Concepts

3.2 General Sinusoidal   What is a sinusoidal graph? What do the terms period, amplitude, cycle, frequency and phase shift mean relative
Graphs                   to the graph of a sinusoid? What is concavity? What are points of Inflection? What are critical points?

                         Skills check, ability to:
                             Correctly graph and state pertinent information given the equation of a sinusoid
                             Correctly state pertinent information and the equation given the graph of a sinusoid
                             Correctly identify the equation and graph a sinusoid given pertinent information

                         Concept check:
                            True/False: The equation of a sinusoid is unique. Why or why not?


3.3 Graphs of Tangent,   What are the graphs of the remaining trig functions? Where do they have asymptotes? What is the Quotient
Cotangent, Secant and    Property for Tangent and Cotangent?
Cosecant Functions
                         Skills check, ability to:
                             Correctly graph and state pertinent information given the equation of any trig function
                             Correctly state pertinent information and the equation of any trig function given its graph
                             Correctly identify the equation and graph any trig function given pertinent information
                             Correctly identify domain and range for all the trig functions
                             Correctly sketch transformations for all the trig functions

                         Concept check:
                            Explain how the domain and range of the trig functions are established
                            Are the discontinuities seen in the graphs of the trig functions step discontinuities? Why or why not?
                            Explain how one could accurately draw the graph of the Cosecant function given the graph of the Sine
                            function.
                            What is the amplitude of the Tangent function? Explain.




                                                         Grade 11 Advanced Precalculus, page 8
                                                       Advanced Precalculus Grade 11
        Topic                                      Learning Objectives: Key Definitions, Skills and Concepts

3.3 Graphs of Tangent,        The graph of y = csc  has the same set of asymptotes as the graph of y =
Cotangent, Secant and    A)    sin 
Cosecant Functions       B)    tan 
(cont’d)                 C)    cot 
                         D)    sec 
                         E)    csc 2 

                              The graph of y = sec  never intersects the graph of y =
                              A) 
                              B)  2
                              C) csc 
                              D) cos 
                              E) sin 

                              If k  0, what is the range of the function y = k csc  ?
                              A) [-k, k]
                              B) (-k, k)
                              C) (-  , -k)  (k,  )
                              D) (-  , -k)  [k,  )
                                         1       1
                              E) (-  ,     ]  [ ,)
                                         k        k




                                                          Grade 11 Advanced Precalculus, page 9
                                               Advanced Precalculus Grade 11
         Topic                             Learning Objectives: Key Definitions, Skills and Concepts
3.4 Radian
Measurement of   What is a radian? How does one convert from radians to degrees and degrees to radians? What is the
Angles           relationship between radian measure and arclength?

                 Skills check, ability to:
                      Correctly convert between degrees and radians
                      Express special angles and quadrantal angles as radians
                      Correctly find the trig functions of angles expressed as radians

                 Concept Check:
                    True/False: The radian measure of all three angles in a triangle can be integers. Justify your answer.

                    If the perimeter of a sector is 4 times its radius then the radian measure of the central angle of the sector is:
                    A) 2
                    B) 4
                         2
                    C)
                       
                       4
                    D)
                       
                    E) Impossible to determine without knowing the radius

                    A central angle in a circle of radius r has a measure of x radians. If the same central angle was drawn is a
                    circle of radius 2r, then its radian measure would be:
                        x
                    A)
                        2
                         x
                    B)
                        2r
                    C) x
                    D) 2x
                    E) 2rx




                                                 Grade 11 Advanced Precalculus, page 10
                                                        Advanced Precalculus Grade 11
         Topic                                     Learning Objectives: Key Definitions, Skills and Concepts
3.5 Circular Functions   What is a circular function and how is it different from a trig function? What is the argument of a circular
                         function?

                         Skills check, ability to:
                            Correctly find the arclength of a circle subtended by an angle in radians
                            Correctly sketch the graph of circular functions (both original and transformed) given its equation
                            Correctly find the equation of a circular function given its graph

                         Concept Check:
                            Draw the BOX diagram for the circular functions showing input, output and rule. Be specific as to the input.
                            Explain why the word wrapping is appropriate.
                            True/False: The values of sin x and sin (x + 2  ) are always the same. Justify your answer.

                            The period of the function f(x) = 210sin(420x+840) is
                                 
                            A)
                                 840
                                 
                            B)
                                 420
                                 
                            C)
                                 210
                                 210
                            D)
                                 
                                 420
                            E)
                                 

                            A sinusoid with amplitude 4 has minimum value of 5. Its maximum value is _______.

                            The graph of y = f(x) is a sinusoid with period 45 passing thru the point (6,0). Which of the following can be
                            determined from the given information?
                            I. f(0)          II. f(6)         III. f(96)
                            A) I only
                            B) II only
                            C) I and III only
                            D) II and III only
                            E) I, II and III
                                                         Grade 11 Advanced Precalculus, page 11
                                                      Advanced Precalculus Grade 11
         Topic                                    Learning Objectives: Key Definitions, Skills and Concepts

3.6 Inverse Circular    What is the Arccosine relation and how is it different from the Inverse Cosine function?
Relations: Given y,
Find x                  Skills check, ability to:
                             Given the equation of a circular or trigonometric function and a particular function value correctly find the
                             value of x or  either graphically, numerically or algebraically

                        Concept Check:
                           Graph each of the following functions and interpret the graph to find the domain, range and period of each
                           function. Which of the three functions has points of discontinuity? Are the discontinuities removable or
                           nonremovable?
                           A) y = sin 1 (sin x)
                           B) y = cos 1 (cos x)
                           C) y = tan 1 (tan x)



3.7 Sinusoidal         What is a mathematical model?
Functions as
Mathematical Models     Skills check, ability to:
                            Correctly interpret real world data to form a mathematical model
                            Use this model to answer questions about the data
                            Use this model to make reasonable predictions about the future

                        Concept check:




                                                        Grade 11 Advanced Precalculus, page 12
                                                    Advanced Precalculus Grade 11
                             Unit 4: Trigonometric Function Properties, Identities and Parametric Functions
        Topic                                 Learning Objectives: Key Definitions, Skills and Concepts

4.2 Pythagorean,      What are the Pythagorean, Reciprocal and Quotient Properties?
Reciprocal and
Quotient Properties   Skills check, ability to:
                          Correctly express one trig function in terms of another or others using the appropriate properties

                      Concept check:
                         On the assumption that one knows sin 2 x + cos 2 x = 1, explain how the other forms of the Pythagorean
                         Identity can be derived.
                         Explain how knowing the Quotient Property will allow one to determine the asymptotes of the Tangent and
                         Cotangent functions.
                         Graph the functions y = sin 2 x and y = -cos 2 x in the same viewing window. Describe the apparent
                         relationship between the two graphs and verify it with a trigonometric property.




                                                      Grade 11 Advanced Precalculus, page 13
                                                   Advanced Precalculus Grade 11
        Topic                                  Learning Objectives: Key Definitions, Skills and Concepts

4.3 Identities and   What is the difference between an equation and identity? How can identities help to transform a complicated trig
Algebraic            expression into a simpler one?
Transformations of
Expressions          Skills check, ability to:
                         Correctly transform a trig expression into another, simpler, trig expression
                         Correctly demonstrate algebraically that a given equation is an identity

                     Concept check:
                        Why is it “illegal” to work on both sides of an equation in an effort to prove it is an identity?
                        True/False: All trig equations are identities.
                                                x2 1 x2 1
                        Consider the equation                 2 . The left hand side of the equation is not defined when x =  1,
                                                 x 1 x  1
                        while the right hand side is defined for all x. What impact does this observation have on the status of the
                        equation as an identity?

                                                            f ( x)
                        If f(x) = g(x) is an identity and          = k, which of the following must be false?
                                                            g ( x)
                        A)   g(x)  0
                        B)   f(x) = 0
                        C)   k=1
                        D)   f(x) – g(x) = 0
                        E)   f(x) · g(x) > 0

                       True/False: sin  = tan  cos  for all real numbers.




                                                     Grade 11 Advanced Precalculus, page 14
                                                      Advanced Precalculus Grade 11
        Topic                                     Learning Objectives: Key Definitions, Skills and Concepts

Solving Trig Equations   How does one solve a trig equation? Are there multiple answers to a trig equation? Are there trig equations with
(NOTE: This material     no solutions? Are there extraneous solutions to trig equations?
is covered in Section
4.4. However, use of     Skills check, ability to:
an alternative text is       Correctly solve a trig equation given a domain algebraically and graphically
recommended)                 Correctly solve a trig equation without a given domain
                             Correctly solve a quadratic trig equation algebraically and graphically
                             Correctly solve a trig equation with a complicated argument algebraically and graphically
                             Correctly solve an equation containing a trig expression and an algebraic expression using a graphing
                             calculator (i.e. sin x = x)

                         Concept check:
                            When it is reasonable to use a graphing calculator to solve a trig equation?
                            How can you solve a trig equation exactly using a graphing calculator?
                            How can you extend the ideas of solving a trig equation to solving a trig inequality?



4.5 Parametric           What is a parametric function? How do you graph a parametric function? How do you convert between
Functions                parametric and rectangular functions?

                         Skills check, ability to:
                             Correctly graph a pair of parametric functions by hand and using the graphing calculator
                             Correctly use the Pythagorean identity to eliminate the parameter from a pair of parametric equations
                             Identify a pair of parametric equations as an ellipse or circle
                             Given the graph of an ellipse or circle, find the correct parametric equation

                         Concept check:
                            Why do you think parametric equations have been introduced at this time?




                                                        Grade 11 Advanced Precalculus, page 15
                                                  Advanced Precalculus Grade 11
        Topic                                 Learning Objectives: Key Definitions, Skills and Concepts

4.6 Inverse Trig   In what ways are the graphs of inverse trig relations and inverse trig functions the same/different?
Relations Graphs
                   Skills check, ability to:
                      Correctly sketch all six inverse trig functions
                      Correctly create the BOX diagram of all six inverse trig function identifying input, output and rule
                      Correctly state the domain and range of all six inverse trig functions
                      Correctly identify the quadrants that correspond to the range for the six inverse trig functions
                      Correctly evaluate functions composed of trig and inverse trig functions (i.e. sin (tan 1 (-1))
                      Correctly evaluate functions composed of a trig function and its inverse function (i.e. sin (sin 1 (4))

                   Concept check:
                      If f(x) = x+3, f ( f 1 (7)) = 7 and f 1 (f(7)) =7, why is sin (sin   1
                                                                                                  (4))  4?
                      Under what circumstances will sin (sin 1 (x)) = x?




                                                    Grade 11 Advanced Precalculus, page 16
                                                       Advanced Precalculus Grade 11
                                                  Unit 5: Properties of Combined Sinusoids
       Topic                                      Learning Objectives: Key Definitions, Skills and Concepts
5.2 Composite            What does the graph of y = A cos x + B sin x look like? Is it periodic? Is it sinusoidal? How can the linear
Argument and Linear      combination of cosine and sine be written as a single cosine function with a phase shift? What is the expansion of
Combination Properties   cos (A-B)?

                         Skills check, ability to:
                           Correctly express a linear combination of cosine and sine as a single cosine function with a phase shift
                           Correctly express a single cosine function with a phase shift as a linear combination of cosine and sine
                           Correctly solve trig equations involving a linear combination of sine and cosine

                         Concept check:
                           If f is a trig function and g is a trig function, is the new function f + g always periodic? Always sinusoidal?
                           Under what circumstances will f + g be sinusoidal?
                           True/False: Cosine distributes over subtraction (that is cos (A-B) = cos A – cos B)




                                                         Grade 11 Advanced Precalculus, page 17
                                                     Advanced Precalculus Grade 11
        Topic                                    Learning Objectives: Key Definitions, Skills and Concepts

5.3 Other Composite   What is the Odd-Even Property for the trig functions? What is the Cofunction Property for the trig Functions?
Argument Properties   What are the expansions of sin (A  B), cos (A  B) and tan (A  B)?

                      Skills check, ability to:
                        For the six trig functions be able to express f (-x) in terms of f(x) and f(90º - x) in terms of f(x)
                        Correctly expand sin (A  B), cos (A  B) and tan (A  B)
                        Correctly use the expansions to verify identities and solve trig equations

                      Concept check:
                        True/False: If cos A + cos B = 0, then A and B are supplementary angles. Justify your answer.

                        If cos A cos B = Sin A sin B, then cos (A + B) =
                        A) 0
                        B) 1
                        C) cos A + cos B
                        D) sin A + sin B
                        E) cos A cos B + sin A sin B

                        Exactly evaluate sin 15º

                        Assume A, B, and C are the three angles of some triangle. Prove sin (A + B) = sin C




                                                       Grade 11 Advanced Precalculus, page 18
                                                      Advanced Precalculus Grade 11
        Topic                                     Learning Objectives: Key Definitions, Skills and Concepts

5.5 The Sum and       What are the sum to product properties?
Product Properties
                      Skills check, ability to:
                         Correctly transform the sum or difference to a product of sines and/or cosines with positive arguments

                      Concept check:
                                                                1
                        Prove the following identity sin u sin v =(cos (u-v) – cos (u+v)). This is called the product-to-sum formula.
                                                                2
                                                                                      uv      u v
                        Using the product-to-sum formula prove cos u - cos v = -2 sin      sin       . This is called the sum to
                                                                                       2         2
                        product formula.


5.6 Double and Half   How can sin 2A, cos 2A and tan 2A be expressed as functions of sin A, cos A and tan A?
Argument Properties                A       A         A
                      How can sin , cos      and tan   be expressed as functions of sin A, cos A and tan A?
                                   2       2         2


                      Skills check, ability to:
                                                                                         A
                          Correctly find the exact values of functions of 2A and           , given the function value of one trig function of A
                                                                                         2
                          and a domain for A
                          Correctly use the expansions to verify identities and solve equations

                      Concept check:
                         Explain how the other forms of cos 2A can be derived if one knows cos 2A = cos 2 A – sin 2 A
                         Recall that we could write exact values of sin  and cos  when  had a reference angle of 0, 30, 45, 60
                         and 90 degrees. Explain how you could now find exact values for  = 15º and  = 75º




                                                       Grade 11 Advanced Precalculus, page 19
                                                       Advanced Precalculus Grade 11
                                                        Unit 6: Triangle Trigonometry
         Topic                                     Learning Objectives: Key Definitions, Skills and Concepts

6.2 Oblique Triangles:   What is an oblique triangle? What is the Law of Cosines and when is its use appropriate?
Law of Cosines
                         Skills check, ability to:
                             Correctly utilize the Law of Cosines to “solve” a triangle

                         Concept check:
                            True/False: If  ABC is any triangle with sides and angles labeled in the usual manner, then
                             b2  c 2  2bc cos A . Justify your answer.



6.3 Area of a Triangle   What is the trigonometric formula for the area of a triangle? What is Hero’s Formula?

                         Skills check, ability to:
                              Correctly compute the area of a triangle given two sides and the included angle
                              Correctly compute the area of a triangle given three sides

                         Concept check:
                            True/False: If a, b and  are two sides and the included angle of a parallelogram, then area of the
                            parallelogram is ab sin  . Justify your answer.


6.4 Oblique Triangles:   What is the Law of Sines and when is its use appropriate?
Law of Sines
                         Skills check, ability to:
                              Correctly use the Law of Sines to “solve” a triangle

                         Concept check:
                            True/False: The perimeter of a triangle with two 10 inch sides and two 40º angles is greater than 36. Justify
                            your answer.




                                                         Grade 11 Advanced Precalculus, page 20
                                                        Advanced Precalculus Grade 11
        Topic                                      Learning Objectives: Key Definitions, Skills and Concepts
6.5 The Ambiguous         When would one expect to find multiple solutions to a triangle? How, in the process of solving a triangle would
Case                      one know that multiple solutions are possible?

                          Skills check, ability to:
                               Correctly find the measure of the third side in a triangle given the measure of two sides and the non included
                               angle

                          Concept check:
                             What is the triangle inequality and what is its application to the Law of Sines?
                             Which of the following three triangle parts do not necessarily fix a triangle:
                             A) AAS
                             B) ASA
                             C) SAS
                             D) SSA
                             E) SSS


Navigation Problems       What is a course? What is a bearing? How is a compass reading expressed in surveying?
(NOTE: Navigation
problems are covered      Skills check, ability to:
in Section 6.6, however        Correctly use information of on the course of a ship or plane to solve problems related to its travel
use of an alternative          Correctly use surveying information to solve problems
text is recommended.
See Brown text:           Concept check:
Applications of Trig to
Navigation and
Surveying)




                                                          Grade 11 Advanced Precalculus, page 21
                                                     Advanced Precalculus Grade 11
         Topic                                   Learning Objectives: Key Definitions, Skills and Concepts
6.7 Real World           What problems can be solved by the creation of a triangle and appropriate techniques to “solve” the triangle?
Triangle Problems
                         Skills check, ability to:
                              Correctly formulate and solve real world problems that are appropriate to this material

                         Concept check:




Vectors (NOTE: Vector    What is a vector? When are two vectors equal? What is the magnitude of a vector? What is vector subtraction?
Addition is covered in   What is multiplication of a vector by a scalar? How are vectors added geometrically? How are vectors
Section 6.6, however     represented algebraically? How are vectors added algebraically?
use of an alternative
text is recommended.     Skills check, ability to:
See Brown text:               Correctly solve problems relating to the geometric interpretation of vectors
Geometric and                 Correctly solve problems relating to the algebraic interpretation of vectors
Algebraic
Representation of        Concept check:
Vectors)




                                                        Grade 11 Advanced Precalculus, page 22
                                                         Advanced Precalculus Grade 11
                                                Unit 7: Polar Coordinates and Complex Numbers
Polar Coordinates         What is the polar coordinate system and how is it used to plot points? What is the procedure to convert between
(NOTE: Polar              rectangular coordinates and polar coordinates? How are polar equations graphed?
coordinates are covered
in Section 13.2,          Skills check, ability to:
however use of an              Correctly plot a point given in polar coordinates
alternative text is            Correctly covert between rectangular and polar points and equations
recommended. See               Correctly graph polar equations by hand and on the graphing calculator
Brown text: Polar
Coordinates)              Concept check:
                             True/False: Polar coordinates are unique. Justify your answer.
                             True/False: If r 1 and r 2 are not 0, and if (r 1 ,  ) and ( r 2 ,  +  ) represent the same point in the plane,
                             then r 1 = -r 2 . Justify your answer.
13.3 Intersection of      When and where do two polar curves have an actual intersection?
Polar Curves
                          Skills check, ability to:
                               Correctly identify actual points of intersection between two polar curves by hand and on the graphing
                               calculator

                          Concept check:




                                                           Grade 11 Advanced Precalculus, page 23
                                                  Advanced Precalculus Grade 11
13.4 Complex Numbers   How are complex numbers expressed in polar form? How are complex numbers multiplied/divided in polar
in Polar Form          form? What is DeMoivre’s Theorem? How can you find all the roots of complex numbers?

                       Skills check, ability to:
                            Correctly express complex numbers in polar form
                            Correctly multiply/divide complex numbers in polar form
                            Correctly find powers of complex numbers in polar form
                            Correctly find all the roots of complex numbers

                       Concept check:
                          Consider the number 4 + 3 i . You need to raise this number to the 5th power. What are the three options you
                          have for performing this operation, and which will be the easiest?




                                                     Grade 11 Advanced Precalculus, page 24
                                                    Advanced Precalculus Grade 11
                                               Unit 8: Properties of Elementary Functions
7.2 Identifying      What are the graphical features of linear, constant, quadratic, power and exponential functions? When are each
Functions from       increasing/decreasing/neither? When is each concave up/down? What are real world examples of each? What
Graphical Patterns   are the procedures for finding a specific equation of each?

                     Skills check, ability to:
                          Correctly identify the type of function from its sketch
                          Correctly find, by hand and using the regression feature on the graphing calculator, the equation of the
                          function given specific information

                     Concept check:
                        What kinds of transformations can make an exponential function appear to be a power function (if viewed
                        over a specific domain)?
                        True/False: Every exponential function is strictly increasing. Justify your answer.
                        Refer to the expression f( a, b, c ) = a · b c (So, for example f( 2, 3 ,x ) = 2 · 3 x , an exponential function)
                              If b = x, state the conditions on a and c under which the expression f ( a, b, c ) is a quadratic function.
                              If b = x, state the conditions on a and c under which the expression f ( a, b, c ) is a decreasing linear
                              function.
                              If c = x, state the conditions on a and b under which the expression f ( a, b, c ) is an increasing
                              exponential function.
                              If c = x, state the conditions on a and b under which the expression f ( a, b, c ) is a decreasing
                              exponential function.




                                                     Grade 11 Advanced Precalculus, page 25
                                                   Advanced Precalculus Grade 11
7.3 Identifying      What numerical patterns are exhibited by linear, quadratic, power and exponential functions? What is direct and
Functions form       inverse variation?
Numerical Patterns
                     Skills check, ability to:
                          Correctly identify a function given its data table
                          Correctly find function values given an initial value and the type of function
                          Solve direct and inverse variation problems


                     Concept check:
                        Is it possible for more than one function to fit the same set of data?
                        If this is the case, how would you decide which function is a better model?




                                                     Grade 11 Advanced Precalculus, page 26
                                                        Advanced Precalculus Grade 11
7.4 Logarithms:           What is a common/natural logarithm? How do you convert between exponential form and logarithmic form?
Definition, Properties,   What are the properties of Logarithms? What is the Change of Base Formula? How do you solve equations
and Equations             involving logarithms? How do you solve exponential equations?

                          Skills check, ability to:
                               Convert between exponential form and logarithmic form
                               Express sum/differences of logs as a single logarithm
                               Express products of constants and logs as a single logarithm
                               Correctly calculate the log of any base using the Change of Base Formula
                               Correctly solve logarithmic and exponential equations
                               Correctly identify extraneous roots of logarithmic equations

                          Concept check:
                             Explain why it is possible for a logarithmic equation to have extraneous roots.
                             Determine if each of the following statements are true or false and then explain why you answered as you
                             did:
                                 log 5 = 2.5 log 2
                                 log 5 > log 2
                                 log 5 = log 10 – log 2
                                 log 5 = 1 – log 2
                                 log 5 < log 10

                              Describe how to transform the graph of f(x) = ln x into the graph of g(x) = log   1   x
                                                                                                                e
                              True/False: The logarithm of a positive number is positive. Justify your answer.
                              True/False: If $100 is invested at 5% annual interest for 1 year, there is no limit to the final value of the
                              investment if it is compounded sufficiently often. Justify your answer.




                                                          Grade 11 Advanced Precalculus, page 27
                                                        Advanced Precalculus Grade 11
7.5 Logarithmic          What is the graphical pattern exhibited by a logarithmic function? What is the domain/range of a logarithmic
Functions                function? What is the numerical pattern exhibited by a logarithmic function? What is the relationship between
                         an exponential function and a logarithmic function?

                         Skills check, ability to:
                              Identify a logarithmic function from its graph and its data table
                              Correctly find, by hand and on the graphing calculator, a specific equation of a logarithmic function
                              Correctly identify real world illustrations of logarithmic functions
                              Correctly sketch a logarithmic function that has been transformed

                         Concept check:
                            Graph the equations y 1 = log(x), y 2 = log (10x) and y 3 = log (100x). How do the graphs compare and why
                            is this happening?
                            Immediately following the gold medal performance of the US Woman’s gymnastic team in the 1996
                            Olympics, an NBC commentator, John Tesh, said of one of the team members: “Her confidence and
                            performance have grown logarithmically.” He clearly thought this was an enormous compliment. Is it a
                            compliment? Is it realistic?

7.6 Logistic Functions   What is a logistic function and when is its use appropriate? What are real world illustrations of logistic growth?
for Restrained Growth    What is the domain/range of a logistic function? What is the significance of the inflection point found in a
                         logistic function?

                         Skills check, ability to:
                              Correctly find, by hand and on the graphing calculator, a specific logistic function that accurately fits data
                              Correctly identify the carrying capacity from anecdotal information

                         Concept check:
                            Is all growth logistic?
                            Compare and contrast logistic and exponential functions.




                                                         Grade 11 Advanced Precalculus, page 28
                                                    Advanced Precalculus Grade 11
                                                     Unit 9: Sequences and Series
14.2 Arithmetic,      What is a sequence? How are sequences categorized? How are sequences defined (explicitly v. recursively)?
Geometric and Other   What does the graph of a sequence look like? What is its domain/range? What is meant by the limit of an
Sequences             infinite sequence?

                      Skills check, ability to:
                      Correctly identify the type of sequence given some consecutive terms
                      Correctly create a formula, both explicit and recursive, for a sequence given some consecutive terms
                      Correctly use a formula to calculate additional terms of the sequence
                      Correctly generate a sequence on the graphing calculator
                      Given a sequence, correctly identify the term number of any term
                      Correctly use sequences to solve problems
                      Identify the Fibonacci sequence
                      Correctly use a graphing calculator in sequence mode to produce the graph of a sequence
                      Correctly identify the limit of an infinite sequence
                      Correctly calculate limits in the form: lim f (n)
                                                             n 
                      Correctly identify when a limit does not exist

                      Concept check:
                      True/False: If a geometric sequence contains all positive terms, then the new sequence formed by taking the log
                      of all the terms in the original sequence is arithmetic. Justify your answer.




                                                     Grade 11 Advanced Precalculus, page 29
                                                          Advanced Precalculus Grade 11
14.3 Series and Partial   What is a series and a partial sum of a series? What is sigma notation and how is it used to indicate partial sums?
Sums                      What is meant by a converging/diverging series? Under what circumstances can an infinite series have a sum?

                          Skills check, ability to:
                          Correctly calculate partial sums of arithmetic/geometric series
                          Correctly evaluate an expression given in sigma notation
                          Correctly write a partial sum of an arithmetic/geometric series in sigma notation
                          Correctly use pattern recognition to write a partial sum in sigma notation
                          Use a graphing calculator to evaluate an expression given in sigma notation
                          Correctly demonstrate when a series converges
                          Correctly find the number of terms in a series given S n
                          Correctly calculate the sum of an infinite geometric series

                          Concept check:
                          Can an infinite arithmetic series have a sum? Why or why not?
                          Do all infinite geometric series have sums?
                          Can infinite series, other than geometric converge? Why or why not?
                          If two infinite geometric series have the same sum are they necessarily the same series? Explain.
                          Can the sum of a convergent infinite geometric series be less than its first term? Explain.

                          When basketball player Patrick Ewing was signed by the NY Knicks, he was given a contract for $30 million: $3
                          million a year for 10 years. Of course, since much of the money was to be paid in the future, the team’s owners
                          did not have to have all $30 million available on the day of the signing. How much money would the owners
                          have to deposit in a bank account on the day of the signing in order to cover all future payments? Assuming the
                          account was earning interest, the owners would have to deposit much less than $30 million. (Assume the account
                          was earning 5% interest per year). (Answer: $24.3 million)

                          Now suppose Patrick Ewing’s contract guaranteed him and his heirs an annual payment of $3 million forever.
                          How much would the owners need to deposit in an account today in order to provide these payments? (Answer:
                          $63 million)




                                                         Grade 11 Advanced Precalculus, page 30
                                                        Advanced Precalculus Grade 11
Mathematical Induction     What is mathematical induction and how is it used?
(NOTE: This material
is not covered in the      Skills check, ability to:
Foerster text. Use of an   Correctly formulate and prove a hypothesis using mathematical induction
alternative text is
recommended)               Concept check:
                           Students often time do not believe that an inductive proof has actually proven anything. Explain why a well
                           constructed inductive proof is valid.




                                                          Grade 11 Advanced Precalculus, page 31
                                                    Advanced Precalculus Grade 11
                                                        Unit 10: Combinatorics
15.1 Venn Diagrams      What is a Venn Diagram and how is it used to illustrate union/intersection of sets and thereby aid in counting?
(See Brown text:
Combinatorics           Skills check, ability to:
Chapter)                     Correctly illustrate sets having no intersection and a finite intersection
                             Correctly illustrate wholly contained subsets
                             Use Venn Diagrams to solve counting problems
                             Use Venn Diagrams to illustrate DeMorgan’s Laws

                        Concept check:
                           Is union distributive over intersection? Justify your answer.
                           Is complement distributive over intersection? Justify your answer.

15.2 The                What are these principles and how do they aid in solving counting problems? What are mutually exclusive
Multiplication,         events? What is the meaning of factorial?
Addition and
Complement Principles   Skills check, ability to:
                             Correctly use these principles to solve counting problems
                             Correctly interpret English language questions into problems that can be solved using these principles
                             Correctly construct a branch diagram
                             Use a graphing calculator to evaluate factorials

                        Concept check:

15.3 Permutations and   What is a permutation? What is a combination? What English words indicate each?
Combinations
                        Skills check, ability to:
                             Correctly evaluate permutations and combinations by hand and using a graphing calculator
                             Correctly solve problems using permutations and combinations
                             Correctly interpret English language into problems involving permutations and combinations


                        Concept check:
                           There are many ways one could justify that 0! = 1 and not 0. Choose one rationale and explain it thoroughly.
                           You have a fresh carton containing one dozen eggs and you need to choose two for breakfast. Give a
                           counting argument based on this scenario to explain why 12 C2  12 C10 .

                                                         Grade 11 Advanced Precalculus, page 32
                                                         Advanced Precalculus Grade 11
15.4 Permutations with   How does the formula for permutations differ when all the elements are not distinguishable? How is the formula
Repetition; Circular     for linear permutations different from circular permutations?
Permutations
                         Skills check, ability to:
                              Correctly evaluate the number of permutations when there is repetition of elements
                              Correctly evaluate the number of circular permutations

                         Concept check:



MIXED                    This is practice in determining which of the many combinatoric principles should be applied to solve problems.
COMBINATORICS
EXCERCISES               Skills check, ability to:

                         Concept check:




                                                        Grade 11 Advanced Precalculus, page 33
                                                 Advanced Precalculus Grade 11
15.5 The Binomial   What is the binomial theorem and how is it related to the study of combinatorics?
Theorem: Pascal’s
Triangle            Skills check, ability to:
                         Correctly expand powers of binomials
                         Correctly find individual terms of a binomial expansion
                         Correctly find binomial coefficients using either combinations or Pascal’s Triangle

                    Concept check:
                       There are two ways to expand a binomial (repeated multiplication and the theorem). Expand (x +  x) 4 and
                       show that the results are the same. Which method do you prefer?

                        Just by looking at patterns in Pascal’s Triangle, guess the answers to the following questions:
                            A) What positive integer appears the least number of times?
                            B) What number appears the greatest number of times?
                            C) Is there any positive integer that does NOT appear in Pascal’s Triangle?
                            D) If you go along any row alternately adding and subtracting the numbers, what is the result?
                            E) If p is a prime number, what do all the interior numbers along the p th row have in common?
                            F) Which rows have all even numbers?
                            G) Which rows have all odd numbers?




                                                   Grade 11 Advanced Precalculus, page 34
                                                       Advanced Precalculus Grade 11
                                                             Unit 11: Probability
16.1 Introduction to     What is an experiment, an event, and a sample space? What is the difference between experimental and
Probability (See Brown   theoretical probability? What information is communicated by quantifying the probability of an event?
text: Probability
Chapter)                 Skills check, ability to:
                              Correctly identify and list the sample space for an experiment
                              Correctly identify events that have a probability of zero and one
                              Correctly evaluate the probability of either of two events (mutually exclusive or not)

                         Concept check:
                              True/False: The sample space for rolling two dice and considering the sum is {2,3,4,5,6,7,8,9,10,11,12}.
                              Why or why not?
16.2 Probability of      How do you compute the probability of two events occurring together? What is conditional probability? What
Events Occurring         are independent events?
Together
                         Skills check, ability to:
                              Correctly find the probability of two events occurring together
                              Correctly determine if two events are independent
                              Create a branch diagram including probabilities

                         Concept check:


16.3 The Binomial        What is the binomial probability theorem and when is its use appropriate? How can the binomial probability
Probability Theorem      theorem be used to approximate probabilities when trials are not independent?

                         Skills check, ability to:
                              Correctly determine when the use the binomial probability theorem is appropriate
                              Correctly use the binomial probability theorem
                              Use the Binomial probability theorem to approximate probabilities when trials are not independent

                         Concept check:




                                                         Grade 11 Advanced Precalculus, page 35
                                                   Advanced Precalculus Grade 11
16.4 Probability       How can combinatorics be used to solve probability problems?
Problems Solved with
Combinations           Skills check, ability to:
                            Correctly use combinations to solve probability problems

                       Concept check:
                          Probability problems can be solved in many ways. Create a problem and show multiple ways to arrive at
                          your solution. Comment upon the merits of each method.



16.5 Conditional       What are the formulas related to conditional probability? How can conditional probability be used to judge the
Probability            accuracy of a prediction?

                       Skills check, ability to:
                            Correctly create a branch diagram including probabilities
                            Correctly use conditional probability to compare two events, one that can be determined with complete
                            confidence and one that is more difficult to determine

                       Concept check:
                          How is conditional probability used to determine the accuracy of the SAT test? Why is this information
                          important?
                          How is conditional probability used to determine the accuracy of a medical test? What is the impact of false
                          positives/false negatives?




                                                      Grade 11 Advanced Precalculus, page 36
                                                     Advanced Precalculus Grade 11
16.6 Expected Value     How can one find the expected value of a given random experiment?

                        Skills check, ability to:
                             Correctly calculate the expected value of a random experiment
                             Correctly identify when a game of chance is fair

                        Concept check:
                            Gladys has a personal rule never to enter the lottery (picking 6 numbers from 1 to 46) until the payoff
                            reaches $4 million. When it does reach $4 million, she buys ten different $1 tickets. Assume that the payoff
                            is exactly $4 million.
                                                                                                                 1
                                A) What is the probability that Gladys holds the winning ticket? (Answer:            , which is
                                                                                                               46 C6

                                     1
                                              .00000010676 )
                                 9,366,819
                                B) Fill-in the probability distribution for Gladys’s possible payoffs in the table below (Note that we
                                subtract $10 from the $4 million, since Gladys has to pay for her tickets even if she wins)
                      Value                                                       Probability
                      -10
                      +3,999,990
                            C) Find the expected value of the game for Gladys.
                            D) In terms of the answer to part B, explain to Gladys the long-term implications of her strategy.




                                                       Grade 11 Advanced Precalculus, page 37
                                              Advanced Precalculus Grade 11
                                   Unit 12: Limits and Introduction to Calculus
19.1 Limits   Are all limits calculated as n   ? What is a working definition of a limit? How can the graph of a function be
              used to calculate a limit? What is a continuous function?

              Skills check, ability to:
                   Correctly evaluate limits in the form: lim f ( x) , lim f ( x) , lim f ( x)
                                                              x c         x           x 
                  Correctly evaluate one sided limits
                  Correctly evaluate limits of piecewise functions
                  Identify intervals where a function is continuous

              Concept check:
                 True/False: If lim f ( x)  L , then f(c) = L. Justify your answer.
                                  x c

                  True/False: If lim f ( x)  L , and f is continuous at x = c, then f(c) = L. Justify your answer.
                                  x c
                  Draw a function f defined on the interval [-4, 5] with the following constraints:
                     lim f ( x)  6 , lim f ( x)  5 , lim f ( x)  8 , f(2) does not exist, lim f ( x) does not exist
                                                          
                       x 0              x 2             x 2                                   x 4




                                                Grade 11 Advanced Precalculus, page 38
                                                        Advanced Precalculus Grade 11
19.2 Graphs of Rational   What is a rational function? Do all rational functions have asymptotes? How do you determine if a rational
Functions                 function has an asymptote or a hole? What is a slant asymptote?

                          Skills check, ability to:
                               Correctly sketch a rational function

                          Concept check:
                             Holes are often called removable discontinuities while asymptotes are called nonremovable discontinuities.
                             Explain why these names make sense.
                                                            x2  9
                             Compare the graph of f ( x)          and g(x) = x + 3.
                                                             x 3
                                   Are the domains equal?
                                   Does f have a vertical asymptote?
                                   Explain why the graphs appear identical.
                                   Are the functions identical?

                                          x2
                              Let f ( x)       Which of the following is true about the graph of f?
                                         x5
                             A) There is no vertical asymptote
                             B) There is a horizontal asymptote but no vertical asymptote
                             C) There is a slant asymptote but no vertical asymptote.
                             D) There is a vertical asymptote and a slant asymptote.
                             E) There is a vertical asymptote and a horizontal asymptote.
20.1 The Slope of A       What is meant by the slope of a curve? Is this constant as it is with a line? What is the derivative of a function?
Curve
                          Skills check, ability to:
                               Find the slope of a curve at a given point
                               Find the function which represents the derivative of a function at all points
                               Write the equation of a line tangent to a curve at a given point

                          Concept check:
                             True/False: All functions have derivatives everywhere. Justify your answer.
                             What geometric characteristics determine if a function is differentiable?




                                                          Grade 11 Advanced Precalculus, page 39
                                               Advanced Precalculus Grade 11

                                 Recommended Unit Sequencing and Pacing Guide
Timeframe

Q1          Unit 1: Functions and Mathematical Models
                    1.1 Functions: Algebraically, Numerically, Graphically, and Verbally
                    1.2 Kinds of Functions
                    1.3 Dilations and Translation of Function Graphs
                    1.4 Composition f Functions
                    1.5 Inverse Functions
                    1.6 Reflections, Absolute Values, and Other Transformations

            Unit 2: Periodic Functions and Right Triangle Problems
                    2.1 Introduction to Periodic Functions
                    2.2 Measurement of Rotation
                    2.3 Sine and Cosine Functions
                    2.4 Values of Six Trigonometric Functions
                    2.5 Inverse Trigonometric Functions and Triangle Problems

            Unit 3: Applications of Trigonometric and Circular Functions
                 3.1 Sinusoids: Amplitude, Period and Cycles
                 3.2 General Sinusoidal Graphs
                 3.3 Graphs of Tangent, Cotangent, Secant and Cosecant Functions
                 3.4 Radian Measure of Angles
                 3.5 Circular Functions
                 3.6 Inverse Circular Relations: Given y, Find x
                 3.7 Sinusoidal Functions As Mathematical Models

            Unit 4: Trigonometric Function Properties, Identities and Parametric Functions
                 4.1 Introduction to the Pythagorean Property
                 4.2 Pythagorean, Reciprocal and Quotient Properties
                 4.3 Identities and Algebraic Transformation of Expressions




                                                Grade 11 Advanced Precalculus, page 40
                                               Advanced Precalculus Grade 11
Timeframe

Q2          Unit 4: Trigonometric Function Properties, Identities and Parametric Functions
                  4.4 Arcsine, Arctangent, Arccosine and Trigonometric Equations
                  4.5 Parametric Functions
                  4.6 Inverse Trigonometric Relation Graphs

            Unit 5: Properties of Combined Sinusoids
                    5.1 Introduction to Combination of Sinusoids
                    5.2 Composite Arguments and Linear Combination Properties
                    5.3 Other Composite Arguments
                    5.6 Double and Half Argument Properties

            Unit 6: Triangle Trigonometry
                    6.1 Introduction to Oblique Triangles
                    6.2 Oblique Triangles: Law of Cosines
                    6.3 Area of a Triangle
                    6.4 Oblique Triangles: Law of Sines
                    6.5 The Ambiguous Case
                    6.7 Real-World Triangle Problems
                        Vectors


            Midterm




                                                Grade 11 Advanced Precalculus, page 41
                                                Advanced Precalculus Grade 11
Timeframe

Q3          Unit 7: Polar Coordinates, Complex Numbers
                     13.1 Introduction to Polar Coordinates
                     13.2 Polar Equations and Other Curves
                     13.3 Intersection f Polar Curves
                     13.4 Complex Numbers in Polar Form

             Unit 8: Properties of Elementary Functions
                     7.1 Shapes of Function Graphs
                     7.2 Identifying Functions from Graphical Patterns
                     7.3 Identifying Functions from Numerical Patterns
                     7.4 Logarithms: Definition, Properties, and Equations
                     7.5 Logarithmic Functions
                     7.6 Logistic Functions for Restrained Growth

             Unit 9: Sequences and Series
                     14.1 Introduction to Sequences and Series
                     14.2 Arithmetic, Geometric and Other Sequences
                     14.3 Series and Partial Sums




                                                 Grade 11 Advanced Precalculus, page 42
                                               Advanced Precalculus Grade 11
Timeframe

Q4          Unit 10: Combinatorics
                   15.1 Venn Diagrams
                   15.2 The Multiplication, Addition and Complement Principles
                   15.3 Permutations and Combinations
                   15.4 Permutations with Repetition; Circular Permutation
                        Mixed Combinatoric Exercises
                   15.5 The Binomial Theorem; Pascal’s Triangle

            Unit 11: Probability
                   16.1 Introduction to Probability
                   16.2 Probability of Events Occurring Together
                   16.3 The Binomial Probability Theorem
                   16.4 Probability Problems Solved With Combinations
                   16.5 Working With conditional Probability
                   16.6 Expected Value

            Unit 12: Limits and Introduction to Calculus
                   19.1 Limits of Functions
                   19.2 Graphs of Rational Functions
                   20.1 The Slope of A Curve



            Final




                                               Grade 11 Advanced Precalculus, page 43

				
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