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Prerequisites:
Advanced Precalculus, Level 5
11
One Year (5 credits)
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
Topic
1.1 Functions:
Algebraically,
Numerically,
Graphically and
Verbally
Advanced Precalculus Grade 11
Unit 1: Functions & Mathematical Models
Learning Objectives: Key Definitions, Skills and Concepts
What is a function? What are the four ways a function can be represented? What are the domain and range of a
function?
Skills check, ability to:
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
Topic
1.3 Dilations and
Translations of
Function Graphs
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What are dilations and translations? How are these accomplished graphically and algebraically?
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
Functions
What is composition of functions? What is the domain of a composed function? How can the graphing calculator
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
Topic
1.5 Inverse of
Functions
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What is the definition of an inverse function? Do all functions have inverse functions? How can you tell if a
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,
Absolute Values and
Other Transformations
How are reflections over the coordinate axes accomplished using transformations? What are even and odd
functions? What is the Greatest Integer Function? What is a piecewise function?
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
Topic
2.2 Measurement of
Rotation
Advanced Precalculus Grade 11
Unit 2: Periodic Functions and Right Triangle Problems
Learning Objectives: Key Definitions, Skills and Concepts
How are angles drawn on the Cartesian Plane? What is standard position of an angle? What does a negative
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
Functions
What is a periodic function? What are the definitions of the Sine and Cosine Functions on the Unit Circle?
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
Topic
2.4 Values of the Six
Trigonometric
Functions
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What are the definitions of the six trig 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
Topic
2.5 Inverse Trig
Functions and Triangle
Problems
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What is the principal branch of the sine and cosine functions? What does the notation sin
What is an angle of elevation/depression?
1
(x) =  mean?
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
Topic
3.2 General Sinusoidal
Graphs
Advanced Precalculus Grade 11
Unit 3: Applications of Trigonometric and Circular Functions
Learning Objectives: Key Definitions, Skills and Concepts
What is a sinusoidal graph? What do the terms period, amplitude, cycle, frequency and phase shift mean relative
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,
Cotangent, Secant and
Cosecant Functions
What are the graphs of the remaining trig functions? Where do they have asymptotes? What is the Quotient
Property for Tangent and Cotangent?
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
Learning Objectives: Key Definitions, Skills and Concepts
Topic
3.3 Graphs of Tangent,
Cotangent, Secant and
Cosecant Functions
(cont’d)
A)
B)
C)
D)
E)
The graph of y = csc  has the same set of asymptotes as the graph of y =
sin 
tan 
cot 
sec 
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
Topic
3.4 Radian
Measurement of
Angles
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What is a radian? How does one convert from radians to degrees and degrees to radians? What is the
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
Topic
3.5 Circular Functions
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
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
Learning Objectives: Key Definitions, Skills and Concepts
Topic
3.6 Inverse Circular
Relations: Given y,
Find x
What is the Arccosine relation and how is it different from the Inverse Cosine function?
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
Functions as
Mathematical Models
What is a mathematical model?
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
Topic
4.2 Pythagorean,
Reciprocal and
Quotient Properties
Advanced Precalculus Grade 11
Unit 4: Trigonometric Function Properties, Identities and Parametric Functions
Learning Objectives: Key Definitions, Skills and Concepts
What are the Pythagorean, 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
Learning Objectives: Key Definitions, Skills and Concepts
Topic
4.3 Identities and
Algebraic
Transformations of
Expressions
What is the difference between an equation and identity? How can identities help to transform a complicated trig
expression into a simpler one?
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?
If f(x) = g(x) is an identity and
A)
B)
C)
D)
E)
f ( x)
= k, which of the following must be false?
g ( x)
g(x)  0
f(x) = 0
k=1
f(x) – g(x) = 0
f(x) · g(x) > 0
True/False: sin  = tan  cos  for all real numbers.
Grade 11 Advanced Precalculus, page 14
Topic
Solving Trig Equations
(NOTE: This material
is covered in Section
4.4. However, use of
an alternative text is
recommended)
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
How does one solve a trig equation? Are there multiple answers to a trig equation? Are there trig equations with
no solutions? Are there extraneous solutions to trig equations?
Skills check, ability to:
Correctly solve a trig equation given a domain algebraically and graphically
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
Functions
What is a parametric function? How do you graph a parametric function? How do you convert between
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
Topic
4.6 Inverse Trig
Relations Graphs
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
In what ways are the graphs of inverse trig relations and inverse trig functions the same/different?
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
Under what circumstances will sin (sin 1 (x)) = x?
Grade 11 Advanced Precalculus, page 16
1
(4))  4?
Topic
5.2 Composite
Argument and Linear
Combination Properties
Advanced Precalculus Grade 11
Unit 5: Properties of Combined Sinusoids
Learning Objectives: Key Definitions, Skills and Concepts
What does the graph of y = A cos x + B sin x look like? Is it periodic? Is it sinusoidal? How can the linear
combination of cosine and sine be written as a single cosine function with a phase shift? What is the expansion of
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
Topic
5.3 Other Composite
Argument Properties
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What is the Odd-Even Property for the trig functions? What is the Cofunction Property for the trig Functions?
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
Learning Objectives: Key Definitions, Skills and Concepts
Topic
5.5 The Sum and
Product Properties
What are the sum to 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
(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.
Prove the following identity sin u sin v =
5.6 Double and Half
Argument Properties
How can sin 2A, cos 2A and tan 2A be expressed as functions of sin A, cos A and tan A?
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:
Correctly find the exact values of functions of 2A and
A
, 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
Topic
6.2 Oblique Triangles:
Law of Cosines
Advanced Precalculus Grade 11
Unit 6: Triangle Trigonometry
Learning Objectives: Key Definitions, Skills and Concepts
What is an oblique triangle? What is the Law of Cosines and when is its use appropriate?
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
b 2  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:
Law of Sines
What is the Law of Sines and when is its use appropriate?
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
Topic
6.5 The Ambiguous
Case
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
When would one expect to find multiple solutions to a triangle? How, in the process of solving a triangle would
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
(NOTE: Navigation
problems are covered
in Section 6.6, however
use of an alternative
text is recommended.
See Brown text:
Applications of Trig to
Navigation and
Surveying)
What is a course? What is a bearing? How is a compass reading expressed in surveying?
Skills check, ability to:
Correctly use information of on the course of a ship or plane to solve problems related to its travel
Correctly use surveying information to solve problems
Concept check:
Grade 11 Advanced Precalculus, page 21
Topic
6.7 Real World
Triangle Problems
Advanced Precalculus Grade 11
Learning Objectives: Key Definitions, Skills and Concepts
What problems can be solved by the creation of a triangle and appropriate techniques to “solve” the triangle?
Skills check, ability to:
Correctly formulate and solve real world problems that are appropriate to this material
Concept check:
Vectors (NOTE: Vector
Addition is covered in
Section 6.6, however
use of an alternative
text is recommended.
See Brown text:
Geometric and
Algebraic
Representation of
Vectors)
What is a vector? When are two vectors equal? What is the magnitude of a vector? What is vector subtraction?
What is multiplication of a vector by a scalar? How are vectors added geometrically? How are vectors
represented algebraically? How are vectors added algebraically?
Skills check, ability to:
Correctly solve problems relating to the geometric interpretation of vectors
Correctly solve problems relating to the algebraic interpretation of vectors
Concept check:
Grade 11 Advanced Precalculus, page 22
Polar Coordinates
(NOTE: Polar
coordinates are covered
in Section 13.2,
however use of an
alternative text is
recommended. See
Brown text: Polar
Coordinates)
13.3 Intersection of
Polar Curves
Advanced Precalculus Grade 11
Unit 7: Polar Coordinates and Complex Numbers
What is the polar coordinate system and how is it used to plot points? What is the procedure to convert between
rectangular coordinates and polar coordinates? How are polar equations graphed?
Skills check, ability to:
Correctly plot a point given in polar coordinates
Correctly covert between rectangular and polar points and equations
Correctly graph polar equations by hand and on the graphing calculator
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.
When and where do two polar curves have an actual intersection?
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
13.4 Complex Numbers
in Polar Form
Advanced Precalculus Grade 11
How are complex numbers expressed in polar form? How are complex numbers multiplied/divided in polar
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
7.2 Identifying
Functions from
Graphical Patterns
Advanced Precalculus Grade 11
Unit 8: Properties of Elementary Functions
What are the graphical features of linear, constant, quadratic, power and exponential functions? When are each
increasing/decreasing/neither? When is each concave up/down? What are real world examples of each? What
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 · bc (So, for example f( 2, 3 ,x ) = 2 · 3x , 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
7.3 Identifying
Functions form
Numerical Patterns
Advanced Precalculus Grade 11
What numerical patterns are exhibited by linear, quadratic, power and exponential functions? What is direct and
inverse variation?
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
7.4 Logarithms:
Definition, Properties,
and Equations
Advanced Precalculus Grade 11
What is a common/natural logarithm? How do you convert between exponential form and logarithmic form?
What are the properties of Logarithms? What is the Change of Base Formula? How do you solve 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
e
x
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
7.5 Logarithmic
Functions
Advanced Precalculus Grade 11
What is the graphical pattern exhibited by a logarithmic function? What is the domain/range of a logarithmic
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
for Restrained Growth
What is a logistic function and when is its use appropriate? What are real world illustrations of logistic 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
14.2 Arithmetic,
Geometric and Other
Sequences
Advanced Precalculus Grade 11
Unit 9: Sequences and Series
What is a sequence? How are sequences categorized? How are sequences defined (explicitly v. recursively)?
What does the graph of a sequence look like? What is its domain/range? What is meant by the limit of an
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
14.3 Series and Partial
Sums
Advanced Precalculus Grade 11
What is a series and a partial sum of a series? What is sigma notation and how is it used to indicate partial 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
Mathematical Induction
(NOTE: This material
is not covered in the
Foerster text. Use of an
alternative text is
recommended)
Advanced Precalculus Grade 11
What is mathematical induction and how is it used?
Skills check, ability to:
Correctly formulate and prove a hypothesis using mathematical induction
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
15.1 Venn Diagrams
(See Brown text:
Combinatorics
Chapter)
Advanced Precalculus Grade 11
Unit 10: Combinatorics
What is a Venn Diagram and how is it used to illustrate union/intersection of sets and thereby aid in counting?
Skills check, ability to:
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
Multiplication,
Addition and
Complement Principles
What are these principles and how do they aid in solving counting problems? What are mutually exclusive
events? What is the meaning of factorial?
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
Combinations
What is a permutation? What is a combination? What English words indicate each?
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
15.4 Permutations with
Repetition; Circular
Permutations
Advanced Precalculus Grade 11
How does the formula for permutations differ when all the elements are not distinguishable? How is the formula
for linear permutations different from circular 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
COMBINATORICS
EXCERCISES
This is practice in determining which of the many combinatoric principles should be applied to solve problems.
Skills check, ability to:
Concept check:
Grade 11 Advanced Precalculus, page 33
15.5 The Binomial
Theorem: Pascal’s
Triangle
Advanced Precalculus Grade 11
What is the binomial theorem and how is it related to the study of combinatorics?
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
16.1 Introduction to
Probability (See Brown
text: Probability
Chapter)
16.2 Probability of
Events Occurring
Together
Advanced Precalculus Grade 11
Unit 11: Probability
What is an experiment, an event, and a sample space? What is the difference between experimental and
theoretical probability? What information is communicated by quantifying the probability of an event?
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?
How do you compute the probability of two events occurring together? What is conditional probability? What
are independent events?
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
Probability Theorem
What is the binomial probability theorem and when is its use appropriate? How can the binomial probability
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
16.4 Probability
Problems Solved with
Combinations
Advanced Precalculus Grade 11
How can combinatorics be used to solve probability problems?
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
Probability
What are the formulas related to conditional probability? How can conditional probability be used to judge the
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
16.6 Expected Value
Advanced Precalculus Grade 11
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
19.1 Limits
Advanced Precalculus Grade 11
Unit 12: Limits and Introduction to Calculus
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
Grade 11 Advanced Precalculus, page 38
x4
19.2 Graphs of Rational
Functions
Advanced Precalculus Grade 11
What is a rational function? Do all rational functions have asymptotes? How do you determine if a rational
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
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.
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?
Let f ( x) 
20.1 The Slope of A
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