Download Unit 2: Functions – continued

Course Title:
Precalculus, Level 4 and Level 3
Grade:
11 or 12
Length of Course:
One Year (5 credits)
Prerequisites:
Algebra, Geometry, Algebra 2
Description:
This Precalculus course aims at preparing students for success in college-level Calculus. To
succeed in Calculus, students must first and foremost acquire a thorough understanding of functions
– particularly the properties, behavior and manipulation of important functions such as polynomial,
exponential, logarithmic and trigonometric functions. Beyond functions, students must also then
have a firm understanding of analytic trigonometry, of sequences and series and of limits. As such,
this course focuses solely on the in-depth treatment of these topics and these topics only, as their
mastery is considered critical for success in Calculus. For the level 3 track, a unit on Probability
and Counting (in lieu of the more rigorous treatment of limits prescribed in the curriculum) is
included to prepare students for college-level Statistics.
This course strives to give students a proper balance between the comprehension of key concepts
and the mastery of skills. 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.
Scope and Sequence:
Unit sequencing is designed to allow a Level 4 student to seamlessly transfer to Level 3 without
gaps in coverage. A pacing guide for both Level 4 and Level 3 is attached.
Text:
Level 4: Precalculus Mathematics for Calculus 5th Edition, James Stewart, Lothar Redlin and Saleem
Watson, Thomson Brooks/Cole 2006 [Stewart]
Level 3: Functions, Statistics and Trigonometry, Rubenstein, Schultz, etal. Scott, Foresman and Company
1992 [FST]
Reference Texts:
Advanced Mathematics, Precalculus with Discrete Mathematics and Data Analysis, Richard G. Brown,
Houghton Mifflin Company 1992 [Brown]
Precalculus with Trigonometry, Concepts and Applications, Paul A. Foerster, Key Curriculum Press 2003
[Foerster]
Precalculus with Limits: A Graphical Approach 4th Edition, Larson, McDougal Littell [Larson]
Precalculus
Unit 1: Fundamentals
Learning Objectives
The student will …
1.a Perform basic algebraic
operations on exponential
expressions, radical
expressions, and polynomial
expressions.
(4.1-B2, B4,
4.3-A3, D1, D3)
1.b Recognize rational
expressions and perform basic
algebraic operations on rational
expressions.
(4.1-B1
4.3-A3, D1, D2, D3)
Content Outline
Key Definitions, Skills and Concepts
What is an exponent? What is a radical? What is a rational exponent? What is a polynomial?
Skills check, ability to:
Simplify expressions with positive, negative and fractional exponents
Simplify expressions with roots of degree 2 or higher
Switch back and forth from radical notation to fractional exponent notation
Express numbers in scientific notation
Add, subtract and multiply polynomials
Factor polynomials using a variety of techniques, such as factoring formulas, trial and error or factoring a
common monomial
Concept check:
How can our understanding of exponent notation be used to prove each of the “exponent rules”?
What is a rational expression?
Skills check, ability to:
Simplify rational expressions by canceling common factors from both numerator and denominator
Multiply, divide, add and subtract rational expressions
Simplify compound fractions
Rationalize numerator or denominator using conjugate radical
Concept check:
How is simple fractional arithmetic similar to manipulating rational expressions?
Are the following statements correct? If not, why not?
Instructional Materials
12-4 and 12-3:
[Stewart] Section 1.2 (Exponents
and Radicals),
[Stewart] 1.3 (Algebraic
Expression)
12-4 and 12-3:
[Stewart] Section 1.4 (Rational
Expressions)
a2  b2  a  b
a 1  b 1  (a  b) 1
1.c Solve linear and quadratic
equations.
(4.1-A3, B1
4.3-A3, D2, D3)
What is a linear equation? What is a quadratic equation?
Skills check, ability to:
Solve a variety of linear equations
Solve multivariate equations for a given variable
Solve quadratic equations by factoring, completing the square and the quadratic formula
Solve equations with rational expressions and radicals
Concept check:
How do you know when something is a solution to an equation?
Derive the quadratic formula.
Why is the discriminant important?
Precalculus, page 2
12-4 and 12-3:
[Stewart] Section 1.5 (Equations)
Precalculus
Unit 2: Functions
Learning Objectives
The student will …
Content Outline
Key Definitions, Skills and Concepts
2.a Recognize and evaluate
functions.
What is a function? What is the domain and range of a function? When is a variable independent? When is it
dependent?
(4.3-A3, B1, B2, D3)
Skills check, ability to:
Evaluate functions (including piecewise defined functions)
Concept check:
What does it mean to say that f is a function of x?
True or false: f (x ) the same as f  x . Explain why.
Give examples of functions in real life. Explain why your examples are functions.
Represent functions using machine diagrams and arrow diagrams.
Represent a given function verbally, algebraically, graphically (visually) and numerically (i.e. using a table
of values).
2.b Graph functions using a
graphing calculator.
What is an ordered pair? What is a graph of a function?
When is a function increasing? When is a function decreasing? What is the average rate of change of a function?
(4.3-A3, B1, B2, D3)
Skills check, ability to:
Graph a function using a graphing calculator (student should be able to enter functions in a graphing
calculator, change the window size and display the corresponding table of values)
Find the values of a function from a graph
Find the domain and range of a function from a graph
Test whether a given equation or a graph is a function (vertical line test)
Graph piecewise defined functions
Use a graph to find intervals where a function increases or decreases.
Calculate the average rate of change of a function
Concept check:
Why does the vertical line test work in telling us whether an equation or a graph is or is not a function?
What is the relationship between the average rate of change of a function to its slope if that function happens
to be linear? What is the relationship if that function happens to be non-linear?
Precalculus, page 3
Instructional Materials
12-4:
[Stewart] Section 2.1 (What is a
function?)
12-3:
[FST] 2-1 (The Language of
Functions)
12-4:
[Stewart] Section 2.2 (Graphs of
Functions),
[Stewart] Section 2.3 (Increasing
and Decreasing Functions)
12-3:
[FST] 3-1 (Using an Automatic
Grapher)
Precalculus
Learning Objectives
The student will …
Content Outline
Key Definitions, Skills and Concepts
2.c Apply transformations to a
given function.
What is a transformation of a function? What different types of transformations are there? What is an odd
function? What is an even function?
(4.3-B3)
Skills check, ability to:
Recognize and graph horizontal translation or shift: i.e. graphing f(x-c) and f(x+c) from f(x)
Recognize and graph vertical translation or shift: i.e. graphing f(x) – c and f(x) + c from f(x)
Recognize and graph reflections: i.e. graphing f(-x) and –f(x)
Recognize and graph vertical and horizontal stretching and shrinking: i.e. graphing cf(x) and f(cx)
Recognize when functions are symmetric and the type of symmetry that they exhibit (even or odd)
Concept check:
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?
Instructional Materials
12-4:
[Stewart] Section 2.4
(Transformations of Functions)
12-3:
[FST] 3-2 (The Graph Translation
Theorem)
[FST] 3-4 (Symmetries of Graphs)
[FST] 3-5 (The Graph Scale
Change Theorem)
Unit 2: Functions – continued
What are the algebraic properties of functions? What is a composite function?
2.d Combine functions to create
new functions and identify their
resulting domains.
(4.3-B4)
2.e Identify one-to-one functions
and determine their corresponding
inverses.
(4.1-B4)
Skills check, ability to:
Perform addition, subtraction, division and multiplication of functions (algebraically and graphically)
Determine the domain of the resulting combined functions
Find composite functions and their corresponding domains
Concept check:
Why is the domain of the combined function the intersection of each respective function’s domain when
functions are added, subtracted or multiplied?
Explain how one finds the domain of a composite function.
Why is the domain of f(g(x) not necessarily the same as the domain of g(f(x))? Give an example.
What is a one-to-one function? What is the definition of an inverse of a function? What is the inverse function
property?
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
Concept check:
Why does a function that is not one-to-one not have an inverse?
Explain how one finds an inverse.
Precalculus, page 4
12-4:
[Stewart] Section 2.7 (Combining
Functions)
12-3:
[FST] 3-7 (Composition of
Functions)
12-4:
[Stewart] Section 2.8 (One-to-One
Functions and their Inverses)
12-3:
[FST] 3-8 (Inverse Functions)
Precalculus
Unit 3: Polynomial and Rational Functions
Learning Objectives
The student will …
3.a Identify the key attributes
(e.g. degree, zeros, extrema,
etc.) of a polynomial function
and its graph
(4.3-A3, B5)
3.b Divide polynomials using
both long and synthetic
division.
(4.3-A3, D1)
3.c Find the zeros of
polynomials using the Rational
Zero Theorem and Descartes
Rule of Signs.
(4.3-A3, D2)
Content Outline
Key Definitions, Skills and Concepts
What is a polynomial function? What is a zero (in the context of a polynomial function)? What are extrema? What
is a local minimum? What is a local maximum? What is meant by end behavior?
Skills check, ability to:
State the degree of a polynomial
Find the end behaviors of polynomials
Find the zeros of a polynomial
Sketch polynomials using the zeros and a table of values
Use a graphing tool to find critical points such as local extrema and zeros
Concept check:
Describe how you go about finding the end behavior of a polynomial function.
What effect does a “multiple zero” have on the graph of a polynomial?
Consider a polynomial with 3 real zeros, how many local extrema do you expect to find. Explain why.
What is polynomial long division? What is synthetic division?
Skills check, ability to:
Divide polynomials using long and synthetic division
Find the value of a polynomial using the remainder theorem
Create a polynomial with a given set of zeros
Concept check:
What do the remainder and factor theorems say and how are they useful in helping us find zeros?
What are the Rational Zeros Theorem and Descartes Rule of Signs?
Skills check, ability to:
Use the Rational Zero Theorem and Descartes Rule of Signs to find the zeros of a polynomial
Concept check:
Explain how the Rational Zero Theorem works. Why is the Rational Zero Theorem primarily concerned with
finding the factors of the leading coefficient and the constant term?
What are all the tools that we can use to find real zeros of polynomials?
Precalculus, page 5
Instructional Materials
12-4:
[Stewart] Section 3.1 (Polynomial
Functions and Their Graphs)
12-3:
[FST] 9.3 (Graphs of Polynomial
Functions)
[FST] 9.5 (The Factor Theorem)
12-4:
[Stewart] Section 3.2 (Dividing
Polynomials)
12-3:
[FST] 9.4 (Division and the
Remainder Theorem),
[FST] 9.5 (The Factor Theorem)
12-4 and 12-3:
[Stewart] Section 3.3 (Real Zeros
of Polynomials)
Precalculus
Learning Objectives
The student will …
Unit 3: Polynomial and
Rational Functions - continued
3.d Perform basic algebraic
operations on complex
numbers and find complex
solutions to quadratic
equations.
(4.3-A3)
3.e Apply the Fundamental
Theorem of Algebra to finding
the zeros of a polynomial.
(4.3-D2)
Content Outline
Key Definitions, Skills and Concepts
What is a complex number?
12-3:
[FST] 9.6 (Complex Numbers)
2
What is the value of i ? What is the value of i ?
How do you determine whether a quadratic equation has complex solutions?
What is the Fundamental Theorem of Algebra?
Skills check, ability to:
Find all (including complex) zeros of a polynomial
Use the Conjugate Zeros Theorem to find some roots of a polynomial
Concept Check:
What does the Fundamental Theorem of Algebra say?
What does the Zeros Theorem say?
In your own words, explain why the Zeros Theorem is true if you assume that the Fundamental Theorem of
Algebra is true?
2
(4.3-B5)
12-4:
[Stewart] Section 3.4 (Complex
Numbers)
Skills check, ability to:
Recognize complex numbers and their parts
Add, subtract, multiply and divide complex numbers
Simplify expressions with square roots of negative numbers
Find complex solutions to quadratic equations
Concept Check:
The Zeros Theorem asserts that y  x has two zeros, and yet the graph of
once. How do you reconcile this apparent conflict?
3.f Identify the key attributes
(e.g. asymptotes, intercepts,
end behavior, etc.) of a rational
function and its graph.
(12-4 only)
Instructional Materials
Precalculus, page 6
12-3:
[FST] 9.7 (The Fundamental
Theorem of Algebra)
y  x 2 only intersects the x-axis
What is a rational function? What is an asymptote? What is a vertical asymptote? What is a horizontal asymptote?
What is a slant asymptote?
Skills check, ability to:
Use transformation rules to sketch graphs of simple rational functions
Sketch rational functions by finding intercepts and asymptotes
Find slant asymptotes by dividing the polynomials
Concept Check:
What accounts for the vertical asymptotes in a rational function?
How does dividing a rational function give an equation for its end behavior?
12-4:
[Stewart] 3.5 (Complex Zeros and
the Fundamental Theorem of
Algebra)
12-4:
[Stewart] 3.6 (Rational Functions)
Precalculus
Unit 4: Exponential and Log Functions
Learning Objectives
The student will …
4.a Recognize, evaluate, graph
and apply transformations to
exponential functions.
(4.3-B5)
4.b Recognize, evaluate, graph
and apply transformations to
logarithmic functions and
convert logarithmic functions
to exponential functions (and
vice versa).
(4.3-B5)
4.c Manipulate (i.e. expand or
combine) and evaluate
logarithmic expressions using
the laws of logarithms.
(4.3-D1)
Content Outline
Key Definitions, Skills and Concepts
What is an exponential function? What is a natural exponential function?
Skills check, ability to:
Express an exponential function in standard form (y=a^x used in Stewart or y=a*b^x used in other texts)
Evaluate exponential functions (including natural exponential functions)
Graph exponential functions (including natural exponential functions)
Identify and distinguish graphs of exponential functions.
Apply transformations of exponential functions
Concept check:
What distinguishes an exponential function from a linear function?
Give a verbal representation of an exponential function.
What is the number e and when is it used? (Or, what is so natural about the number e?)
What is a logarithmic function? What is a common logarithm? What is a natural logarithm?
Skills check, ability to:
Switch back and forth from logarithmic to exponential expressions.
Evaluate logarithms using basic properties of logarithms
Graph logarithmic functions
Apply transformations of logarithmic functions (reflections, vertical translation and horizontal translation)
Evaluate common logarithms
Evaluate natural logarithms
Find the domain of a logarithmic function
Concept check:
How are logarithmic functions related to exponential functions?
Why is the domain of a logarithmic function restricted?
What are the laws of logarithms?
Skills check, ability to:
Use the laws of logarithms to evaluate logarithmic expressions
Expand and combine logarithmic expressions
Evaluate logarithms using the change of base formula
Use the change of base formula to graph a logarithmic function.
Concept check:
How do the laws of exponents give rise to the laws of logarithms?
Precalculus, page 7
Instructional Materials
12-4:
[Stewart] Section 4.1 (Exponential
Functions)
12-3:
[FST] 4.3 (Exponential Functions,
note: natural exponentials
covered in FST 4.6)
12-4:
[Stewart] Section 4.2 (Logarithmic
Functions)
12-3:
[FST] 4.5 (Logarithmic Functions)
[FST] 4.6 (e and Natural
Logarithms)
12-4:
[Stewart] Section 4.3 (Laws of
Logarithms)
12-3:
[FST] 4.7 (Properties of
Logarithms)
Precalculus
Learning Objectives
The student will …
Unit 4: Exponential and Log
Functions - continued
4.d Solve exponential and
logarithmic equations.
(4.3-A3, B1, B4, D3)
4.e Apply exponential and
logarithmic functions to reallife situations.
(4.3-A3, B1, B2, B4, D3)
Content Outline
Key Definitions, Skills and Concepts
What is an exponential equation? What is a logarithmic equation?
Skills check, ability to:
Solve equations that involve variables in the exponent (algebraically and graphically)
Solve equations that involve logarithms of a variable (algebraically and graphically).
Solve more complicated compound interest problems (e.g. finding the term for an investment to double)
Concept check:
Why are logarithms useful in solving exponential equations?
Describe the steps involved in solving a typical logarithmic equation.
What is an exponential growth model? What is an exponential decay model? What are logarithmic scales?
Skills check, ability to:
Apply exponential growth models to real life situations: e.g. predicting the future (and past) size of a population
growing exponentially
Apply exponential decay models to real life situations: e.g. calculating the amount of mass remaining of a
radioactively decaying substance after t units of time.
Convert relative magnitudes measured in logarithmic scales to relative magnitudes measured in linear scales
Concept check:
What does it mean for something to grow or decay exponentially?
How do we know we can use an exponential growth or decay function to model physical phenomena?
Can a half-life decay model be alternatively expressed using a different decay factor? Give an example.
Why are logarithmic scales useful?
Precalculus, page 8
Instructional Materials
12-4:
[Stewart] Section 4.4 (Exponential
and Logarithmic Equations)
12-3:
[FST] 4.8 (Solving Exponential
Equations)
12-4:
[Stewart] Section 4.5 (Modeling
with Exponential and
Logarithmic Functions)
12-3:
[FST] 4.4 (Finding Exponential
Models)
[FST] 4.9 (Exponential and
Logarithmic Modeling)
Precalculus
Unit 5: Trigonometric Ratios and Functions
Learning Objectives
The student will …
5.a Use radian measure to
measure the size of an angle (or
amount of rotation) and convert
between degree and radian
measure.
(4.3-B4, D1, D2)
5.b Find the trigonometric ratio
of an acute angle inside a right
triangle.
(4.3-B4, D1, D2)
5.c Find the value of the
trigonometric function of an
angle (of any size).
(4.3-B4, D1, D2)
Content Outline
Key Definitions, Skills and Concepts
What are radian and degree measures for angles? What are coterminal angles? What are arc length and sector areas?
Skills check, ability to:
Convert between degree and radian measure
Find coterminal angles
Find arc length and sector areas
Concept check:
Using a piece of string, demonstrate how to create an angle of measure 1, 2 and 3 radians on a circle. Hint: how
is the radius of circle related to its circumference?
In your own words, explain the concept of radian measure. Hint: Think about a circle of radius 1.
Derive the formulas for the arc length and sector area when using an angle in radian measure.
What are the six right triangle trigonometric ratios? What are special triangles?
Skills check, ability to:
Find exact values of the trigonometric ratios when given two lengths of a right triangle
Use the trigonometric ratios to solve right triangles
Find the trigonometric values of special right triangles (45-45-90 and 30-60-90) without the use of a calculator
Use the inverse function on the calculator to solve for angles in applications problems
Concept check:
Justify why trigonometric ratios within a right triangle makes sense using the geometric theorem of similarity.
What are the six trigonometric ratios as defined when the angle is placed in standard position? What is a reference
angle? What are the basic trig identities?
Skills check, ability to:
Find reference angle for any angle in standard position
Find the exact value of any special angle, including nonacute special angles
Determine in what quadrant an angle must lie given the signs of the trigonometric functions
Find the exact values of the trig functions when given one of the values
Find the area of a triangle using the SAS formula
Concept check:
To determine sin 150, sin 210, sin 330 and sin 570, I only need to know the value of sin 30. Is this true or false
and why?
Prove the Pythagorean theorem.
Precalculus, page 9
Instructional Materials
12-4:
[Stewart] Section 6.1 (Angle
Measure)
12-3:
[FST] 5.1 (Measures of Angles
and Rotations),
[FST] 5.2 (Lengths of Arcs and
Areas of Sectors)
12-4:
[Stewart] Section 6.2
(Trigonometry of Right
Triangles)
12-3:
[FST] 5.3 (Trigonometric Ratios
of Acute Angles)
12-4:
[Stewart] Section 6.3
(Trigonometric Functions of
Angles)
12-3:
[FST] 5.4 (The Sine, Cosine and
Tangent Functions),
[FST] 5.5 (Exact Values of
Trigonometric Functions)
Precalculus
Learning Objectives
The student will …
Unit 5: Trigonometric Ratios
and Functions - continued
5.d Use the unit circle to find
the trigonometric ratios of a
given angle (of any size) and
find the terminal point of a
given rotation around the unit
circle.
(4.3-B4, D1, D2)
5.e Recognize and sketch
graphs of trigonometric
functions and identify their key
attributes (e.g. amplitude,
period, horizontal translation,
etc.).
(4.3-B4, D1, D2)
5.f Use the Law of Sines and
the Law of Cosines to solve
triangles and triangle
applications.
(4.3-B4, D1, D2)
Content Outline
Key Definitions, Skills and Concepts
What is the unit circle? What are the even/odd properties?
Skills check, ability to:
Use the unit circle to find the values of the six trig functions for special angles
Find a terminal point on the unit circle when given a variety of information
Determine whether a trigonometric function is even or odd
Concept Check:
How are the trig functions defined for the unit circle and how is this consistent with the definitions we have
seen so far?
If you are told to compare cos 77 and cos 82 and say which one is bigger without using a calculator. How
would you do it?
What is a periodic graph? What is meant by amplitude, period and phase shift?
Skills check, ability to:
Recognize the graphs of the six trigonometric functions
Graph transformations of the six trigonometric functions
State the amplitude, period and phase shift of a given trigonometric function
Write the trigonometric function for a given graph
Concept Check:
Why is the graph of any trignometric function periodic?
Which graphs have asymptotes and why do they exist?
How do the period, amplitude and phase shift relate to our earlier studies of transforming functions?
What is the Law of Sines? What is the Law of Cosines? What is the ambiguous case?
Skills check, ability to:
Use the Law of Sines and Law of Cosines to solve for all possible triangles when given a set of conditions
Use Law of Sines and Cosines to solve problems involving bearing and direction
Concept Check:
How do we prove the Law of Sines and Law of Cosines?
How is the Law of Cosines related to the Pythagorean theorem?
Why is it possible for more than one triangle to exist when given two sides lengths and a non-included angle
measure? How do you check for other possibilities?
Precalculus, page 10
Instructional Materials
12-4:
[Stewart] Section 5.1 (The Unit
Circle),
[Stewart] 5.2 (The Trigonometric
Functions of Real Numbers)
12-3:
[FST] 5.5 (Exact Values of
Trigonometric Functions),
[FST] 5.7 (Properties of Sines,
Cosines and Tangents)
12-4:
[Stewart] 5.3 (Trigonometric
Graphs),
[Stewart] 5.4 (More Trigonometric
Graphs)
12-3:
[FST] 5.6 (Graphs of the Sine,
Cosine and Tangent Functions),
[Stewart] 5.3 (Trigonometric
Graphs)
12-4:
[Stewart] 6.4 (The Law of Sines),
[Stewart] 6.5 (The Law of
Cosines)
12-3:
[FST] 5.8 (The Law of Cosines),
[FST] 5.9 (The Law of Sines)
Precalculus
Unit 6: Analytic Trigonometry
Learning Objectives
The student will …
6.a Recognize and prove
trigonometric identities and
simplify trigonometric
expressions using trigonometric
identities.
(4.3-A3, B4, D1, D2, D3)
Content Outline
Key Definitions, Skills and Concepts
What is a trigonometric identity? What are the following identities: reciprocal identities, Pythagorean identities, and
even and odd identities?
12-4 only: What are the cofunction identitites?
Skills check, ability to:
Simplify trigonometric expressions using factoring, common denominators and trigonometric identities
Prove trigonometric identities
Concept check:
When is a trigonometric equation not an identity? Give an example.
True or false:
6.b Evaluate and graph inverse
trigonometric functions.
sin 2 x  cos 2 y  1 .
Give your reasoning.
What is the definition of the inverse sine function? The inverse cosine function? And the inverse tangent function?
12-4 only: What is the definition of the inverse secant function? The inverse cosecant function? The inverse
cotangent function?
Instructional Materials
12-4: [Stewart] Section 7.1
(Trigonometric Identities)
12-3:
[FST] 5-7 (Properties of Sines,
Cosines, and Tangents)
[FST] 6-7 (The Secant, Cosecant
and Cotangent Functions)
12-4:
[Stewart] Section 7.4 (Inverse
Trigonometric Functions)
(4.3-B4, D1, D2, D3)
Skills check, ability to:
Evaluate inverse sine, inverse cosine and inverse tangent expressions
Graph inverse sine, inverse cosine and inverse tangent functions
Evaluate trigonometric functions composed of trigonometric and inverse trigonometric functions
Concept check:
Why are the domains and ranges of inverse functions restricted?
6.c Solve trigonometric
equations.
(4.3-B4, D1, D2, D3)
Skills check, ability to:
Solve trigonometric equations by isolating the trigonometric function to find a single solution and using the
known periodicity of the function to find other solutions
Solve trigonometric equations by graphing the trigonometric function to find a single solution and using the
known periodicity of the function to find other solutions
Solve trigonometric equations by factoring the quadratic equivalents of the trigonometric equation
Solve trigonometric equations by using trigonometric identities
Concept check:
Describe the steps involved in solving a trigonometric equation.
Why does the calculator only give you one solution when you use an inverse trigonometric function? Why is
this sometimes not the right solution?
Precalculus, page 11
12-3:
[FST] 6-5 (Inverse Circular
Functions)
12-4:
[Stewart] Section 7.5
(Trigonometric Equations)
12-3:
[FST] 6-6 (Solving Trigonometric
Equations)
Precalculus
Learning Objectives
The student will …
Unit 6: Analytic Trigonometry continued
6.d Use the angle addition and
angle subtraction formulas for
sine, cosine and tangent to
evaluate trigonometric
equations.
(12-4 only topic)
Content Outline
Key Definitions, Skills and Concepts
What are the angle addition and angle subtraction formulas for sine, cosine and tangent?
Skills check, ability to:
Use angle addition and angle subtraction formulas to evaluate trigonometric expressions
Convert sums of sines and cosines into an angle sum of sine or cosine
Concept check:
How are the angle addition formulas derived? (optional)
Instructional Materials
12-4:
[Stewart] Section 7.2 (Addition
and Subtraction Formulas)
12-3: Not applicable
(4.3-A3, D1, D2, D3)
6.e Use the double angle and
half-angle formulas for sine,
cosine and tangent to evaluate
trigonometric equations.
(12-4 only topic)
(4.3-A3, D1, D2, D3)
What are the double angle and half-angle formulas for sine, cosine and tangent?
Skills check, ability to:
Use half-angle formulas to lower the power of a trigonometric expression
Concept check:
How are the angle addition, double angle and half-angle formulas related?
Derive double angle formulas for sine, cosine and tangent from the corresponding angle sum formula
Derive the half-angle formula for sine, cosine and tangent from the corresponding double angle formula
Precalculus, page 12
12-4:
[Stewart] Section 7.3 (DoubleAngle, Half-Angle, and SumProduct Formulas)
12-3: Not applicable
Precalculus
Unit 7: Sequences and Series
Learning Objectives
The student will …
7.a Recognize sequences and
summation notation and
express sequences in both
explicit and recursive forms.
(4.3-A3)
7.b Distinguish between
arithmetic and geometric
sequences and find their
formulas and partial/infinite
sums
(4.3-A3)
7.c Expand binomial
expressions using the Binomial
Theorem
Note: 12-3 students do not need
to cover this topic as part of
this unit, this topic is included
in the unit on Probability and
Counting
Content Outline
Key Definitions, Skills and Concepts
What is a sequence? What are the terms of a sequence? What are the partial sums? What are explicit and recursive
formulas? What is sigma notation?
12-3 only: What is the limit of a sequence?
Skills check, ability to:
Use a sequence's explicit rule to find any term of a sequence
Find a sequence's explicit rule when given the terms of the sequence
Use a recursive rule to find the first five terms of a sequence
Write the recursive rule for a given sequence
Find the limit of a sequence
Find the partial sums of a sequence when given either its rule or the first few terms of the sequence
Evaluate an expression using sigma notation
Write a series in sigma notation
Concept check:
When a sequence is expressed with an explicit formula, what is the domain of that sequence?
Is it possible to define a sequence both recursively and explicitly?
What are arithmetic and geometric sequences? What are the formulas for the partial sums of arithmetic and
geometric series? What is the formula for the sum of an infinite geometric series?
Skills check, ability to:
Find explicit and recursive formulas for given arithmetic and geometric sequences
Find the partial sums of arithmetic and geometric series
Recognize when an infinite geometric series has a sum and find it
Concept check:
Derive the formulas for the partial sums of arithmetic and geometric series
How does the formula for the partial sum of a geometric series give rise to the formula for the sum of an
infinite geometric series?
What is the binomial theorem? What is Pascal's Triangle? What is factorial? What is nCr?
Skills check, ability to:
Expand a binomial expression using the binomial theorem and either Pascal's triangle or nCr
Use the binomial theorem to find just one specified term of the expansion of a given binomial expression
Factor an expression using the binomial theorem
Concept check:
Why is the binomial theorem an easier way to expand binomial expressions?
How does one construct Pascal's Triangle and what are some of the other interesting properties of the triangle?
(4.3-A3)
Precalculus, page 13
Instructional Materials
12-4:
[Stewart] Section 11.1 (Sequences
and Summation Notation)
12-3:
[FST] 8.1 (Formulas for
Sequences),
[FST] 8.2 (Limits of Sequences)
12-4:
[Stewart] Section 11.2 (Arithmetic
Sequences),
[Stewart] 11.3 (Geometric
Sequences)
12-3:
[FST] 8.3 (Arithmetic Series),
[FST} 8.4 Geometric Series),
[FST} 8.5 Infinite Series
12-4:
[Stewart] Section 11.6 (The
Binomial Theorem)
Precalculus
Unit 8: Limits: A Preview of Calculus
(Recommended for 12-4 only)
Learning Objectives
The student will …
8.a Understand and explain the
concept of a limit and estimate the
value of a given limit numerically
and graphically.
(4.3-A2)
8.b Find the value of a given limit
algebraically using limit laws.
(4.3-A2)
8.c Understand and explain the
concept of a derivative and find
the derivative of a given function
using the definition of a
derivative.
Content Outline
Key Definitions, Skills and Concepts
What is the definition of a limit? What is the definition of a one-sided limit?
Skills check, ability to:
Estimate limits using a calculator (by way of table of values or a graph)
Identify situations when the limit fails to exist
Concept check:
Is the calculator always a reliable way to find limits?
When does a limit fail to exist and why?
What are the limit laws?
Skills check, ability to:
Evaluate limits using limit laws
Evaluate limits by algebraic simplification, by rationalization of terms and by direct substitution
Evaluate limits of piecewise defined functions
Concept check:
Is it possible for a function to have a limit at a given point and yet not be defined at that point? Give
examples.
Under what conditions can you use direct substitution to evaluate a limit?
What is the definition of a tangent line? What is the definition of a derivative?
Skills check, ability to:
Find the tangent line of a curve at a given point
Find the derivative of a function at a given point
Find the derivative of a function
Calculate instantaneous velocity or instantaneous rate of change
Concept check:
In your own words, explain how the definition of a derivative describes a way to measure instantaneous
slope?
What is the difference between the average slope of a curve and the instantaneous slope of a curve?
How are the tangent line of a curve at a given point and the derivative of a curve at a given point related to
each other?
What is the difference between the derivative of a function at a given point and the derivative of a function?
Why can’t you use direct substitution when using the definition of a derivative to evaluate the derivative of a
function?
Precalculus, page 14
Instructional Materials
12-4:
[Stewart] Section 12.1 (Finding
Limits Numerically and
Graphically)
12-4:
[Stewart] Section 12.2 (Finding
Limits Algebraically)
12-4:
[Stewart] Section 12.3 (Tangent
Lines and Derivatives)
Precalculus
Unit 9: Probability and Counting
(Recommended for 12-3 only – to be covered if time permits)
Learning Objectives
The student will …
9.a Compute simple probabilities
by counting the number of
favorable events and the size of
the sample space.
(4.4-B1, B3, B4, B5, B6)
9.b Compute probabilities using
addition and/or multiplication
counting principles.
(4.4-B1, B3, B4, B5, B6)
9.c Compute probabilities of
compound events.
(4.4- B4)
9.d Distinguish between
permutations and combinations
and compute permutations and
combinations.
(4.4-C1, C2, C3, C4)
Content Outline
Key Definitions, Skills and Concepts
What is the definition of sample space? What is the definition of a favorable event? What is meant by a fair
event?
Instructional Materials
12-3:
[FST] 7-1 (Probability of a
Simple, Discrete Event)
Skills check, ability to:
List all the outcomes of sample space for a given probabilistic experiment
List all the favorable events for a given probabilistic experiment
Compute simple probabilities from knowledge of the size of favorable events and size of the sample space
Concept check:
Why are values of probabilities always inclusively between zero and one?
Explain what is wrong, if anything, with the following statement: I flip a coin three times and it comes up
head two times, the probability of getting heads must be 2/3?
What is the addition counting principle? What is the multiplication counting principle? What is a mutually
exclusive event? What is a complementary event?
Skills check, ability to:
Compute probabilities using addition counting principles
Determine whether events are mutually exclusive
Identify complementary events
Concept check:
Why does the addition counting principle require the subtraction of the intersection of the added events?
What is an independent event? What is a dependent event? What is a conditional probability?
12-3:
[FST] 7-2 (Addition Counting
Principles)
[FST] 7-3 (Multiplication
Counting Principles)
12-3:
[FST] 7-4 (Compound Events)
Skills check, ability to:
Determine whether events are independent or dependent
Use Baye’s theorem to calculate conditional probabilities.
Concept check:
When is P(A)*P(B/A) equivalent to finding P(A)*P(B)?
What is a permutation? What is a combination?
Skills check, ability to:
Find the number of ways to arrange or select objects when order matters
Find the number of ways to arrange or select objects when order does not matter
Concept check:
What is the difference between a permutation and a combination?
Why does the formula for combination involve an extra division by n factorial?
Precalculus, page 15
12-3:
[FST] 7-5 (Advanced Counting:
Permutations)
[FST] 8-6 (Advanced Counting:
Combinations)
Precalculus
Learning Objectives
The student will …
Unit 5: Probability and Counting
- continued
Content Outline
Key Definitions, Skills and Concepts
What is Pascal’s Triangle?
12-3:
[FST] 8-7 (Pascal’s Triangle)
9.e Construct Pascal’s Triangle
and recognize its properties.
(4.4-C4)
Skills check, ability to:
Locate the numerical properties represented by the pattern in Pascal’s triangle.
9.f Expand binomial expressions
using the binomial theorem and
determine probabilities in
binomial experiments.
What is the binomial theorem? What is a binomial probability?
(4.4-D2)
Instructional Materials
Skills check, ability to:
Expand binomials using “n choose r”
Use the binomial theorem to solve counting problems
Determine probabilities in situations involving binomial experiments
Precalculus, page 16
12-3:
[FST] 8-8 (Binomial Theorem)
[FST] 8-9 (Binomial Probabilities)
Precalculus
Unit Sequencing and Pacing
Timeframe
Quarter 1
Quarter 2
12-4 Unit Sequencing and Pacing
12-3 Unit Sequencing and Pacing
Unit 1: Fundamentals
1.a Algebraic Notation and Manipulation (w/ focus on
factoring)
1.b Rational Expressions
1.c Linear and Quadratic Equations
Unit 1: Fundamentals
1.a Algebraic Notation and Manipulation (w/ focus on
factoring)
1.b Rational Expressions
1.c Linear and Quadratic Equations
Unit 2: Functions
2.a Recognizing and evaluating functions
2.b Graphs of Functions
2.c Transformations of Functions
2.d Combinations and compositions of functions
2.e One-to-One Functions and their Inverses
Unit 2: Functions
2.a Recognizing and evaluating functions
2.b Graphs of Functions
2.c Transformations of Functions
2.d Combinations and compositions of functions
2.e One-to-One Functions and their Inverses
Unit 3: Polynomial Functions
3.a Attributes of Polynomial Functions and their Graphs
3.b Long and Synthetic Polynomial Division
3.c Finding Zeros of Polynomials: Rational Zero
Theorem and Descartes Rule of Signs
3.d Complex Number Operations and Solutions
3.e Fundamental Theorem of Algebra
3.f Attributes of Rational Functions and their Graphs
Unit 3: Polynomial Functions
3.a Attributes of Polynomial Functions and their Graphs
3.b Long and Synthetic Polynomial Division
Unit 4: Exponential and Log Functions
4.a Exponential Functions
4.b Logarithmic Functions
4.c Laws of Logarithms
4.d Solving Exponential and Logarithmic Equations
4.e Modeling with Exponential and Logarithmic Functions
Unit 3: Polynomial Functions
3.c Finding Zeros of Polynomials: Rational Zero
Theorem and Descartes Rule of Signs
3.d Complex Number Operations and Solutions
3.e Fundamental Theorem of Algebra
Unit 5: Trigonometric Ratios and Functions
5.a Angle Measure: Radians and Degrees
5.b Right Triangle Trigonometry
5.c Trigonometric Functions of Angles (of Any Size)
5.d The Unit Circle
Unit 4: Exponential and Log Functions
4.a Exponential Functions
4.b Logarithmic Functions
4.c Laws of Logarithms
4.d Solving Exponential and Logarithmic Equations
4.e Modeling with Exponential and Logarithmic Functions
Unit 5: Trigonometric Ratios and Functions
5.a Angle Measure: Radians and Degrees
5.b Right Triangle Trigonometry
Midterm
Midterm
Precalculus, page 17
Precalculus
Timeframe
Quarter 3
12-4 Unit Sequencing and Pacing
12-3 Unit Sequencing and Pacing
Unit 5: Trigonometric Ratios and Functions
5.e Graphing Trigonometric Functions
5.f Law of Sines and Law of Cosines
Unit 5: Trigonometric Ratios and Functions
5.c Trigonometric Functions of Angles (of Any Size)
5.d The Unit Circle
5.e Graphing Trigonometric Functions
5.f Law of Sines and Law of Cosines
Unit 6: Analytic Trigonometry
6.a Trigonometric Identities
6.b Inverse Trigonometric Functions
6.c Solving Trigonometric Equations
6.d Addition and Subtraction Formulas
6.e Double Angle and Half-Angle Formulas
Quarter 4
Unit 6: Analytic Trigonometry
6.a Trigonometric Identities
6.b Inverse Trigonometric Functions
6.c Solving Trigonometric Equations
Unit 7: Sequences and Series
7.a Sequences and Summation Notation
7.b Arithmetic and Geometric Sequences and Series
7.c Binomial Theorem
Unit 7: Sequences and Series
7.a Sequence Formulas and Limits of a Sequence
7.b Arithmetic, Geometric and Infinite Series
Unit 9: Probability and Counting (time permitting)
9.a Probability of a Simple, Discrete Event
9.b Addition and Multiplication Counting Principles
9.c Compound Events
9.d Advanced Counting: Permutations and Combinations
9.e Pascal’s Triangle
9.f Binomial Theorem and Binomial Probabilities
Unit 8: Limits: A Preview of Calculus
8.a Finding Limits Numerically and Graphically
8.b Finding Limits Algebraically
8.c Instantaneous Slope and Tangent Lines
Final
Final
Precalculus, page 18