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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSIntroduction

The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn.

The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, Teacher Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below.

Curriculum Information:This section includes the objective, focus or topic, and in some, not all, foundational objectives that are being built upon.

Essential Knowledge and Skills:Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective.

Key Vocabulary:This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.

Essential Questions and Understandings:This section delineates the key concepts, ideas, and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives.

Teacher Notes and Elaborations:This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.

Resources:This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.

Sample Instructional Strategies and Activities:This section lists ideas and suggestions that teachers may use when planning instruction.

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The following chart lists the objectives for the Prince William County Functions/Trigonometry Curriculum. Some objectives from the Virginia Department of Education Mathematical Analysis Standards and all the objectives from the Virginia Department of Education Trigonometry Standards meet the objectives of Functions/Trigonometry. The chart organizes the objectives by topic. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.

Topic ObjectivesFunctions MA 1, MA 2, MA 3, MA 7, MA 9Discrete Mathematics MA 4, MA 5, MA 6Analytic Geometry MA 8, MA 13Equations MA 14

Triangular and Circular Trigonometric Functions T1, T2, T3Inverse Trigonometric Functions T4, T7Trigonometric Identities T5Trigonometric Equations, Graphs, and Practical Problems T6, T8, T9

*NOTE: Objective MA 14 (which includes operations with matrices) should be taught before Objective MA 8 (which uses translation and rotation matrices).

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Essential Knowledge and Skills

Key VocabularyEssential Questions and Understandings

Teacher Notes and ElaborationsTopicFunctions

Functions/TrigonometryVirginia SOL MA.1The student will investigate and identify the characteristics of polynomial and rational functions and use these to sketch the graphs of the functions. This will include determining zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, and maximum or minimum points. Graphing utilities will be used to investigate and verify these characteristics.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Identify a polynomial function,

given an equation or graph. Identify rational functions,

given an equation or graph. Identify domain, range, zeros,

upper and lower bounds, y-intercepts, symmetry, asymptotes (horizontal and vertical), intervals for which the function is increasing or decreasing, points of discontinuity, end behavior, and maximum and minimum points, given a graph of a function.

Sketch the graph of a polynomial function.

Sketch the graph of a rational function.

Investigate and verify characteristics of a polynomial or rational function, using a graphing calculator.

Key Vocabularyasymptotecontinuousdiscontinuousdomainhorizontal asymptoteslocal (relative) maximumlocal (relative) minimumlower boundpolynomial functionrangerational functionslant asymptotes

Essential Questions What is a polynomial function? What is a rational function? What are the characteristics and components of polynomial and rational functions?

Essential Understandings The graphs of polynomial and rational functions can be determined by exploring

characteristics and components of the functions.

Teacher Notes and ElaborationsA function is a correspondence or rule that assigns to every element in a set D (domain) exactly one element in a set R (range). It describes a dependent relationship between quantities. The domain of a function is the set of values for which the function is defined. The range of a function is the set of output values of the function. Range is the set of elements to which the elements of the domain are assigned by the function. For example, the value of the expression depends of the value of x. Therefore, this is a function of

x and is written . So, if , then . Because ,

is referred to as a zero for the function f. In general, if , then a is called a zero of the function f.

End behavior is the appearance of a graph as it is followed farther and farther in either direction. For polynomials, the end behavior is indicated by drawing the positions of the ends of the graph, which may be pointed up or down. Other graphs may also have end behavior indicated in terms of the ends of the graph, or in terms of asymptotes or limits.

Polynomial end behavior:1. If the degree of n of a polynomial is even, then the ends of the graph are either both up

or both down.2. If the degree n is odd, then one end of the graph is up and one is down.3. If the leading coefficient an is positive, the right end of the graph is up.4. If the leading coefficient an is negative, the right end of the graph is down.

An asymptote of a curve is a line that the curve approaches without end. That is, the distance between the curve and its asymptote becomes less and less as becomes large. A line that the graph of a function approaches and to which it gets arbitrarily close as x or y approaches positive or negative infinity is an asymptote. Instruction in end behavior asymptotes should include horizontal asymptotes, vertical asymptotes, and slant asymptotes.

(continued)

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSupper boundvertical asymptoteszero of the function

Curriculum Information Essential Questions and UnderstandingsTeacher Notes and Elaborations

TopicFunctions


Teacher Notes and Elaborations (continued)Vertical Asymptotes:

- The rational function has a point of discontinuity for each real zero of Q(x).

- If P(x) and Q(x) have no common real zeros, then the graph of f(x) has a vertical asymptote at each real zero of Q(x).- If P(x) and Q(x) have a common real zero a, then there is a hole in the graph or a vertical asymptote at x = a.

Horizontal Asymptotes:- The graph of a rational function has at most one horizontal asymptote. - The graph of a rational function has a horizontal asymptote at y = 0 if the degree of the denominator is greater than the degree of

the numerator.

- If the degrees of the numerator and the denominator are equal, then the graph has a horizontal asymptote at . a is the

coefficient of the term of highest degree in the numerator and b is the coefficient of the term of highest degree in the denominator.- If the degree of the numerator is greater than the degree of the denominator then the graph has no horizontal asymptote.

Slant Asymptotes- The graph of a rational function (having no common factors and whose denominator is of degree 1 or greater) has a slant asymptote

if the degree of the numerator exceeds the degree of the denominator by exactly 1.

A polynomial is a monomial or the sum of monomials. For any polynomial, you can write the corresponding polynomial function as:

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSwhere n is a nonnegative integer and the coefficients ,… are real numbers.

(continued)


TopicFunctions


Teacher Notes and Elaborations (continued)

A rational function has the form where f(x) and g(x) are polynomials and g(x) 0.

Special tests for symmetry of a graph include symmetry in the x-axis, symmetry in the y-axis, symmetry in the line y = x, and symmetry in the origin.

If a graph has an axis of symmetry, then when the graph is folded along the axis, the two halves of the graph coincide. For example, the graph of a quadratic function has a vertical axis of symmetry. The vertex of the parabola is the point where the axis of symmetry intersects the parabola. If , the parabola opens upward and the function has a minimum point at the vertex. If , the parabola opens downward, and the function has a maximum point at the vertex.

A figure has a point of symmetry if a rotation of 180º about the point produces the same figure.

A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f. A function f is bounded above if there is some number b that is greater than or equal to every number in the range of f. Any such number b is called upper bound of f. A function f is bounded if it is bounded both above and below.

A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x). A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

The points at which a function changes its increasing, decreasing, or constant behavior are helpful in determining the relative maximum or relative minimum values of the function. A function value f(a) is called a local (relative) minimum of f if there exists an interval that contains a such that A function value f(a) is called a local (relative) maximum of f if there exists an interval that contains a such that

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicFunctions


Teacher Notes and Elaborations (continued)A function is continuous if the graph can be drawn without lifting the pencil from the paper. A graph is discontinuous if it has jumps, breaks, or holes in it.

Set builder notation may be used to represent solution sets. For example, if the solution is y = 10, then in set notation the answer is written {y: y = 10} or { }.

Curriculum Information Resources Sample Instructional Strategies and Activities

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TopicFunctions

Functions/TrigonometryVirginia SOL MA.1

Text:Precalculus Enhanced with Graphing Utilities, 4th Edition, Sullivan and Sullivan, Pearson Prentice Hall

Precalculus with Limits, A Graphing Approach, Larson, Hostetler, and Edwards, Houghten Mifflin

Precalculus: Graphical, Numerical, Algebraic, 7th Edition 2007, Demana, Waits, Foley and Kennedy, Pearson/Prentice Hall

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

PWC Mathematics websitehttp://pwcs.math.schoolfusion.us/

Virginia Department of Education website http://www.doe.virginia.gov/instruction/mathematics/index.shtml

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http://www.doe.virginia.gov/instruction/mathematics/index.shtml


http://www.pwcsmath.com/



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Teacher Notes and ElaborationsTopicFunctions

Functions/TrigonometryVirginia SOL MA.2The student will apply compositions of functions and inverses of functions to real-world situations. Analytical methods and graphing utilities will be used to investigate and verify the domain and range of resulting functions.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find the composition of

functions. Find the inverse of a function

algebraically and graphically. Determine the domain and

range of the composite functions.

Determine the domain and range of the inverse of a function.

Verify the accuracy of sketches of functions, using a graphing utility.

Key Vocabularycomposition of functionsdomain inverse functions one-to-one function range

Essential Questions What is a composition of functions? What are inverse functions?

Essential Understandings In composition of functions, a function serves as input for another

function. A graph of a function and its inverse are symmetric about the line y =

x.

Teacher Notes and ElaborationsA function is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable. An equation in function notation is y = f(x) where f is the name of the function, y is the dependent variable (output value), x is the independent variable (input value), and f(x) is the value of the function at x.

The domain of a function is the set of all values (inputs) of the independent variable for which the function is defined. If x is in the domain of f, f is defined at x. If x is not in the domain of f, f is undefined at x. The range of a function is the set of all values (outputs) assumed by the independent variable (the set of all function values).

A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).

Characteristics of a function from set A to set B1. Each element of A must be matched with an element of B.2. Some elements of B may not be matched with any element of A.3. Two or more elements of A may be matched with the same element of B.4. An element of A (domain) can not be matched with two different elements of B

(which contains the range).

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLS(continued)


TopicFunctions

Functions/TrigonometryVirginia SOL MA.2The student will apply compositions of functions and inverses of functions to real-world situations. Analytical methods and graphing utilities will be used to investigate and verify the domain and range of resulting functions.

Teacher Notes and Elaborations (continued)The composite of f and g, denoted , is defined by two conditions:

1. , which is read “f dot g of x equals f of g of x”;2. x is in the domain of g and g(x) is in the domain of f.

g(x) x f(g(x)) g f Domain of g Domain of f

The domain of is the set of x satisfying condition (2) above. The operation that combines f and g to produce their composite is called the composition of functions.

A function f is one-to-one if, for a and b in its domain, f(a) = f(b) implies that a = b. A function f has an inverse function if and only if f is one-to-one.

To find an inverse function:1. Use the Horizontal Line Test to decide whether f has an inverse function.2. In the equation for f(x), replace f(x) by y.3. Interchange the roles of x and y, and solve for y.4. Replace y by in the new equation.5. Verify that f and are inverse functions of each other by showing that the domain of f is equal to the range of , the range of f

is equal to the domain of , and

When a function f has an inverse, the graph of can be obtained from the graph of f by changing every point on the graph to the point . The rule for can be obtained by interchanging x and y in the equation and solve for the new y.

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TopicFunctions








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Curriculum Information Essential Knowledge and SkillsKey Vocabulary

Essential Questions and UnderstandingsTeacher Notes and Elaborations

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Functions/TrigonometryVirginia SOL MA.3The student will investigate and describe the continuity of functions, using graphs and algebraic methods.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Describe continuity of a function. Investigate the continuity of absolute

value, step, rational, and piece-wise-defined functions.

Use transformations to sketch absolute value, step, and rational functions.

Verify the accuracy of sketches of functions, using a graphing utility.

Key Vocabularyabsolute value functioncontinuous functiondiscontinuous functionpiecewise functionrational functionstep function

Essential Questions What are the characteristics of a continuous function? What are the characteristics of a discontinuous function?

Essential Understandings Continuous and discontinuous functions can be identified by their equations or graphs.

Teacher Notes and ElaborationsA function is a continuous function if the graph can be drawn without lifting the pencil from the paper. The graph has no breaks, holes, or gaps. A discontinuous function has jumps, breaks, or holes.

These graphs give examples of functions that are discontinuous at .

c c c

If , the function fails to be continuous at x = 0.

Many rational and piecewise functions are examples of discontinuous functions.(continued)


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Teacher Notes and Elaborations (continued)A rational function is a function defined by a rational expression.

It has the form where f(x) and g(x) are polynomials and .

The simplest type of rational function is the reciprocal function .

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. Given the following rational function:

, x cannot be 6 or –1, because the denominator cannot equal zero.

The graph of this rational function has two vertical asymptotes, .

A piecewise function consists of different function rules for different parts of the domain.

(continued)

Curriculum Information Essential Questions and Understandings

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTeacher Notes and Elaborations

TopicFunctions


Teacher Notes and Elaborations (continued)A step function is a special type of function whose graph is a series of line segments. The graph of a step function looks like a series of small steps. The figure below shows the graph of the step function , which is a greatest integer function.

Another name for this kind of graph is a piecewise linear graph, because the graph consists of small line segments.

The absolute value function, denoted by , can be defined as a piecewise function as follows: .

The graph of the absolute value function has a characteristic V-shape.

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-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Resources Sample Instructional Strategies and Activities

TopicFunctions








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TopicDiscrete Mathematics

Functions/TrigonometryVirginia SOL MA.4The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for combinations, and Pascal’s Triangle.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Expand binomials having positive

integral exponents. Use the Binomial Theorem, the formula

for combinations, and Pascal’s Triangle to expand binomials.

Key VocabularyBinomial TheoremcombinationPascal’s triangle

Essential Questions How are binomial expressions expanded?

Essential Understandings The Binomial Theorem provides a formula for calculating the product for any

positive integer n. Pascal’s Triangle is a triangular array of binomial coefficients.

Teacher Notes and ElaborationsA combination is an arrangement of a set of objects in which order is not important. The formula for determining the number of selections in any combination is:

where .

Pascal’s triangle can be utilized to help determine the numerical coefficients in binomial expansion.

Patterns in Pascal’s Triangle- Row n of Pascal’s triangle contains entries.- The kth entry in row n of Pascal’s triangle is .- The sum of all the entries in row n of Pascal’s triangle equals 2n.- .

The Binomial Theorem, using combinatorics, tells how to expand a positive integer power of a binomial.

For any positive integer n,

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Functions/TrigonometryVirginia SOL MA.5The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to an intuitive approach to a limit.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use and interpret the notation: Σ, n, nth,

and an. Given the formula, find the nth term, an ,

for an arithmetic or geometric sequence.

Given the formula, find the sum, Sn , if it exists, of an arithmetic or geometric series.

Model and solve problems, using sequence and series information.

Distinguish between a convergent and divergent series.

Discuss convergent series in relation to the concept of a limit.

Key Vocabularyarithmetic sequenceconvergedivergefinite seriesgeometric sequenceinfinite geometric seriesinfinite sequenceinfinite serieslimitsequence

Essential Questions What are arithmetic and geometric sequences? What are arithmetic and geometric series? How can the nth term in a sequence be found? How can the sums of sequences and series be found? What is a limit? What is the difference between a convergent and divergent series?

Essential Understandings Examination of infinite sequences and series may lead to a “limiting”

process. Arithmetic sequences have a common difference between any two

consecutive terms. Geometric sequences have a common factor between any two

consecutive terms.

Teacher Notes and ElaborationsA sequence is an ordered list of numbers, called terms. Two specific types of sequences are arithmetic and geometric.

Formulas for the nth term of general arithmetic and geometric sequences are:Arithmetic Sequence: where n is the nth term, d is the common

difference, and a1 is the first term.Geometric Sequence: where n is the nth term, r is the common ratio,

and a1 is the first term, r ≠ 1.

Formulas for sums of finite series are:

Arithmetic Series:

Geometric Series: where r is the common ratio and r ≠ 1.

An infinite geometric series is a geometric series with infinitely many terms. A partial sum of an infinite series is the sum of a given number of terms and not the sum of the entire series. When the partial sums of an infinite series approach a fixed number as n increases, the infinite geometric series is said to converge. When the partial sums of an infinite series do not approach a fixed number as n increases, the infinite geometric series is said to

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSdiverge.

(continued)


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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicDiscrete Mathematics

Functions/TrigonometryVirginia SOL MA.5The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to an intuitive approach to a limit.

Teacher Notes and Elaborations (continued)In some instances the sum of an infinite geometric series can be computed. For example if 0 < r ≤ 1.

An infinite sequence continues without end and a finite sequence has a distinct number of terms. A finite series is the sum of the terms of a finite sequence. An infinite series is the sum of the terms of an infinite sequence.

A limit is a central concept indicating a number that a sequence of numbers approaches.

For example, given the infinite sequence of numbers, 1, , , , , ,…, the sequence approaches the number 0 as its limit, since the

values of the fractions are becoming smaller and smaller. The limit of this sequence is 0.


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Teacher Notes and ElaborationsTopicDiscrete Mathematics

Functions/TrigonometryVirginia SOL MA.6The student will use mathematical induction to prove formulas and mathematical statements.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Compare inductive and deductive

reasoning. Prove formulas/statements, using

mathematical induction.

Key Vocabularydeductive reasoninginductive reasoningmathematical induction

Essential Questions What is inductive reasoning and when is it used? What is deductive reasoning and when is it used? How is inductive reasoning used to prove formulas and math statements? How is deductive reasoning used to prove formulas and math statements? What is mathematical induction?

Essential Understandings Mathematical induction is a method of proof that depends on a recursive process. Mathematical induction allows reasoning from specific true values of the variable to

general values of the variable.

Teacher Notes and ElaborationsThe words induction and deduction are usually used to contrast two patterns of logical thought. Reasoning by induction uses evidence derived from particular examples to draw conclusions about general principles. When reasoning by deduction general principles are used to draw conclusions about specific cases.

Inductive reasoning works from observation toward generalizations and theories. This is also called a “bottom-up” approach. Inductive reasoning starts from specific observations (or measurements); looks for patterns (or no patterns), regularities (or irregularities); formulates a hypothesis that can be worked with; and finally ends with developing general theories or drawing conclusions. In a conclusion, using induction, a number of specific instances are observed and from them infer a general principle or law.

Deductive reasoning works from the “general” to the “specific”. This is also called a “top-down” approach. The deductive reasoning works as follows: think of a theory about a topic and narrow it down to a specific hypothesis that can be tested. Narrow it down further to collect observations for hypothesis. It should be noted that observations are collected to accept or reject hypothesis and the reason this is done is to confirm or refute the original theory. In a conclusion, when using deduction, reason from general principles to specific cases, as in applying a mathematical theorem to a particular problem or in citing a law to predict the outcome of an experiment.

Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises. A deductive argument is valid if the conclusion does follow necessarily from the premises, (e.g., if the conclusion must be true then the provided premises are true). A deductive argument is sound if its premises are true.

(continued)

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Functions/TrigonometryVirginia SOL MA.6The student will use mathematical induction to prove formulas and mathematical statements.

Teacher Notes and Elaborations (continued)Deductive arguments are valid or invalid, sound or unsound, but are never true or false.

Deductive reasoning can be contrasted with inductive reasoning. In cases of inductive reasoning, it is possible for the conclusion to be false even though the premises are true.

Inductive reasoning is open-ended and exploratory especially at the beginning. On the other hand, deductive reasoning is narrow in nature and is concerned with testing or confirming hypothesis. Inductive reasoning allows for the possibility that the conclusion is false even where all the premises are true. The premises of an inductive logical argument indicate some degree of support for the conclusion but do not ensure truth.

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. This method can be extended to prove statements about more general well-founded structures. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics. Mathematical induction is a form of rigorous deductive reasoning.

The Principle of Mathematical InductionLet be a statement about integer n then is true for all positive integers n provided the following conditions are satisfied:

1. (the anchor) is true; and2. (the inductive step) if is true, then is true.

Example: Using mathematical induction, prove that is true for all positive integers n.

(the anchor) For n = 1, the equation reduces to , which is true.

(the inductive hypothesis) Assume that the equation is true for n = k. That is, assume is true.

(the inductive step) The next term on the left-hand side would be . Add this to both sides of and get

This is exactly the statement , so the equation is true for n = k + 1. Therefore, is true for all positive integers by mathematical induction.

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TopicFunctions

Functions/TrigonometryVirginia SOL MA.7The student will find the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity. A graphing utility will be used to verify intuitive reasoning, algebraic methods, and numerical substitution.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Verify intuitive reasoning about the

limit of a function, using a graphing utility.

Find the limit of a function algebraically, and verify with a graphing utility.

Find the limit of a function numerically, and verify with a graphing utility.

Use limit notation when describing end behavior of a function.

Key Vocabularyend behaviorlimit limit of a function

Essential Questions How is end behavior of a function described? What is the limit of an algebraic function?

Essential Understandings The limit of a function is the value approached by f(x) as x approaches

a given value or infinity.

Teacher Notes and ElaborationsThe limit of a function is a fundamental concept concerning the behavior of that function near a particular input. Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is “close” to L whenever x is close to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L.

End behavior is the appearance of a graph as it is followed farther and farther in either direction. For polynomials, the end behavior is indicated by drawing the positions of the ends of the graph, which may be pointed up or down. Other graphs may also have end behavior indicated in terms of the ends of the graph, or in terms of asymptotes or limits.

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as .

Some of the easiest limits to evaluate are those involving a continuous function. A function is a continuous function if the graph can be drawn without lifting the pencil from the paper. A function f(x) is continuous at a real number c if . A discontinuous function has jumps, breaks, or holes.

In this definition, there are three conditions for continuity at :1. must exist.2. f(c) must exist.3. .

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(continued)


TopicFunctions


Teacher Notes and Elaborations (continued)These are graphical examples of limits that fail to exist.

a----

b---- b-----

c c

and does not exist

does not exist f(c) is undefined because f(x) does not approach a

because specific value

A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit.

Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement means that for each M > 0 there exists a such that whenever .

Similarly, the statement means that for each whenever .

To define the infinite limit from the left, replace by . To define the infinite limit from the right, replace

by .

The equal sign in the statement does not mean that the limit exists; it means the limit fails to exist by denoting the unbounded behavior of f(x) as x approaches c.

Limits can be found graphically, numerically, or algebraically as the following example on the next page demonstrates.

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TopicFunctions



Find

Solve Graphically Solve Numerically Solve Algebraically

= 1 + 1 + 1 = 3

The graph suggests that the limit exists The table gives evidence that the and is about 3. limit is 3.

Techniques for Evaluating the limit of a Quotient of Two Functions1. If possible, use the quotient theorem for limits. If both exist, and the

2. If , try the following techniques.

a. Factor to lowest terms.

b. If involves a square root, try multiplying both by the conjugate of the square root expression.3. If , then either statement (a) or (b) below is true.

a.

b.

4. If x is approaching infinity or negative infinity, divide the numerator and denominator by the highest power of x in the denominator.

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X Y1 .997 2.991.998 2.994.999 2.997

1 ERROR1.001 3.0031.002 3.0061.003 3.009

Y1 = (X3 - 1)/(X - 1)


(continued)


TopicFunctions


Teacher Notes and Elaborations (continued)Techniques for Evaluating the limit of a Quotient of Two Functions (continued)

5. If all else fails, guess by evaluating for very large values of x, and guess by evaluating for

x-values very near . These limits can also be guessed by using a graphing calculator to examine the graph of for

very large values of x, or for x-values very near . The graphing calculator might not show points of discontinuity.

Examples of evaluating limits.

Example 1: Divide the numerator and denominator by (the highest power in the denominator).

When n is very large, are very near 0.


When n is very large, are very near 0.

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(continued)


TopicFunctions




When n is very large, is very near 0. Therefore when n is very large. Thus,

since increases without bound as n does, then and .

Example 4: Multiply by the conjugate of the square root expression.

Divide the numerator and denominator by (the highest power in the

denominator).

n approaches 0.

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TopicFunctions








35







TopicAnalytical Geometry

Functions/TrigonometryVirginia SOL MA.8The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. Transformations in the coordinate plane will be used to graph conic sections.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Given a translation or rotation

matrix, find an equation for the transformed function or conic section.

Investigate and verify graphs of transformed conic sections, using a graphing utility.

Key Vocabularydirectrixellipsefocal axisfocushyperbolaparabolarotation matrixtranslation matrix

Essential Questions What is a conic section and what are the four essential conic sections? What are the characteristics of the four essential conic sections? Given an equation, how can the conic section be determined? How can an equation of a conic be determined if the center is not at the origin?

Essential Understandings Matrices can be used to represent transformations of figures in the plane.

Teacher Notes and ElaborationsIf double cones are extended indefinitely up and down and sliced by a plane tilted at various angles, the resulting cross sections are called conic sections. Conic sections include a circle, an ellipse, a hyperbola, and a parabola.

The focus is a fixed point on the concave side of a conic section. The directrix is a fixed line on the convex side of a conic section.

A circle is the set of all points in a plane that are equidistant from a fixed point, called the center.

An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, , called the foci, is a constant.

A hyperbola is the set of points P(x, y) in a plane such that the absolute value of the difference between the distances from P to two fixed points in the plane, , called the foci, is a constant.

A parabola is the set of all points P(x, y) in the plane whose distance to a fixed point, called the focus, equals its distance to a fixed line, called the directrix.

Conics with a vertex of (0 ,0):Circle:

Ellipse: For x-axis

36


For y-axis

(continued)


37

FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicAnalytical Geometry



Hyperbola: For x-axis

For y-axis

Each hyperbola has two asymptotes that intersect at the center of the hyperbola. The asymptotes pass through the corners of a rectangle of dimensions 2a by 2b, with its center at (0, 0).

The equations of the asymptotes are: for the horizontal transverse axis

for the vertical transverse axis

Parabola: For x-axis: where the graph opens to the right or the leftFor y-axis: where the graph opens upward or downward

The focal axis contains the two focus points. For an ellipse it is called the major axis. The major axis is the longest axis of an ellipse. The shorter axis is called the minor axis. For a hyperbola the focal axis is called the transverse axis.

Conics with a vertex of (h, k):Circle: In general, if P(x, y) is on the circle with center C(h, k) and radius r, then the equation of the circle is:

Ellipse: For focal (major) axis :

For focal (major) axis :

Hyperbola: For focal (transverse) axis :

For focal (transverse) axis :

(continued)


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Teacher Notes and Elaborations (continued)Each hyperbola has two asymptotes that intersect at the center of the hyperbola. The asymptotes pass through the corners of a rectangle of dimensions 2a by 2b, with its center at (h, k).

The equations of the asymptotes are: for the horizontal transverse axis

for the vertical transverse axis

Parabola: For axis : where the graph opens to the right or the leftFor axis : where the graph opens upward or downward

Matrices have many uses in the real world. One use is transforming geometric figures using translations or rotations.

Matrix multiplication can be used in combination with sum and difference identities to determine the coordinates of points rotated on a plane about the origin. A rotation is a transformation that turns a figure about a fixed point called the center of rotation. A figure can be rotated as much as .

If P(x, y) is any point in a plane, then the coordinates of the image of point after a rotation of θ degrees about the origin can be

found by using a rotation matrix as follows .

The following example rotates a rectangular figure 30º about the origin. The rectangular figure has vertices at A(2, 1), B(5, 1), C(5, ), and D(2, ). Write matrices for a 30º rotation and for the vertices of figure ABCD.

Find the matrix product (to the nearest hundredth).

(continued)

Curriculum Information Essential Questions and Understandings39




Teacher Notes and Elaborations (continued)The approximate coordinates of the vertices for the image of figure ABCD are A'(1.23, 1.87), B'(3.83, 3.37), C'(4.83, 1.63), and D'(2.23, 0.13).

A translation is an operation that shifts a figure horizontally, vertically, or both without changing the size or shape of the figure. The transformed figure is called the image and the original figure is called the preimage. Matrix [M] represents a rectangle. The rectangle is translated in a plane a units in the x-direction and b units in the y-direction. Matrix addition can be used to translate all the vertices of a figure in one step. The following example shows the translation matrix of rectangle M translated a units in the x-direction and b units in the y-direction.

[M] =

+ =

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41






42




TopicFunctions

Functions/TrigonometryVirginia SOL MA.9The student will investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and logarithms, and the solution of logarithmic and exponential equations.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Identify exponential functions

from an equation or a graph. Identify logarithmic functions

from an equation or a graph. Define e, and know its

approximate value. Write logarithmic equations in

exponential form and vice versa.

Identify common and natural logarithms.

Use laws of exponents and logarithms to simplify expressions and solve equations.

Model real-world problems, using exponential and logarithmic functions.

Graph exponential and logarithmic functions, using a graphing utility, and identify asymptotes, intercepts, domain, and range.

Key Vocabularycommon logarithmeexponential functionlogarithmic functionnatural logarithm

Essential Questions What is e? When are logarithmic functions used in real-world situations? When are exponential functions used in real-world situations? How are logarithmic and exponential functions related algebraically, graphically and

numerically?

Essential Understandings Exponential and logarithmic functions are inverse functions. Some examples of appropriate models or situations for exponential

and logarithmic functions are:- population growth;- compound interest;- depreciation/appreciation;- Richter scale for earthquakes; and- radioactive decay.

Teacher Notes and ElaborationsAn exponential function is a function of the form , where a is a positive constant and not equal to 1. Population growth and viral growth are among examples of exponential functions. Exponential and logarithmic functions are either strictly increasing or strictly decreasing. Logarithmic functions are inverses of exponential functions. Exponential and logarithmic functions also have asymptotes.

Logarithms to base 10 are called common logarithms. A common logarithm of any positive real number x is defined to be the exponent that results when x is written as a power of 10.

Example:

Another form of logarithm is In x, the natural logarithm, with base e ≈ 2.718281828. The natural logarithm of x is usually denoted In x although it is sometimes written .

In x = k if and only if Example: In 5 ≈ 1.6 because

Common logarithmic functions are used to define many real-world measurement scales, such as the decibel scale for the relative intensities of sounds.

(continued)43



TopicFunctions


Teacher Notes and Elaborations (continued)Properties of Logarithms

1. loga1 = 0 because a0 = 1.2. logaa = 1 because a1 = a.3. logaax = x and . Inverse Properties4. If logax = logay, then x = y. One-to-one Property

Properties of Natural Logarithms1. In 1 = 0 because 2. In e = 1 because 3. In Inverse Properties4. If In x = In y, then x = y One-to-one Property

Laws of LogarithmsIf M and N are positive real numbers and b is a positive number other than 1, then:

1. Log of a Product

2. Log of a Quotient

3. if and only if M = N Identity (one-to-one)4. , for any real number k Log of a Power

The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of Algebra. This is true because they convert complicated products, quotients, and exponential forms into simpler sums, differences, and products. The following examples use the properties and laws of logarithms.

Example 1 uses the properties to expand the expression:Log of a Product

Log of a Power

Example 2 uses properties to condense the expression:

Log of a Power

Log of a Product

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TopicFunctions


Teacher Notes and Elaborations (continued)Example 3 solves an equation with a natural logarithm:

Take the natural log of each side Inverse Property

Example 4 solves an equation with a natural logarithm:

Add 3 to each side

Divide each side by 4

Take the natural log of each side

Inverse Property

Divide each side by 2

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TopicFunctions








46








Functions/TrigonometryVirginia SOL MA.13The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Solve and create problems,

using trigonometric functions. Solve and create problems,

using the Pythagorean Theorem.

Solve and create problems, using the Law of Sines and the Law of Cosines.

Solve real-world problems using vectors.

Key VocabularyLaw of CosinesLaw of Sinesvector

Essential Questions When is the Pythagorean Theorem not used to find missing sides of a triangle? How can the Law of Sines be utilized to find the missing side/angle of a triangle? Under what conditions would the Law of Cosines be utilized to find the missing side of

a triangle?

Essential Understandings Real-life real-world problems can be modeled using trigonometry and vectors.

Teacher Notes and ElaborationsA vector is a directed segment having both magnitude (length) and direction.

A resultant vector is the sum of two or more vectors.

Vectors are frequently used in problems of navigation or road work construction.

Other formulas for the area of a triangle are .

The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides of lengths a, b, and c (a opposite , , and ).

This law is often used if two angles and a side are known (AAS or ASA).

The Law of Cosines states that for any triangle with sides of lengths a, b, and c then .

This law is often used when at least two sides are known (SAS or SSS).

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicEquations

Functions/TrigonometryVirginia SOL MA.14The student will use matrices to organize data and will add and subtract matrices, multiply matrices, multiply matrices by a scalar, and use matrices to solve systems of equations.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Add, subtract, and multiply

matrices and multiply matrices by a scalar.

Model problems with a system of no more than three linear equations.

Express a system of linear equations as a matrix equation.

Solve a matrix equation. Find the inverse of a matrix. Verify the commutative and

associative properties for matrix addition and multiplication.

Key Vocabularycolumnelementinverse matrixmatrixrowscalar

Essential Questions What is a matrix? How are matrices used to organize and describe data? How and when would a matrix be used? What do the numbers in a matrix represent? How are matrices added, subtracted, and multiplied? How are problems with a system modeled? How is a matrix used to solve a system of linear equations? How is an inverse of a matrix determined? How are commutative and associative properties of matrix addition and multiplication

verified?

Essential Understandings Matrices are a convenient shorthand for solving systems of equations. Matrices can model a variety of linear systems. Solutions of a linear system are values that satisfy every equation in the system. Matrices can be used to model and solve real-world problems.

Teacher Notes and ElaborationsA matrix is a rectangular array of terms (elements) in rows and columns that are enclosed with brackets. A matrix organizes a data set visually.

A matrix is identified by its dimensions, rows and columns (e.g., a 2 × 3 matrix has two rows and three columns). Each element of the matrix represents a combination of the characteristics defined by the rows and columns. Matrices are usually named by an uppercase letter and the matrix dimensions with the letter name. The dimension or order of a matrix is essential to determining which operations can be performed.

Matrices are used to sort, list, and organize data such as sorting by gender the number of soccer, football, basketball, and softball players in a set. They make large amounts of data easy to read and evaluate for the decision-making process. A matrix is an efficient method of organizing real-world data for the purpose of interpreting, analyzing, and performing calculations with data.

A relationship exists between arithmetic operations and operations with matrices. Computing with matrices is different from computing with real numbers. Matrices do not follow all of the properties of the real number system. Matrix multiplication is associative, but not commutative. Two matrices may have a product of zero even if neither of the factors equals zero.

(continued)


50



Teacher Notes and Elaborations (continued)Operations with MatricesAdditionMatrices have special algebraic rules. Two matrices having the same dimensions can be added to produce a new matrix by finding the sums of the corresponding elements of the matrices. This operation is called matrix addition. Since corresponding elements are added two matrices can not be added if they have different dimensions.

Example 1 (Addition):

SubtractionMatrix subtraction is similar to real number subtraction: to subtract a matrix, add the additive inverse of the matrix. In simpler terms, two matrices can be subtracted by finding the differences of the corresponding elements of the matrices.

The additive inverse of a matrix A, denoted by –A, is the matrix in which each element is the opposite of its corresponding element in A.

For example: If

Example 2 (Subtraction):

Scalar MultiplicationThe operation of multiplying a matrix A by a real number c is called scalar multiplication. In matrix Algebra, any real number is called a scalar. The new matrix, cA, is the result of multiplying each element in A by c.

(continued)


51



Teacher Notes and Elaborations (continued)Operations with MatricesExample 3 (Scalar multiplication):

MultiplicationThe definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the ith row and the jth column of the product AB is obtained by multiplying the entries in the ith row of A by the corresponding entries in the jth column of B and then adding the results.For example: If

The product AB of two matrices is defined if and only if the number of columns in equals the number of rows in .

Example 4 (Matrix multiplication):

Any matrix whose elements are all zero is called a zero matrix, denoted by . The following matrices are zero matrices.

(continued)


52



Teacher Notes and Elaborations (continued)A square matrix is any matrix having the same number of rows as columns. The main diagonal of a square matrix is the diagonal that extends from upper left to lower right. Any matrix whose main diagonal elements are 1 and whose other elements are 0 is called an identity matrix, denoted by . The following matrices are identity matrices.

The real numbers 5 and are called multiplicative inverses since . Similarly, a matrix can have a multiplicative inverse. If the product of two square matrices is the identity matrix then those two matrices are inverses of each other. The inverse of a matrix is

denoted (inverse matrix). The following examples illustrate the identity and inverse properties for multiplication.

To find an inverse of a square matrix use the following formula.

.

(continued)


53



Teacher Notes and Elaborations (continued)Example 5 (Finding the inverse matrix):

A matrix equation is an equation in which a variable is a matrix. Using algebraic properties of addition, subtraction, and scalar multiplication the value of the unknown matrix can be found. Solving a linear equation such as 3x + 7 = 20 is similar to solving any matrix equation in the form AX + B = C

Example 6 (Solving a matrix equation):

Given this equation:

Multiply by scalar 3

Add the inverse to both sides

Solve

(continued)


54


TopicEquations


Teacher Notes and Elaborations (continued)Example 7 (Solving a system using matrices):

Given this system of equations

Replace the equations by a single matrix equation:

Find the inverse of matrix A ( )

Multiply both sides by the inverse

The solution to the system is .

Real-world applications of matrices should include data manipulation and solutions to systems of equations done manually and with a graphing calculator.

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Resources Sample Instructional Strategies and Activities

TopicEquations








Develop concept of matrix multiplication from a problem-solving approach. Once this concept is understood, use graphing calculators or computer programs with matrix capabilities to solve problems that are more interesting.

Students will enter two 4 × 4 matrices, A and B, into a graphing calculator, multiply A × B, multiply B × A, and compare the results.

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Topic Triangular and Circular Trigonometric Functions

Functions/TrigonometryVirginia SOL T.1 The student, given a point other than the origin on the terminal side of the angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Define the six triangular trigonometric

functions of an angle in a right triangle. Define the six circular trigonometric

functions of an angle in standard position.

Make the connection between the triangular and circular trigonometric functions.

Recognize and draw an angle in standard position.

Show how a point on the terminal side of an angle determines a reference triangle.

Key Vocabularycircular trigonometric functiondegreesinitial sideradiansreference triangleterminal sidetriangular trigonometric functionunit circle

Essential Questions What is the standard position of an angle? Given a point on the terminal side of an angle, how are the values of the six

trigonometric functions determined? What is the relationship between trigonometric and circular functions?

Essential Understandings Triangular trigonometric function definitions are related to circular trigonometric

function definitions. Both degrees and radians are units for measuring angles. Drawing an angle in standard position will force the terminal side to lie in a specific

quadrant. A point on the terminal side of an angle determines a reference triangle from which the

values of the six trigonometric functions may be derived.

Teacher Notes and ElaborationsAs derived from the Greek language, the word trigonometry means “measurement of triangles”.

An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side.

The six trigonometric functions of an angle θ are called sine, cosine, tangent, cotangent, secant and cosecant. The functions are defined with the angle θ (the Greek letter theta) in standard position.

In the rectangular coordinate system an angle with its vertex at the origin and with its initial side along the positive x-axis is in standard position. For any point P(x, y) on the terminal side of an angle θ in standard position, r is defined as the distance from the vertex to P

. A point on the terminal side of an angle determines a reference triangle from which the values of the six trigonometric functions may be derived.

(continued)58



Topic Triangular and Circular Trigonometric Functions

Functions/TrigonometryVirginia SOL T.1 The student, given a point other than the origin on the terminal side of the angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.

Teacher Notes and Elaborations (continued)The six triangular trigonometric functions of θ are:

The properties of the trigonometric functions are connected with the circular function definitions by using a unit circle (a circle with the radius of one).

If the terminal side of an angle θ in standard position intersects the unit circle at P(x, y), then the six circular trigonometric functions are defined as:

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. Degrees and radians are equivalent units for angle measurement. One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

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TopicTriangular and Circular Trigonometric Functions

Functions/TrigonometryVirginia SOL T.1

Text:Trigonometry, Sixth Edition, 2006,McDougal Littell/Houghten Mifflin



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61






Functions/TrigonometryVirginia SOL T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Given one trigonometric function value,

find the other five trigonometric function values.

Develop the unit circle, using both degrees and radians.

Solve problems, using the circular function definitions and the properties of the unit circle.

Recognize the connections between the coordinates of points on a unit circle and- coordinate geometry;- cosine and sine values; and - lengths of sides of special right

triangles (30° - 60° - 90° and 45° - 45° - 90°).

Key VocabularydegreesPythagorean identitiesradiansratio (quotient) identitiesreciprocal identitiesunit circle

Essential Questions What are the Pythagorean, ratio, and reciprocal identities? Given the value of one trigonometric function, how are the remaining functions

determined?

Essential Understandings If one trigonometric function value is known, then a triangle can be formed to use in

finding the other five trigonometric function values. Knowledge of the unit circle is a useful tool for finding all six trigonometric values for

special angles.

Teacher Notes and ElaborationsGiven the value of one trigonometric function, a triangle can be formed to use in finding the other five trigonometric function values or the remaining functions may also be found using one of the following methods:Definitions of the trigonometric functions:

and the

and the

and the

Relationships between trigonometric functions are identities.Reciprocal Identities:

Since and the , then and .

Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities hold for any angle θ that does not lead to a zero denominator.

Pythagorean Identities:

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(continued)



Functions/TrigonometryVirginia SOL T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.

Teacher Notes and Elaborations (continued)Ratio or Quotient Identities:

Degrees and radians are equivalent units for angle measurement. A central angle with sides and intercepted arcs all the same length measures 1 radian.

A unit circle is one that lies on the x-axis, has origin (0, 0), and a radius of 1.

Knowledge of the unit circle is a useful tool for finding all six trigonometric values for special angles.

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Text:Trigonometry, Sixth Edition, 2006, McDougal Littell/Houghten Mifflin



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Functions/TrigonometryVirginia SOL T.3 The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find trigonometric function values of

specials angles and their related angles in both degrees and radians.

Apply the properties of the unit circle without using a calculator.

Use a conversion factor to convert from radians to degrees and vice versa without using a calculator.

Key Vocabularycoterminal anglesdegreesquadrantal anglesradianrevolutionunit circle

Essential Questions What is the relationship between radians and degrees? What is the relationship between families of coterminal angles? What is meant by the special angles?

Essential Understandings Special angles are widely used in mathematics. Unit circle properties will allow special-angle and related-angle trigonometric values to

be found without the aid of a calculator. Degrees and radians are units of angle measure. A radian is the measure of the central angle that is determined by an arc whose length is

the same as the radius of the circle.

Teacher Notes and ElaborationsThe two most common units used to measure angles are radians and degrees. The radian measure of an angle in standard position is defined as the length of the corresponding arc

divided by the radius of the circle ( ). One degree, 1°, is the result from a rotation of

of a complete revolution about the vertex in the positive direction. A full revolution

(counterclockwise) corresponds to 360º.

To convert radians to degrees and vice versa, multiply by the appropriate conversion factor

.

Multiples, between 0 and 2π, of first quadrant special angles are found without the aid of a calculator.

Angles that measure greater than 2π can be formed by adding or subtracting a multiple of 2π to its coterminal angle measuring between 0 and 2π.

Two angles in standard position with the same initial and terminal sides are called coterminal angles.

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(continued)



Functions/TrigonometryVirginia SOL T.3 The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.


Special angles are widely used in mathematics. The first quadrant special angles of a unit circle (a circle with a radius of one) are , ,

. The quadrantal angles (any angle with the terminal side on the x-axis or y-axis) of a unit circle are 0, , π, , 2π.

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicInverse Trigonometric Functions

Functions/TrigonometryVirginia SOL T.4 The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use a calculator to find the

trigonometric function values of any angle in either degrees or radians.

Define inverse trigonometric functions. Find angle measures by using the

inverse trigonometric functions when the trigonometric function values are given.

Key Vocabularyinverse trigonometric functions

Essential Questions What are inverse trigonometric functions?

Essential Understandings The trigonometric function values of any angle can be found by using a calculator. The inverse trigonometric functions can be used to find angle measures whose

trigonometric function values are known. Calculations of inverse trigonometric function values can be related to the triangular

definitions of the trigonometric functions.

Teacher Notes and ElaborationsThe values of the trigonometric functions of any angle can be approximated using a calculator. Most values are approximated to four decimal places. Depending upon the problem, calculators must be in the appropriate mode, whether radian or degree.

The inverse trigonometric functions can be used to find angle measures whose trigonometric function values are known. Given the value of any trigonometric function, the angle may be determined by using the appropriate inverse function key on the calculator. Values of inverse trigonometric functions are always in radians.

Definitions of the Inverse Trigonometric Functions:


Function Domain Range

if and only if sin y = x

if and only if cos y = x

if and only if tan y = x

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TopicInverse Trigonometric Functions





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Teacher Notes and ElaborationsTopicTrigonometric Identities

Functions/TrigonometryVirginia SOL T.5The student will verify basic trigonometric identities and make substitutions, using the basic identities.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Use trigonometric identities to make

algebraic substitutions to simplify and verify trigonometric identities. The basic trigonometric identities include - reciprocal identities; - Pythagorean identities; - sum and difference identities;- double-angle identities; and- half-angle identities.

Key Vocabularyidentitydouble-angle identitieshalf-angle identitiesPythagorean identitiesreciprocal identitiessum and difference identitiestrigonometric identitiesverify

Essential Questions What is an identity? What is the difference between solving equations and verifying identities?

Essential Understandings Trigonometric identities can be used to simplify trigonometric expressions, equations, or

identities. Trigonometric identity substitutions can help solve trigonometric equations, verify

another identity, or simplify trigonometric expressions.

Teacher Notes and ElaborationsAn identity is an equation that is true for all possible replacements of the variables. An identity involving trigonometric expressions is a trigonometric identity. Trigonometric identities can be used to simplify trigonometric expressions, equations, or identities. The fundamental trigonometric identities are the following:

- reciprocal identities,- Pythagorean identities,- sum and difference identities,- half angle identities, and- double angle identities.

Reciprocal Identities:

Since and the , then and .

Also, cos θ and sec θ are reciprocals as are tan θ and cot θ. The reciprocal identities hold for any angle θ that does not lead to a zero denominator.

Pythagorean Identities:

Ratio or Quotient Identities:

(continued)

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSCurriculum Information Essential Questions and Understandings

Teacher Notes and ElaborationsTopicTrigonometric Identities

Functions/TrigonometryVirginia SOL T.5The student will verify basic trigonometric identities and make substitutions, using the basic identities.


Double-Angle Identities:

= =

Sum and Difference Identities:

Half-Angle Identities:

The signs of depend on the quadrant in which lies.

To verify a trigonometric identity, either the left or the right side of the equation may be used to deduce the other side. Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding the same terms to both sides, are not valid when working with identities since the statement to be verified may not be true. To verify an identity, show that one side of the identity can be simplified so that it is identical to the other side.

Guidelines for Verifying Trigonometric Identities1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and

cosines pair up well, as do secants and tangents, and cosecants and cotangents.4. If the preceding guidelines do not help, try converting all terms to sines and cosines.5. Always try something. Even making an attempt that leads to a dead end provides insight.6. Try working backwards from the solution, as it can provide great insight.

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TopicTrigonometric Identities





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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSThis page is intentionally left blank.



TopicTrigonometric Equations, Graphs, and Practical Problems

Functions/TrigonometryVirginia SOL T.6 The student, given one of the six trigonometric functions in standard form, willa. state the domain and the range of

the function;b. determine the amplitude, period,

phase shift, vertical shift; and asymptotes;

c. sketch the graph of the function by using transformations for at least a two-period interval; and

d. investigate the effect of changing the parameters in a trigonometric function on the graph of the function.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Determine the amplitude, period, phase

shift, and vertical shift of a trigonometric function from the equation of the function and from the graph of the function.

Describe the effect of changing A, B, C, or D in the standard form of a trigonometric equation {e.g., y = A sin(Bx + C) + D, or y = A cos[B(x + C)] + D}.

State the domain and the range of a function written in standard form {e.g., y = A sin(Bx + C) + D ory = A cos[B(x + C)] + D}.

Sketch the graph of a function written in standard form {e.g., y = A sin(Bx + C) + D or y = A cos[B(x + C)] + D } by using transformations for at least a two period interval.

Key Vocabularyamplitudeasymptotehorizontal phase shiftperiod of the functionperiodic functionrangevertical phase shift

Essential Questions What effect does the change in the values A, B, C, and D in the equation

y = A sin(Bx - C) + D, have on the graph of the function? Why are the terms: phase shift, period, amplitude, vertical shift and asymptote

important to curve sketching?

Essential Understandings The domain and range of a trigonometric function determine the scales of the axes for

the graph of the trigonometric function. The amplitude, period, phase shift, and vertical shift are important characteristics of the

graph of a trigonometric function, and each has a specific purpose in applications using trigonometric equations.

The graph of a trigonometric function can be used to display information about the periodic behavior of a real-world situation, such as wave motion or the motion of a Ferris wheel.

Teacher Notes and ElaborationsEach of the six trigonometric functions is a periodic function whose graph is based on repetition. A periodic function is a function such that for every real number in the domain of and for some positive real number . The smallest possible positive value of is the period of the function. The period of the sine, cosine, secant, and cosecant function is 2π. The period of the tangent and cotangent function is π.

The amplitude of a function can be interpreted as half the difference between its maximum and minimum values. The amplitude is half the range (difference between maximum and minimum values).

Suggested five steps to sketch the parent graph of y = A sin Bx or y = A cos Bx, with are:

1. Determine the period of repeat, . Start at 0 on the x-axis and mark off that

distance.2. Divide the interval into four equivalent parts.3. Evaluate the function for each of the five x values resulting from Step 2. The points

will be maximum points, minimum points, and x intercepts.4. Plot those points found in Step 3 and join them with a curve.5. Draw additional cycles to the left and right of the curve.

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Functions/TrigonometryVirginia SOL T.6 The student, given one of the six trigonometric functions in standard form, willa. state the domain and the range of

the function;b. determine the amplitude, period,

phase shift, vertical shift; and asymptotes;

c. sketch the graph of the function by using transformations for at least a two-period interval; and

d. investigate the effect of changing the parameters in a trigonometric function on the graph of the function.

Teacher Notes and Elaborations (continued)Transformations to the original graph can be done through phase shifts. The vertical phase shift moves the horizontal axis of the graph along the y-axis. The horizontal phase shift moves the graph along the x-axis.

Steps to sketch the graph of y = A sin(Bx – C) + D or y = A cos(Bx – C) + D, with are:1. Determine D the vertical phase shift. This will be the new horizontal axis at y = D.2. Determine C the horizontal phase shift. This will lie on the x-axis.3. Follow steps 1 - 5 above.

The asymptote is a straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases without limit.

Reciprocal identities are used to obtain the graphs of the secant and cosecant functions. The cosecant and secant functions will have vertical asymptotes. The asymptotes will have equations of the form , where k is the x-intercept of the sine or cosine function.

Sketching the graphs of the variations of the tangent and cotangent is similar to sketching the graphs of the transformations of sine and cosine functions. Key differences are the period of repeat, asymptotes, and the shape of the graph. Tangent and cotangent graphs do not have amplitude.

The graphing calculator can provide a visual look at how the constants A, B, C, and D affect the graph of a function. Be sure the calculator is set for radians. Most calculators have a trig window with domain [-2π, 2π], range [-4, 4], π, and . Other settings may be preferable for different equations.

Graphs of trigonometric functions model periodic behavior of real world situations such as wave motion, biorhythms, seasonal temperatures, or the motion of a Ferris wheel.

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FUNCTIONS/TRIGONOMETRY CURRICULUM GUIDE (Revised 2011) PRINCE WILLIAM COUNTY SCHOOLSTopicInverse Trigonometric Functions

Functions/TrigonometryVirginia SOL T.7The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Find the domain and range of the

inverse trigonometric functions. Use the restrictions on the domains of

the inverse trigonometric functions in finding the values of the inverse trigonometric functions.

Identify the graphs of the inverse trigonometric functions.

Key Vocabularyinverse trigonometric functionrestrictions on domains

Essential Questions and Understandings What are the domains and ranges of the inverse trigonometric functions? What are the restrictions on the domain of the inverse trigonometric functions?

Essential Understandings Restrictions on the domains of some inverse trigonometric functions exist.

Teacher Notes and ElaborationsThe trigonometric functions are not one-to-one, so it is necessary to determine the restrictions on domains to regions that pass the horizontal line test. The inverse trigonometric functions can be denoted in two ways. For example, the inverse of may be written as or .

Function Domain Range

y = arcsin x [-1,1]

y = arccos x [-1,1] [0, π]

y = arctan x [-∞,∞]

y = arccot x [-∞,∞] [0, π]

Function Domainy = arcsec x [-∞, -1] [1, ∞]y = arccsc x [-∞, -1] [1, ∞]

Function Range

y = arcsec x [0,π],

y = arccsc x ,

The graphs of the inverse trigonometric functions are obtained by interchanging the x- and y- coordinates of the key points of the basic graphs.


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TopicInverse Trigonometric Functions





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Teacher Notes and ElaborationsTopicTrigonometric Equations, Graphs, and Practical Problems

Functions/TrigonometryVirginia SOL T.8 The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Solve trigonometric equations with

restricted domains algebraically and by using a graphing utility.

Solve trigonometric equations with infinite solutions algebraically and by using a graphing utility.

Check for reasonableness of results, and verify algebraic solutions, using a graphing utility.

Key Vocabularytrigonometric equationtrigonometric identities

Essential Questions and Understandings Do trigonometric equations have unique solutions? Why or why not? What is the relationship of the domain and range to the solution of trigonometric

equations?

Essential Understandings Solutions for trigonometric equations will depend on the domains. A calculator can be used to find the solution of a trigonometric equation as the points of

intersection of the graphs when one side of the equation is entered in the calculator as Y1

and the other side is entered as Y2.

Teacher Notes and ElaborationsTrigonometric equations, like most algebraic equations, are true for some, but not for all values of the variable. Trigonometric equations do not have unique solutions. Solutions for trigonometric equations will depend on the domains. They have infinitely many solutions, differing by the period of the function. If the domain of the equations is restricted to one revolution then only those solutions between 0 and 2π will be determined.

To solve a trigonometric equation, use standard algebraic techniques and fundamental trigonometric identities.

The fundamental trigonometric identities are the following:- reciprocal identities,- Pythagorean identities,- sum and difference identities,- half angle identities, and- double angle identities.

Standard algebraic techniques are used to solve trigonometric inequalities.

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Functions/TrigonometryVirginia SOL T.9 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Write a real-world problem involving

triangles. Solve real-world problems involving

triangles. Use the trigonometric functions,

Pythagorean Theorem, Law of Sines, and Law of Cosines to solve real-world problems.

Use the trigonometric functions to model real-world situations.

Identify a solution technique that could be used with a given problem.

Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Key Vocabularydirected line segmentLaw of CosinesLaw of SinesmagnitudesobliquePythagorean Theoremscalarsum and difference formulasvectorvector quantity

Essential Questions and Understandings How are practical problems involving triangles and vectors solved? What is the relationship of a vector to right triangles and trigonometric functions? What is meant by an ambiguous case when determining parts of a triangle?

Essential Understandings A real-world problem may be solved by using one of a variety of techniques associated

with triangles.

Teacher Notes and ElaborationsPractical problems involving right triangles can be solved by applying the right triangle definitions of trigonometric functions and the Pythagorean Theorem. Problems involving oblique (non-right) triangles are solved using the Law of Sines or the Law of Cosines depending upon the given information.

The Law of Sines states that for any triangle with angles of measures A, B, and C, and sides of lengths a, b, and c (a opposite , , and ).

The Law of Cosines states that for any triangle with sides of lengths a, b, and c then .

To solve an oblique triangle, the measure of at least one side and any two other parts of the triangle need to be known. This breaks down into the following cases.

GivenAAS Law of SinesASA Law of SinesSSA Law of Sines (ambiguous case)SAS Law of CosinesSSS Law of Cosines

Heron’s area formula is used if the lengths of the sides of the triangle are known. If two sides of a triangle and the angle between the two sides are known then the area formula below is used:

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Functions/TrigonometryVirginia SOL T.9 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

Teacher Notes and Elaborations (continued)Many quantities in mathematics involve magnitudes. These quantities are called scalar. Other quantities called vector quantities, involve both magnitude and direction. A vector quantity is often represented with a directed line segment, which is called a vector. The length of the vector represents the magnitude of the vector quantity. Each vector has a horizontal and vertical component. Vectors may be added and subtracted.

Sum and Difference Formulas:

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