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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 08: Trigonometric Functions SUGGESTED DURATION : 15 days

Unit Overview

Introduction
This unit bundles student expectations that address rotation angles, the unit circle, and periodic functions. Graphs, attributes, transformations, and inverses of trigonometric functions are analyzed and applied in mathematical and real-world problem situations. These topics are studied using multiple representations, including graphical, tabular, verbal, and algebraic methods. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Geometry Unit 05, students determined the lengths of sides and the measures of angles in a right triangle by applying special right triangle relationships, the Pythagorean Theorem, Pythagorean triples, and the trigonometric ratios sine, cosine, and tangent. In Geometry Unit 06, students described the relationship between the radian measure of an angle, the length of an arc intercepted by a central angle, and the radius of a circle. In Precalculus Unit 07, students determined the values of the trigonometric functions at special angles in mathematical and real-world problem situations and used trigonometric ratios to solve problems.

During this Unit
Students explore the concept of radian measure by wrapping radii around a circle (starting in standard position) to develop the concept that one complete rotation includes 360 degrees or 2π radians. Students represent angles in radians and degrees based on the concept of rotation and find the measure of reference angles and angles in standard position in both mathematical and real-world problems. Students convert angles between degrees and radians and solve mathematical and real-world problems (including linear and angular velocity) using these representations. Students determine the values of the trigonometric functions at the special angles, use them to define and label the unit circle, and describe the relationship between degree and radian measure on the unit circle. Students relate those values of the trigonometric functions at the special angles in mathematical and real-world problems. Students make connections between the unit circle and the periodic function and use the relationships to evaluate trigonometric functions in mathematical and real-world situations. Students graph the trigonometric functions (including sine, cosine, tangent, cosecant, secant, and cotangent) and determine and analyze the key features, including domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing. Students graph transformations of trigonometric functions (sine and cosine), including af(x), f(x) + d, f(xc), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problem situations. Students apply their understanding of transformations to develop and use sinusoidal functions to model mathematical and real-world problem situations and analyze the key features of these functions in context of the problem situation. Students determine inverse functions of sine and cosine functions using graphs, tables, and algebraic representations. Students graph inverse trigonometric functions (including arcsin and arccos), describe the limitations on the domains of these functions, and determine and analyze the key features of these graphs.

After this Unit
In Units 09 – 12, students will continue to apply trigonometric functions and relationships when studying trigonometric equations and identities, vectors, parametric equations, and polar equations. In subsequent mathematics courses, students will continue to apply these concepts as they arise in problem situations.

Function analysis serves as the foundation for college readiness. Focusing on real world function analysis and representation is emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning C1, D1, D2; III. Geometric Reasoning A1, A3, C1; IV. Measurement Reasoning A1, B1; VI. Statistical Reasoning B2, C3; VII. Functions A1, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to a 2007 report published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) notes the necessity for high school students to create and interpret models of complex phenomena by identifying the essential attributes of a situation and selecting a mathematical relationship with similar attributes. Specifically, students should recognize that phenomena with periodic features are often best modeled by trigonometric functions. As natural and fundamental examples of periodic functions, trigonometric functions can be used to approximate any periodic function and a variety of real-world phenomena (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the Texas College and Career Readiness Standards (2009), which call for students to be able to understand and analyze features of a function to model real-world situations. Algebraic models allow us to efficiently visualize and analyze the vast amount of interconnected information that is contained in a functional relationship; these tools are particularly helpful as the mathematical models become increasingly complex (National Research Council, 2005). In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve analysis and understanding of elementary functions, including trigonometric and inverse trigonometric functions. Specifically, students must be familiar with the properties, algebra, graphs, and language of trigonometric functions, while also knowing the values of the trigonometric functions at multiples of   , and .

College Board.  (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2005). How students learn: Mathematics in the classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist within rotation angles and the unit circle?
• What invariant (unchanging) and variant (changing) relationships exist within situations involving rotation?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent rotation?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships about …
• rotation?
• the unit circle?
• What tools and processes can be used to …
• represent rotation?
• develop and describe the unit circle?
• What relationships exist between right triangle trigonometry, rotation, and the unit circle?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can …
• rotation concepts
• the unit circle
… be applied when solving problem situations?
• How can measurable attributes related to rotation angles be distinguished and described in order to generalize geometric relationships?
• What processes can be used to determine the …
• trigonometric ratios in right triangles?
• values of trigonometric functions at special angles?
• measure of rotation angles?
• linear and angular velocity in a rotation situation?
• Number and Measure
• Patterns, Operations, and Properties
• Unit Circle
• Degree measure
• Reference angles
• Linear velocity
• Angular velocity
• Trigonometric ratios
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Trigonometric functions are characterized by a periodic rate of change, are based on ratios within right triangles, and can be used to describe, model, and make predictions about situations.
• Inverse trigonometric functions are characterized as inverses of trigonometric functions (under appropriate domain restrictions) and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can trigonometric and inverse trigonometric functions model?
• What graphs, key attributes, and characteristics are unique to trigonometric and inverse trigonometric functions?
• What patterns of covariation are associated with trigonometric and inverse trigonometric functions?
• How can the key attributes of trigonometric and inverse trigonometric functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of trigonometric and inverse trigonometric functions?
• What are the real-world meanings of the key attributes of trigonometric and inverse trigonometric function models?
• How can the key attributes of trigonometric and inverse trigonometric functions be used to make predictions and critical judgments?
• What relationships exist between the unit circle and trigonometric functions?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• What relationships exist between a function and its inverse?
• How are the key attributes of a function related to the key attributes of its inverse?
• How can the inverse of a function be determined and represented?
• Functions can be represented in various ways (including algebraically, graphically, verbally, and numerically) with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Asymptotes
• Symmetries
• Increasing or decreasing
• Functions
• Trigonometric
• Inverse
• Patterns, Operations, and Properties
• Relations and Generalizations
• Transformations
• Parent functions
• Transformation effects
• Number and Measure
• Unit Circle
• Degree measure
• Reference angles
• Trigonometric ratios
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that inverse trigonometric functions are the same as the reciprocalsof the trigonometric functions. For example, sin–1 is not the same as .
• Some students may incorrectly evaluate trigonometric functions or inverse trigonometric functions if they use a calculator in the wrong angle mode.
• Some students may incorrectly determine the horizontal translation of a trigonometric function by not first factoring out a horizontal stretch or compression parameter change. For example, some students might think that the horizontal shift of the function f(x) = cos(3x ), is , rather than since f(x) = cos(3(x )).
• Some students may think that the period of a trigonometric function of the form is b rather than b = , where P = period.

Underdeveloped Concepts:

• Some students may think that radians are a unit of angle measure, rather than realizing that angles measured in radians are unitless.
• Some students may not realize that the cosine function is a sinusoidal function.

Unit Vocabulary

• Angular velocity – rate of change of angular displacement, measured in radians per unit of time
• Covariation – pattern of related change between two variables in a function
• Degree measure – measured as a central angle with respect to the unit circle
• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.
• Linear velocity – the speed, measured in distance per unit of time, with which an object moves along a circular or rotational path
• Radian measure – measured as the directed length of an intercepted arc on the unit circle
• Reference angle – If A is an angle in standard position, the reference angle to angle A is an acute angle created by the terminal side of angle A and the x-axis.
• Quadrantal angle – an angle whose terminal side lies on the x-axis or y-axis
• Special right triangles – right triangles which have angles that measure 30°– 60°– 90° or 45°– 45°– 90°
• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle

Related Vocabulary:

 Amplitude Angle Arccosine Arcsine Asymptote Circular function Clockwise Compression Cosecant Cosine Cotangent Coterminal angles Counterclockwise Decreasing Degrees Discontinuities Domain End behaviors Extrema Increasing Initial side Maximum Minimum Negative rotation Parent function Period Periodic function Positive rotation Quadrant Radians Range Reciprocal function Reflection Revolution Root Rotation angle Secant Sine Sinusoidal Solution Special angles Standard position Stretch Symmetry Tangent Terminal side Transformation Translation Trigonometric ratios Unit circle x-intercept y-intercept Zeros
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2E Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

Determine

AN INVERSE FUNCTION, WHEN IT EXISTS, FOR A GIVEN FUNCTION OVER ITS DOMAIN OR A SUBSET OF ITS DOMAIN

Represent

THE INVERSE OF A FUNCTION USING MULTIPLE REPRESENTATIONS

Including, but not limited to:

• Inverse of a function – function that undoes the original function. When composed f(f –1(x)) = x and f –1(f(x)) = x.
• Characteristics of inverse functions
• Domain of the function becomes an appropriate range of the inverse function.
• Range of the function becomes an appropriate domain of the inverse function.
• Composed as f(f –1(x)) = x and f –1(f(x)) = x
• Multiple representations
• Inverse function notation
• When a function f(x) has an inverse that is also a function, the inverse can be written with f –1(x).
• For the function f(x) = x + 4, the inverse function is f –1(x) = x – 4.
• For the function g(x) = x2:
• If the restricted domain of g(x) is x ≥ 0, then the inverse function is g1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g1(x) = – .
• Algebraic
• The inverse of a function can be found algebraically by:
• Writing the original function in “y = ” form
• Interchanging the x and y variables
• Solving for y
• A function’s inverse can be confirmed algebraically if both of the following are true: f(f –1(x)) = x and f –1(f(x)) = x.
• Tabular
• From the table of values for a given function, the tabular values of the inverse function can be found by switching the x- and y-values of each ordered pair.
• Graphical
• The graphs of a function and its inverse are reflections over the line y = x.
• Verbal description of the relationships between the domain and range of a function and its inverse
• Restrictions on the domain of the original function to maintain functionality
• Inverse functions over a subset of the domain of the original function

Note(s):

• Algebra II analyzed the relationship between functions and inverses, such as quadratic and square root, or logarithmic and exponential, including necessary restrictions on the domain.
• Precalculus extends the analysis of inverses to include other types of functions, such as trigonometric and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VI Statistical Reasoning
• B2 – Select and apply appropriate visual representations of data.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2F

Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph

TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS

Including, but not limited to:

• Graphs of the parent functions
• Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
• Various methods for graphing
• Curve sketching
• Plotting points from a table of values
• Transformations of parent functions (parameter changes abc, and d)
• Using graphing technology

Note(s):

• Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2G

Graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems.

Graph

FUNCTIONS, INCLUDING SINE and COSINE FUNCTIONS AND THEIR TRANSFORMATIONS, INCLUDING af(x), f(x) + d, f(xc), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Sine and cosine functions: f(x) = sin x, f(x) = cos x
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters a, b, c, and d on graphs
• Effects of a on f(x) in af(x)
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the horizontal axis (x-axis)
• Effects of d on f(x) in f(x) + d
• d = 0, no vertical shift
• Translation, vertical shift up or down by |d| units
• Effects of c on f(x) in f(xc)
• c = 0, no horizontal shift
• Translation, horizontal shift left or right by |c| units
• Effects of b on f(x) in f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the vertical axis or y-axis
• Combined transformations of parent functions
• Transforming a portion of a graph
• Illustrating the results of transformations of the stated functions in mathematical problems using a variety of representations
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II graphed transformations of various types of functions, including square root, cube, cube root, absolute value, rational, exponential, and logarithmic functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2H Graph arcsin x and arccos x and describe the limitations on the domain.

Graph

arcsin x AND arccos x

Including, but not limited to:

• Graph of sine function,f(x) = sin x, and inverse sine function, f(x) = arcsin x
• Graph of cosine function, f(x) = cos x, and inverse cosine function, f(x) = arccos x
• Inverse relationships
• x and y values of the ordered pairs are switched.
• Graphs are reflections over the line y = x.
• f(x) = sin x and f(x) = arcsin x, which can also be written as f(x) = sin–1x, are inverses.
• f(x) = cos x and f(x) = arccos x, which can also be written as f(x) = cos–1x, are inverses.
• Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations

Describe

THE LIMITATIONS ON THE DOMAIN OF arcsin x AND arccos x

Including, but not limited to:

• Limitations
• Comparison of domain and range of f(x) = sin x and f(x) = arcsin x
• The range of f(x) = sin x is limited so that the graph of f(x) = arcsin x is a function
• Comparison of domain and range of f(x) = cos x and f(x) = arccosx
• The range of f(x) = cos x is limited so that the graph of f(x) = arccos x is a function
• Domain
• f(x) = arcsin x has a domain of –1 ≤ x ≤ 1
• f(x) = arccos x has a domain of –1 ≤ x ≤ 1
• Range
• f(x) = arcsin x has a range of – f(x) ≤ • f(x) = arccos x has a range of 0 ≤ f(x) ≤ Note(s):

• Algebra II graphed inverse functions and wrote the inverse of a function using notation such as f–1(x).
• Precalculus extends the analysis of inverses to include trigonometric functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• A1 – Recognize whether a relation is a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2I

Determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

Determine, Analyze

THE KEY FEATURES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS, INCLUDING STEP FUNCTIONS SUCH AS DOMAIN, RANGE, SYMMETRY, RELATIVE MAXIMUM, RELATIVE MINIMUM, ZEROS, ASYMPTOTES, AND INTERVALS OVER WHICH THE FUNCTION IS INCREASING OR DECREASING

Including, but not limited to:

• Covariation – pattern of related change between two variables in a function
• Periodic patterns
• Trigonometric functions
• Domain and range
• Represented as a set of values
• {0, 1, 2, 3, 4}
• Represented verbally
• All real numbers greater than or equal to zero
• All real numbers less than one
• Represented with inequality notation
• x ≥ 0
• y < 1
• Represented with set notation
• {x|x ∈ ℜ, x ≥ 0}
• {y|y ∈ ℜ, y < 1}
• Represented with interval notation
• [0, ∞)
• (–∞, 1)
• Symmetry
• Reflectional
• Rotational
• Symmetric with respect to the origin (180° rotational symmetry)
• Relative extrema
• Relative maximum
• Relative minimum
• Zeros
• Roots/solutions
• x-intercepts
• Asymptotes
• Vertical asymptotes (x = h)
• Horizontal asymptotes (y = k)
• Slant asymptotes (y = mx + b)
• Intervals where the function is increasing or decreasing
• Represented with inequality notation, –1 <  ≤ 3
• Represented with set notation, {x|x ∈ ℜ, –1 < x ≤ 3}
• Represented with interval notation, (–1, 3]
• Connections among multiple representations of key features
• Graphs
• Tables
• Algebraic
• Verbal

Note(s):

• Algebra II analyzed functions according to key attributes, such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum values over an interval.
• Precalculus extends the analysis of key attributes of functions to include zeros and intervals where the function is increasing or decreasing.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2O Develop and use a sinusoidal function that models a situation in mathematical and real-world problems.

Develop, Use

A SINUSOIDAL FUNCTION THAT MODELS A SITUATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Sinusoidal functions to generate models
• General forms of sinusoidal equations
• Sine: f(x) = a • sin(b(xc)) + d
• Cosine: f(x) = a • cos(b(xc)) + d
• Characteristics of sinusoidal functions
• Amplitude, a
• Period, P = • Horizontal axis (middle axis), y = d
• Maximum value, d + a
• Minimum value, da
• Local maxima (high points)
• If a sinusoidal function has a maximum at x = k, then other maxima occur at x = k + nP (where P is the period and n is an integer).
• Local minima (low points)
• If a sinusoidal function has a minimum at x = k, then other minima occur at x = k + nP (where P is the period and n is an integer).
• Sinusoidal model from data
• Analyzing data
• Table
• Graph
• Verbal description
• Determining characteristics (amplitude, period, etc.)
• Developing the model
• General forms of sinusoidal equations
• Using transformations
• Using attributes of functions
• Sinusoidal regression equations (using technology)

Note(s):

• Algebra I analyzed and investigated quadratic and exponential functions and their applications.
• Algebra II analyzed and investigated logarithmic, exponential, absolute value, rational, square root, cube root, and cubic functions.
• Algebra I and Algebra II analyzed and described the effects of transformations on the parent functions with changes in abc, and d parameters.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2P Determine the values of the trigonometric functions at the special angles and relate them in mathematical and real-world problems.

Determine

THE VALUES OF THE TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES

Including, but not limited to:

• Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle
• Trigonometric functions are called circular functions because they are based on the unit circle (circle with radius of 1 unit and center at the origin) and right triangle relationships within the circle.
• Sine (sin x or sin θ)
• Cosine (cos x or cos θ)
• Tangent (tan x or tan θ)
• Cotangent (cot& x or cot θ)
• Secant (sec x or sec θ)
• Cosecant (csc x or csc θ)
• Special angles
• Degrees
• Multiples of 30°
• Multiples of 60°
• Multiples of 45°
• Multiples of • Multiples of • Multiples of • Special right triangles – right triangles which have angles that measure 30°– 60°– 90° or 45°– 45°– 90°
• In a 45°– 45°– 90° triangle, the proportions of the sides are 1x:1x: x.
• In a 30°– 60°– 90° triangle, the proportions of the sides are 1x: x:2x.
• Special angles as reference angles

Relate

THE VALUES OF TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Mathematical problem situations
• Real-world problem situations

Note(s):

• Geometry determined side lengths of right triangles using the Pythagorean Theorem and the relationships of special right triangles (30°– 60°– 90° and 45°– 45°– 90°).
• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
• Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure.
• Precalculus generalizes the relationships of special right triangles to determine the values of the trigonometric functions of any multiple of 30° or 45°.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4 Number and measure. The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and real-world problems. The student is expected to:
P.4A Determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and real-world problems.

Determine, To Evaluate

THE RELATIONSHIP BETWEEN THE UNIT CIRCLE AND THE DEFINITION OF A PERIODIC FUNCTION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Periodic functions
• Definition of a periodic function
• Verbal: A function is periodic if it repeats in regular intervals (or periods).
• Symbolic: A function f(x) is periodic if f(x + P) = f(x) for some positive value P. If P is the smallest such value for which this is true, then P is the period.
• Unit circle
• Unit circle on the coordinate grid
• Radius: r = 1
• Center: (0, 0)
• Equation: x2 + y2 = 1
• Rotation angles in standard position
• Vertex: (0, 0)
• Initial side (ray): the positive x-axis
• Direction of rotation
• Positive rotation: counter-clockwise
• Negative rotation: clockwise
• Trigonometric functions of an angle in standard position whose terminal side intersects the unit circle at the point (x, y)
• sinθ = y
• cosθ = x
• tanθ = • cotθ = • secθ = • cscθ = • Relationship between degree measures in relation to the points on the unit circle
• Periodic functions on the unit circle
• Rotation angles for positive and negative rotation
• If θ is a standard position rotation angle that intersects the unit circle at the point (x, y), then x(θ) and y(θ) are both periodic functions with a period of 360°.
• The relationship between the unit circle and period functions is every 360° the function values repeat themselves after equal intervals.
• Graphs of trigonometric functions and angle measures to determine periodic relationships
• Rotation angles
• If θ is a standard position rotation angle that intersects the unit circle at the point (x, y), then x(θ) and y(θ) are both periodic functions with a period of 360°.
• The relationship between the unit circle and periodic functions is that every 360° the function values repeat themselves on equal intervals.
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra I and Algebra II analyzed the symmetry of various graphs, including quadratic functions.
• Algebra II analyzed the symmetry of various graphs, including quadratic, absolute value, and rational functions.
• Precalculus uses symmetry to analyze features of periodic functions.
• Geometry determined side lengths in right triangles using the Pythagorean Theorem and the relationships in special right triangles (30° – 60° – 90° and 45° – 45° – 90°).
• Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
• Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure or degree measure.
• Precalculus generalizes the relationships in right triangles to determine the values of the trigonometric functions for any angle measure.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• III. Geometric Reasoning
• A3 – Recognize and apply right triangle relationships including basic trigonometry.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4B Describe the relationship between degree and radian measure on the unit circle.

Describe

THE RELATIONSHIP BETWEEN DEGREE AND RADIAN MEASURE ON THE UNIT CIRCLE

Including, but not limited to:

• Defining the unit circle
• Unit circle on the coordinate grid
• Radius: r = 1
• Center: (0, 0)
• Rotation angles in standard position
• Vertex: (0, 0)
• Initial side (ray): the positive x-axis
• Direction of rotation
• Positive rotation: counter-clockwise
• Negative rotation: clockwise
• Labeling the unit circle
• Degree measure – measured as a central angle with respect to the unit circle
• Radian measure – measured as the directed length of an intercepted arc on the unit circle
• Coordinates of the unit circle
• Unit Circle
• Computing common unit conversions
• Relationship between degrees and radians
• 180° = π
• Degrees into radians: multiply by • Radians into degrees: multiply by Note(s):

• Students began measuring angles in Grade 4 and continued solving problems with angle measures throughout middle school.
• Grade 6 converted units within a measurement system.
• Geometry used the Cartesian equations for circles and solved problems with central angles and arc lengths.
• Geometry described radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.
• Precalculus combines these skills together to describe the relationship between degree and radian measure.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• IV. Measurement Reasoning
• A1 – Select or use the appropriate type of unit for the attribute being measured.
• B1 – Convert from one measurement system to another.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4C Represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position.

Represent

ANGLES IN RADIANS OR DEGREES BASED ON THE CONCEPT OF ROTATION

Including, but not limited to:

• Rotation angles in standard position
• Vertex: (0, 0)
• Initial side (ray): the positive x-axis
• Direction of rotation
• Positive rotation: counter-clockwise
• Negative rotation: clockwise
• Describing rotation of angles in radians and degrees
• x- and y-axes (quadrants on the coordinate grid)
• Initial and terminal sides (rays)
• Arcs with direction arrows to indicate rotation

Find

THE MEASURE OF REFERENCE ANGLES AND ANGLES IN STANDARD POSITION

Including, but not limited to:

• Reference angle – If A is an angle in standard position, the reference angle to angle A is an acute angle created by the terminal side of angle A and the x-axis.
• Quadrantal angle – an angle whose terminal side lies on the x-axis or y-axis
• Degrees in standard position: 0°, 90°, 180°, 270°, 360°
• Radians in standard position: 0, , π, , 2π
• Interval for angle A in standard position, [0°, 360°] or [0, 2π]
• If angle A is in 1st quadrant, (0°, 90°) or , then the reference angle is A.
• If angle A is in 2nd quadrant, (90°, 180°) or , then the reference angle is 180° – A or πA
• If angle A is in 3rd quadrant, (180°, 270°) or , then the reference angle is A – 180° or Aπ
• If angle A is in 4th quadrant, (270°, 360°) or , then the reference angle is 360° – A or 2πA
• Given an angle not in standard position, use addition or subtraction of multiples of 360° or 2π in order to determine an angle with the same terminal side in standard position, [0°, 360°] or [0, 2π].
• Mathematical problem situations

Note(s):

• Students began measuring angles in Grade 4 and continued solving problems with angle measures throughout middle school.
• Grade 7 solved equations based on geometry concepts, including angle relationships.
• Precalculus represents rotation and reference angles on a coordinate grid and describes the relationship between the two verbally and symbolically.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space features.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.4D Represent angles in radians or degrees based on the concept of rotation in mathematical and real-world problems, including linear and angular velocity.

Represent

ANGLES IN RADIANS OR DEGREES BASED ON THE CONCEPT OF ROTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS, INCLUDING LINEAR AND ANGULAR VELOCITY

Including, but not limited to:

• Describing rotation in radians and degrees
• x- and y-axes (quadrants on the coordinate grid)
• Initial and terminal sides (rays)
• Arcs with direction arrows to indicate rotation
• Describing rotation in terms of revolutions
• 1 revolution = 360° = 2π
• Describing angular velocity
• Angular velocity – rate of change of angular displacement, measured in radians per unit of time
• Formula: angular velocity, ω = • ω = angular velocity
• θ = angle measure in radians
• t = unit of time
• Writing and simplifying ratios of rotation and time
• Converting between and among various measures of angular velocity
• Computing linear velocity
• Linear velocity – the speed, measured in distance per unit of time, with which an object moves along a circular or rotational path
• Formula: linear velocity, • v = linear velocity
• r = radius
• = angular velocity (ω), where θ is angle measure in radians and t is unit of time

Note(s):

• Grade 6 converted units within a measurement system.
• Grade 7 solved problems with circumference of a circle.
• Geometry described radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.
• Precalculus combines these skills together to determine linear and angular velocities in a variety of units of measure.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• IV. Measurement Reasoning
• A1 – Select or use the appropriate type of unit for the attribute being measured.
• B1 – Convert from one measurement system to another.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 