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

#### Unit Overview

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 2  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(x – c), 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 Precalculus 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.

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 .

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
• How do different systems of measure relate to one another?

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Inverses of functions create new functions.

• What relationships and characteristics exist between a function and its inverse?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data, and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?

Transformations of a parent function create a new function within that family of functions.

• Why are transformations of parent functions necessary?
• How do transformations affect a function?
• How can transformations be interpreted from various representations?
• Why does a transformation of a function create a new function?
• How do the attributes of an original function compare to the attributes of a transformed function?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Multiple Representations
• Patterns/Rules
• Ratios
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Geometric Reasoning

• Geometric Attributes/Properties
• Geometric Relationships
• Trigonometric Relationships

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Geometric tools such as radian and degree measures are used to represent angles based on the concept of rotation in mathematical and real-world problems.

• How are degree and radian measures related?
• What relationships exist between degree and radian measures on the unit circle?
• How can angles be represented in degrees using the concept of rotation?
• How can angles be represented in radians using the concept of rotation?
• What role do reference angles play in solving problems with rotational angles?
• How are the measures of reference angles and angles in standard position determined?
• What problem situations can be modeled and solved using rotational angles?
• How can rotational rates of change be represented and used to solve problem situations?
• How do special right triangles relate to the unit circle?
• What methods can be used to determine the values of the trigonometric functions at the special angles?
• How can the values of the trigonometric functions at the special angles be applied in mathematical and real-world problems?

 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.

Algebraic Reasoning

• Multiple Representations
• Patterns/Rules
• Ratios
• Solve

Functions

• Attributes of Functions
• Inverses of Functions
• Non-Linear Functions

Geometric Reasoning

• Geometric Attributes/Properties
• Geometric Relationships
• Transformations
• Trigonometric Relationships

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Trigonometric functions have key attributes, including domain, range, symmetry, relative extrema, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, discontinuities, and left- and right-sided behavior of the graph around discontinuities.

• What are the key attributes of trigonometric functions?
• How can the key attributes of a trigonometric function be determined from multiple representations of the function?
• How can knowledge of the key attributes of a trigonometric function be used to sketch the graph of the function?
• How can the domain and range of a trigonometric function be determined and described?
• How can the relative extrema of a trigonometric function be determined and described?
• How can the zeros of a trigonometric function be determined and described?
• Which trigonometric functions display asymptotic behavior, and how can the asymptotic behavior be described?
• How can the intervals of increasing and decreasing of a trigonometric function be determined and described?
• How can the end behavior of a trigonometric function be determined and described using infinity notation?
• Which trigonometric functions would be classified as even or odd functions, and how can the classification of even or odd be determined graphically and symbolically?
• How can the discontinuities of a trigonometric function and the left- and right-side behavior of the function near these discontinuities be determined and described?
• Why are trigonometric functions referred to as periodic functions?
• How do the trigonometric functions relate to the unit circle?
• What methods can be used to determine the values of the trigonometric functions at the special angles?
• How can the values of the trigonometric functions at the special angles be applied in mathematical and real-world problems?

Inverse trigonometric functions have key attributes, including domain, range, symmetry, relative extrema, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, discontinuities, and left- and right-sided behavior of the graph around discontinuities.

• What are the key attributes of inverse trigonometric functions?
• How can the key attributes of an inverse trigonometric function be determined from multiple representations of the function?
• How can knowledge of the key attributes of an inverse trigonometric function be used to sketch the graph of the function?
• How can the domain and range of an inverse trigonometric function be determined and described?
• How can the relative extrema of an inverse trigonometric function be determined and described?
• How can the zeros of an inverse trigonometric function be determined and described?
• How can the asymptotes of an inverse trigonometric function be determined and described?
• How can the intervals of increasing and decreasing of an inverse trigonometric function be determined and described?
• How can the end behavior of an inverse trigonometric function be determined and described using infinity notation?
• Which inverse trigonometric functions would be classified as even or odd functions, and how can the classification of even or odd be determined graphically and symbolically?
• How can the discontinuities of an inverse trigonometric function and the left- and right-side behavior of the function near these discontinuities be determined and described?
• Why do limitations exist on the domain and range of inverse trigonometric functions, and how can the limitations on the domain and range be described?

The inverse of a function can be determined from multiple representations.

• How can the inverse of a function be determined from the graph of the function?
• How can the inverse of a function be determined from a table of coordinate points of the function?
• How can the inverse of a function be determined from the equation of the function?
• How are a function and its inverse distinguished symbolically?
• How do the attributes of inverse functions compare to the attributes of original functions?

The domain and range of the inverse of a function may need to be restricted in order for the inverse to also be a function.

• When must the domain of an inverse function be restricted?
• How does the relationship between a function and its inverse, including the restriction(s) on the domain, affect the restriction(s) on its range?

Sinusoidal functions can be used to model real-world problem situations by analyzing collected data, key attributes, and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display sinusoidal function models?
• What key attributes identify a sinusoidal function model?
• How do the domain and range of the function compare to the domain and range of the problem situation?
• What are the connections between the key attributes of a sinusoidal function model and the real-world problem situation?
• How can sinusoidal function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

Transformations of sinusoidal functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of sinusoidal functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of sinusoidal functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of sinusoidal functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of sinusoidal functions when f(x) is replaced by f(x) + d, for specific values of d?

#### 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 (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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 g-1(x) = .
• If the restricted domain of g(x) is x ≤ 0, then the inverse function is g-1(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(x – c), f(bx) FOR SPECIFIC VALUES OF abc, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Sine and cosine functions: f(x) = sinxf(x) = cosx
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters abc, 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(x – c)
• 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  and  f(x) = arcsin x
• The range of  f(x) = sin  is limited so that the graph of f(x) = arcsin  is a function
• Comparison of domain and range of  f(x) = cos  and  f(x) = arccos x
• The range of  f(x) = cos  is limited so that the graph of  f(x) = arccos  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, 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 ≥ 0}
• {y| 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  , –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(x – c)) + d
• Cosine: f(x) = a • cos(b(x – c)) + d
• Characteristics of sinusoidal functions
• Amplitude, a
• Period, P =
• Horizontal axis (middle axis), y = d
• Maximum value, d + a
• Minimum value, d – a
• 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 )
• Cosine (cos x or cos )
• Tangent (tan x or tan )
• Cotangent (cot x or cot )
• Secant (sec x or sec )
• Cosecant (csc x or csc )
• 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
• 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 (xy)
• sin = y
• cos = x
• tan =
• cot =
• sec =
• csc =
• 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  is a standard position rotation angle that intersects the unit circle at the point (xy), then x() and y() 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  is a standard position rotation angle that intersects the unit circle at the point (xy), then x() and y() 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
• 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° =
• 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, , , 2
• Interval for angle A in standard position, [0°, 360°] or [0, 2]
• 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  – A
• If angle A is in 3rd quadrant, (180°, 270°) or , then the reference angle is A – 180° or A –
• If angle A is in 4th quadrant, (270°, 360°) or , then the reference angle is 360° – A or 2 – A
• Given an angle not in standard position, use addition or subtraction of multiples of 360° or 2 in order to determine an angle with the same terminal side in standard position, [0°, 360°] or [0, 2].
• 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° = 2(radians)
• Describing angular velocity
• Angular velocity – rate of change of angular displacement, measured in radians per unit of time
• Formula: angular velocity,  =
•  = angular velocity
•  = 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 = r •   OR v = r •
• = linear velocity
•  = angular velocity (), where  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