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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 01: Graphs, Attributes, and Applications of Functions SUGGESTED DURATION : 14 days

Unit Overview

Introduction
This unit bundles student expectations that address investigating families of functions, key attributes, and characteristics including end behavior and left-side and right-side behavior around discontinuities. Concepts are incorporated into both mathematical and real-world problem situations. 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 Algebra I Units 02 – 04 and 08 – 09, students studied the key attributes and characteristics of linear, quadratic, and exponential functions. In Algebra II Units 01 – 02 and 05 – 11, students extended the study of the families of functions to include the key attributes and characteristics of square root, exponential, logarithmic, rational, cubic, cube root, and absolute value functions.

During this Unit
Students graph and analyze families of functions that will be studied in Precalculus, including exponential, logarithmic, rational, polynomial, power, and piecewise defined functions, including step functions. Students determine and analyze key attributes of these functions, including domain and range (interval, inequality, set notation), symmetry, relative maximum, relative minimum, y-intercept, zeros, asymptotes, and intervals over which the function is increasing or decreasing. Students explore even and odd functions and describe the symmetry of graphs of even and odd functions, using graphs, tables, and algebraic properties. Students analyze end behavior of functions in mathematical and real-world problems using tables and graphs and describe this end behavior using infinity notation. Students determine various types of discontinuities, including jump, infinite, and removable discontinuities, in the interval, using graphical and tabular methods. Students explore the limitations of the graphing calculator as it relates to the behavior of the function around the discontinuities. Students describe the left-sided and right-sided behavior of the graph of the function around the discontinuities using verbal and symbolic notation. Students analyze situations modeled by exponential, logarithmic, rational, polynomial, and power functions to solve real-world problems.

After this Unit
In Units 02 – 05 and 08, students will study each family of functions in greater depth and complexity. 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 D1, D2; VI. Statistical Reasoning B2, C3; VII. Functions; 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 Texas, the importance of these skills is emphasized in the TxCCRS, which calls 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). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (NRC, 2005, p. 353). Lastly, students need to be involved in metacognitive engagement in mathematics as they problem solve and reflect on their solutions and strategies; this is particularly important as students transition into more abstract mathematics, where fewer “clues” may exist warning students of a mathematical misstep (NRC, 2005). This particular introductory unit focuses on the language and vocabulary of functions. The National Council of Teachers of Mathematics (2000) supports such a focus, calling for students to use the language of mathematics to express mathematical ideas more precisely. Research also supports an emphasis on language and vocabulary. In Teaching Mathematics in Context (2004), Murray summarizes several research studies that conclude mathematical vocabulary, studied in context, has a profound effect on performance and contributes significantly to better mathematical word problem-solving ability.

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.
Murray, Miki. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann Press.
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

 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 strengths and 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 relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by …
• exponential functions?
• logarithmic functions?
• rational functions?
• polynomial functions?
• power functions?
• piecewise functions?
• step functions?
• What graphs, key attributes, and characteristics are unique to different families of functions?
• What patterns of covariation are associated with different families of functions?
• How can the key attributes of functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of functions?
• What relationships exist between the mathematical and real-world meanings of the key attributes of function models?
• How can key attributes be used to make predictions and critical judgments about the problem situation?
• 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
• End behavior
• Discontinuities
• Even and odd
• Functions
• Exponential
• Logarithmic
• Rational
• Polynomial
• Power
• Piecewise
• Step
• 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 strengths and 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 relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by …
• exponential functions?
• logarithmic functions?
• rational functions?
• polynomial functions?
• power functions?
• piecewise functions?
• step functions?
• What graphs, key attributes, and characteristics are unique to different families of functions?
• What patterns of covariation are associated with different families of functions?
• How can the key attributes of functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of functions?
• What relationships exist between the mathematical and real-world meanings of the key attributes of function models?
• How can key attributes be used to make predictions and critical judgments about the problem situation?
• 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
• End behavior
• Discontinuities
• Even and odd
• Functions
• Exponential
• Logarithmic
• Rational
• Polynomial
• Power
• Piecewise
• Step
• 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 because the end behavior points down, the graph is decreasing. Consider the cases where, as x → –∞, the function also approaches –∞. When describing the left end behavior of a function, we typically say that the graph “points down on the left”. However, students should be advised that the function is NOT decreasing in this interval (it is increasing.). The misconception arises because the symbols ask students to move from right to left (when we normally go left to right).
• Some students may think that they name portions of the graph with intervals of y-values (instead of x-values). e.g., A student might say, “The function y = x2 is increasing when y = 4”. While the function is increasing at the point (2, 4), it is decreasing at the point (–2, 4). So, it is better to say the function is increasing at x = 2.
• Some students may think that the symbolic notation for an open interval (a, b) represents the coordinates for a point. Remind them that the difference between the two depends on the context. For example, the domain of a function is an interval, but a maximum could be given as a point.
• Students may have misconceptions concerning functions in function notation even though they understand the concepts from a graphical standpoint. e.g., A student may assume something like, “Since 7 > 3, then it must be true that f(7) > f(3)”. However, this is not true of all functions, only those that are increasing. e.g., A student may assume something like, “If f(3) = 8, then it must be true that f(–3) must equal –8”. However, this is not true of all functions, only those that are odd.
• Some students may think that the graph of a function is only what appears on the calculator screen, when actually the calculator obscures the details or hidden behavior of a function when the window is not set over an appropriate interval.
• Students may have misconceptions and incorrect assumptions or generalizations about a graph’s behavior when constructing graphs through point-plotting techniques (particularly when too few points are used). e.g., A student might correctly use the points (1, 1) and (2, 0.5) and (4, 0.25) to graph the function f(x) = . However, when sketching the function through these points, the student may extend the graph to the left of the y-axis or below the x-axis, unaware of the asymptotic behavior of the function.
• Some students may misinterpret the graph of a function with discontinuities based solely on its table. Since tables show only discrete values of x and y, the tables often do not fully describe the behavior of a function. • Some students may overlook removable discontinuities based on the graph produced by a graphing calculator. Some calculators “graph over” removable discontinuities. Unit Vocabulary

• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Covariation – pattern of related change between two variables in a function
• Even functions – functions of the form f(x) with the property that f(–x) = f(x)
• Infinite discontinuities – values of x where vertical asymptotes occur, function has an infinite discontinuity at x = c, as xc, f(x) → ±∞
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or when x < c), but approaches a different y-value on the right side of x = c (or when x > c).
• Odd functions – functions of the form f(x) with the property that f(–x) = –f(x)
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).

Related Vocabulary:

 Ceiling function Decreasing Discontinuities Domain End behavior Exponential decay Exponential function Exponential growth Extrema Floor function Greatest integer function Horizontal asymptote Increasing Inequality Infinity notation Interval notation Left-Sided behavior Logarithmic function Maximum Minimum Ordered pairs Piecewise defined function Polynomial function Power function Rational function Range Reflectional symmetry Relative extrema Relative maximum Relative minimum Right-Sided behavior Roots Rotational symmetry Step function Symmetry Vertical asymptote x-intercepts 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.2D Describe symmetry of graphs of even and odd functions.

Describe

SYMMETRY OF GRAPHS OF EVEN AND ODD FUNCTIONS

Including, but not limited to:

• Symmetry of graphs
• Even functions – functions of the form f(x) with the property that f(–x) = f(x)
• Reflectional symmetry
• Graphs of even functions are symmetric with respect to the y-axis.
• Ordered pairs
• If (a, b) is an ordered pair on the graph of an even function, then (–a, b) is also on the graph.
• Algebraic properties
• If f(x) is an even function, then f(–x) = f(x).
• Odd functions – functions of the form f(x) with the property that f(–x) = –f(x)
• Rotational symmetry
• Graphs of odd functions are symmetric with respect to the origin (or have 180° rotational symmetry with respect to the origin).
• Ordered pairs
• If (a, b) is an ordered pair on the graph of an odd function, then (–a, –b) is also on the graph.
• Algebraic properties
• If f(x) is an odd function, then f(–x) = –f(x).

Note(s):

• Algebra II identified symmetry in parabolas.
• Precalculus uses symmetry to describe general functions as being even or odd.
• 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.2F

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

Graph

EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, POWER, AND PIECEWISE DEFINED FUNCTIONS, INCLUDING STEP FUNCTIONS

Including, but not limited to:

• Graphs of the parent functions
• Graphs of piecewise defined functions
• Functions specifically defined over given domains
• Functions that exhibit piecewise behavior
• Step 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 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.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 EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, POWER, 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

Including, but not limited to:

• Covariation – pattern of related change between two variables in a function
• Multiplicative patterns
• Exponential functions
• Logarithmic functions
• Rational functions
• Patterns in the nth differences
• Polynomial functions
• Power 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.2J Analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems.

Analyze, Describe

END BEHAVIOR OF FUNCTIONS, INCLUDING EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, AND POWER FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Describing end behavior with infinity notation
• Right end behavior
• As x → ∞ (or as x approaches infinity) the function becomes infinitely large; f(x) → ∞.
• As x → ∞ (or as x approaches infinity) the function becomes infinitely small; f(x) → –∞.
• As x → ∞ (or as x approaches infinity) the function approaches a constant value, c; f(x) → c.
• Left end behavior
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely large; f(x) → ∞.
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely small; f(x) → –∞.
• As x → –∞ (or as x approaches negative infinity) the function approaches a constant value, c; f(x) → c.
• Determining end behavior from multiple representations
• Tables: evaluating the function for extreme negative (left end) and positive (right end) values of x
• Graphs: analyzing behavior on the left and right sides of the graph
• Determining end behavior from analysis of the function type and the constants used
• Exponential: f(x) = abx
• Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → 0.
• Ex: When a > 0 and 0 < b < 1, as x → ∞ (on the right), f(x) → 0, and as x → –∞ (on the left), f(x) → ∞.
• Logarithmic: f(x) = alogb(x)
• Ex: When a > 0 and b > 1, as x → ∞ (on the right), f(x) → ∞.
• Ex: When a > 0 and b > 1, as x → 0 (on the left), f(x) → –∞.
• Rational: f(x) = , where p(x) and q(x) are polynomials in terms of x, q(x) ≠ 0
• Ex: If the degree of p(x) is greater than the degree of q(x), as x → –∞, f(x) → ±∞ on the left, and as x → ∞, f(x) → ± on the right.
• Ex: If the degree of p(x) is less than the degree of q(x), as x → –∞, f(x) → 0 on the left, and as x → ∞, f(x) → 0 on the right.
• Ex: If the degree of p(x) and q(x) are the same, as x → –∞, f(x) → k on the left, and as x → ∞, f(x) → k on the right, where k is a constant determined by the leading coefficients of p(x) and q(x).
• Polynomial: f(x) = anxn + an–1xn–1 + ... + a2x2 + a1x + a0, where n is a positive integer
• The leading coefficient (an) determines the right end behavior.
• Ex: If an > 0, as x → ∞ (on the right), f(x) → ∞.
• Ex: If an < 0, as x → ∞ (on the right), f(x) → –∞.
• The degree of the polynomial (n) determines whether the left and right end behaviors are the same or different.
• Ex: When an > 0, if n is even, then as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → ∞.
• Ex: When an > 0, if n is odd, then as x → ∞ (on the right), f(x) → ∞, and as x → –∞ (on the left), f(x) → –∞.
• Power: f(x) = axn, where n is a real number
• Ex: If a > 0 and n > 0, as x → ∞ (on the right), f(x) → ∞.
• Ex: If a > 0 and n < 0, as x → ∞ (on the right), f(x) → 0.
• Interpreting end behavior in real-world situations

Note(s):

• Algebra II analyzed the domains and ranges of quadratic, square root, exponential, logarithmic, and rational functions.
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends analysis of domain, range, and asymptotic restrictions to determine the end behavior of functions and describes this behavior using infinity notation.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• 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.2L Determine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities.

Determine

VARIOUS TYPES OF DISCONTINUITIES IN THE INTERVAL (–∞, ∞) AS THEY RELATE TO FUNCTIONS

Including, but not limited to:

• Determining whether a discontinuity is a removable discontinuity or a non-removable discontinuity
• Behavior of function around discontinuities
• Non-removable discontinuities
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or when x < c), but approaches a different y-value on the right side of x = c (or when x > c).
• Graphical
• Tabular
• Algebraic
• Piecewise defined functions
• Evaluate both parts of the function to the left and right at breaks in the domain, then check to see if the values agree.
• General functions
• Jump discontinuities can occur at values of x where the function is not defined due to limits on the domain.
• Infinite discontinuities – values of x where vertical asymptotes occur, function has an infinite discontinuity at x = c, as xc, f(x) → ±∞
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, vertical asymptotes (or infinite discontinuities) occur at values of x where q(x) = 0 but p(x) ≠ 0.
• Trigonometric functions
• For trigonometric functions, vertical asymptotes can occur at values of x where the function is undefined.
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.

Explore

THE LIMITATIONS OF THE GRAPHING CALCULATOR AS IT RELATES TO THE BEHAVIOR OF THE FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Tables
• Hidden behavior of a function
• Because tables show only discrete values of x and y, the tables often do not fully describe the behavior of a function.
• Values of x that get skipped
• Because tables default to integer values of x and y, the tables often skip important features of a function that occur at the rational (decimal) values in between.
• Values of x where a function is undefined
• While tables can locate values of x where a function is undefined, tables do not identify the type of discontinuity that has occurred.
• Graphing functions with graphing calculators
• Evaluating functions at specific x-values
• Setting a window
• Screen width = (maximum x-value) – (minimum x-value)
• Resolution = number of pixels in the screen width
• Δx = (screen width) ÷ (resolution)
• Behavior of calculator graphs around discontinuities
• Jump discontinuities
• Infinite discontinuities
• Removable discontinuities

Note(s):

• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the idea of domain restrictions to include various types of discontinuities: removable, infinite, and jump.
• 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.2M Describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities.

Describe

THE LEFT-SIDED BEHAVIOR AND THE RIGHT-SIDED BEHAVIOR OF THE GRAPH OF A FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Verbal and symbolic
• Left-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the left
• Symbols: xc
• Right-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the right
• Symbols: xc+
• Function behavior near a discontinuity at x = c
• As x approaches c (from the left or right), the function values can approach a constant, k.
• Words: As x approaches c, the function approaches k.
• Symbols: As xc, f(x) → k (or yk)
• As x approaches c (from the left or right), the function values can continue to increase without limit.
• Words: As x approaches c, the function approaches infinity.
• Symbols: As xc, f(x) → ∞ (or y → ∞)
• As x approaches c (from the left or right), the function values can continue to decrease without limit.
• Words: As x approaches c, the function approaches negative infinity.
• Symbols: As xc, f(x) → –∞ (or y → –∞)
• Graphical
• Left-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x < c from left to right
• Right-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x > c from right to left
• Tabular
• Left-sided behavior near a discontinuity at x = c
• Consider values in the table where x < c, such as c – 0.1, c – 0.01, c – 0.001, etc.
• Right-sided behavior near a discontinuity at x = c
• Consider values in the table where x > c, such as c + 0.1, c + 0.01, c + 0.001, etc.
• Use left- and right-sided behavior of a function to determine whether a discontinuity is a removable discontinuity or a non-removable discontinuity

Note(s):

• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the concept of domain restrictions around asymptotes to include other types of discontinuities and analyzes the left-sided and right-sided behavior of functions near these discontinuities.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• 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.2N Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.

Analyze, To Solve

SITUATIONS MODELED BY FUNCTIONS, INCLUDING EXPONENTIAL, LOGARITHMIC, RATIONAL, POLYNOMIAL, AND POWER FUNCTIONS

Including, but not limited to:

• Models that represent problem situations
• Understanding the meaning of the variables (both independent and dependent)
• Evaluating the function when independent quantities (x-values) are given
• Solving equations when dependent quantities (y-values) are given
• Appropriateness of given models for a situation
• Analyzing the attributes of a problem situation
• Determining which type of function models the situation
• Determining a function to model the situation
• Using transformations
• Using attributes of functions
• Using technology
• Describing the reasonable domain and range values
• Comparing the behavior of the function and the real-world relationship
• Exponential functions
• Exponential growth (e.g., accrued interest, population growth, etc.)
• Exponential decay (e.g., half-life, cooling rate, etc.)
• Logarithmic functions (e.g., pH, sound (decibel measures), earthquakes (Richter scale), etc.)
• Rational functions (e.g., averages, temperature/pressure/volume relationships (Boyle’s Law), etc.)
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Polynomial functions (e.g., area, volume, motion, etc.)
• Power functions

Note(s): 