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:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

P.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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 nonexamples, patterns, etc.
 Current knowledge to new learning

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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 realworld problems. The student is expected to:


P.2F 
Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Note(s):
 Grade Level(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, piecewisedefined, and others.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
Graph
RATIONAL 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 a, b, c, and d)
 Using graphing technology
Note(s):
 Grade Level(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, piecewisedefined, and others.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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 realworld problems.

Graph
FUNCTIONS, INCLUDING RATIONAL 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 REALWORLD PROBLEMS
Including, but not limited to:
 General form of parent function
 Rational functions: f(x) =
 Representations with and without technology
 Graphs
 Verbal descriptions
 Algebraic generalizations (including equation and function notation)
 Changes in parameters a, b, c, and d on graphs
 Effects of a on f(x) in af(x)
 a ≠ 0
 a > 1, the graph stretches vertically
 0 < a < 1, the graph compresses vertically
 Opposite of a reflects vertically over the horizontal axis (xaxis)
 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 yaxis
 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
 Realworld problem situations
Note(s):
 Grade Level(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, piecewisedefined, and others.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VI.B.2. Algebraically construct and analyze new functions.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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 RATIONAL 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
 Domain and range
 Represented as a set of values
 Represented verbally
 All real numbers greater than or equal to zero
 All real numbers less than one
 Represented with inequality notation
 Represented with set notation
 {xx ∈ ℜ, x ≥ 0}
 {yy ∈ ℜ, y < 1}
 Represented with interval notation
 Symmetry
 Reflectional
 Rotational
 Symmetric with respect to the origin (180° rotational symmetry)
 Relative extrema
 Relative maximum
 Relative minimum
 Zeros
 Roots/solutions
 xintercepts
 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 < x ≤ 3
 Represented with set notation, {xx ∈ ℜ, –1 < x ≤ 3}
 Represented with interval notation, (–1, 3]
 Connections among multiple representations of key features
 Graphs
 Tables
 Algebraic
 Verbal
Note(s):
 Grade Level(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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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 realworld problems.

Analyze, Describe
END BEHAVIOR OF FUNCTIONS, INCLUDING RATIONAL FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REALWORLD 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
 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).
 Interpreting end behavior in realworld situations
Note(s):
 Grade Level(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:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

P.2K 
Analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes.

Analyze
CHARACTERISTICS OF RATIONAL FUNCTIONS
Including, but not limited to:
 Analyzing algebraically rational functions of the form f(x) =, where p(x) and q(x) are polynomials in x, q(x) ≠ 0
 Degrees (highest power of x) of p(x) and q(x)
 Leading coefficients of p(x) and q(x)
 Factors of p(x) and q(x)
 Zeros of p(x) and q(x) (or where p(x) = 0 and q(x) = 0)
 Quotient of p(x) and q(x) (or p(x) ÷ q(x), obtained through long division of polynomials)
 Determining discontinuities algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in x, q(x) ≠ 0
 Behavior of function around discontinuities
 Determining whether a discontinuity is a removable discontinuity or a nonremovable discontinuity
 Nonremovable discontinuities
 Vertical asymptotes
 Vertical asymptotes occur at values of x where q(x) = 0, but p(x) ≠ 0.
 Removable discontinuities
 Removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.
 Determining end behavior algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, q(x) ≠ 0
 Horizontal asymptotes
 If the degree of p(x) is less than the degree of q(x), f(x) has a horizontal asymptote at y = 0.
 If the degrees of p(x) and q(x) are the same, f(x) has a horizontal asymptote at y = k where k is a constant determined by the leading coefficients of p(x) and q(x).
 Oblique (or slant) asymptotes
 If the degree of p(x) is one more than the degree of q(x), then f(x) has an oblique asymptote of the form y = mx + b determined by the quotient of p(x) and q(x) through long division.
Analyze
THE BEHAVIOR OF THE RATIONAL FUNCTION AROUND THE ASYMPTOTES, INCLUDING HORIZONTAL, VERTICAL, AND OBLIQUE ASYMPTOTES
Including, but not limited to:
 Behavior around horizontal asymptotes
 Verbal and symbolic
 If a rational function f(x) has a horizontal asymptote at y = k, then as the xvalues increase or decrease without bound (or as x → ±∞), the yvalues of the function approach k (or f(x) → k).
 Behavior around vertical asymptotes
 Verbal and symbolic
 If a rational function f(x) has a vertical asymptote at x = h, then as xvalues approach h (or as x → h), the yvalues of the function either increase or decrease without bound (or f(x) → ±∞).
 Tabular
 Graphical
 Behavior around oblique (or slant) asymptotes
 Verbal and symbolic
 If a rational function f(x) has an oblique (or slant) linear asymptote at y = mx + b, then as xvalues increase or decrease without bound (or as x → ±∞), the yvalues of the function approach the line y = mx + b (or f(x) → mx + b).
 Tabular
 Graphical
Note(s):
 Grade Level(s):
 Algebra II determined any asymptotic restrictions on the domain of a rational function.
 Algebra II determined the quotient of polynomials using algebraic methods.
 Precalculus analyzes the behavior of rational functions and describes this behavior around asymptotes.
 Precalculus uses the quotient of polynomials to determine oblique (or other) asymptotes of rational functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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 nonremovable discontinuity
 Behavior of function around discontinuities
 Nonremovable 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 yvalue on the left of x = c (or when x < c), but approaches a different yvalue 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 x → c, 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 yvalue 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 xvalues
 Setting a window
 Screen width = (maximum xvalue) – (minimum xvalue)
 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):
 Grade Level(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:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

P.2M 
Describe the leftsided behavior and the rightsided behavior of the graph of a function around discontinuities.

Describe
THE LEFTSIDED BEHAVIOR AND THE RIGHTSIDED BEHAVIOR OF THE GRAPH OF A FUNCTION AROUND DISCONTINUITIES
Including, but not limited to:
 Verbal and symbolic
 Leftsided behavior near a discontinuity at x = c
 Words: As x approaches c from the left
 Symbols: x → c^{–}
 Rightsided behavior near a discontinuity at x = c
 Words: As x approaches c from the right
 Symbols: x → c^{+}
 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 x → c, f(x) → k (or y → k)
 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 x → c, 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 x → c, f(x) → –∞ (or y → –∞)
 Graphical
 Leftsided behavior near a discontinuity at x = c
 Move along the graph on the interval x < c from left to right
 Rightsided behavior near a discontinuity at x = c
 Move along the graph on the interval x > c from right to left
 Tabular
 Leftsided 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.
 Rightsided 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 rightsided behavior of a function to determine whether a discontinuity is a removable discontinuity or a nonremovable discontinuity
Note(s):
 Grade Level(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 leftsided and rightsided 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:
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

P.2N 
Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve realworld problems.

Analyze, To Solve
SITUATIONS MODELED BY FUNCTIONS, INCLUDING RATIONAL 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 (xvalues) are given
 Solving equations when dependent quantities (yvalues) 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 realworld relationship
 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
Note(s):
 Grade Level(s):
 Algebra II analyzed situations involving exponential, logarithmic, and rational functions.
 Precalculus extends function analysis to include polynomial and power functions and expects students to solve realworld problems and interpret solutions to those problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

P.5 
Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:


P.5L 
Solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and realworld problems.

Solve
RATIONAL INEQUALITIES WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Relating solutions of rational inequalities to the solutions of the related rational equations
 Solving related rational equations using algebraic methods
 Factoring
 Quadratic formula
 Solving related rational equations with technology
 Relating solutions of rational inequalities to the discontinuities of the related rational functions
 Identifying types of discontinuities
 Vertical asymptotes
 Removable discontinuities
 Locating the discontinuities of rational functions using algebraic methods
 Finding values where the denominator of a rational expression is equal to zero
 Factoring
 Quadratic formula
 Checking to see if these values are also zeros of the numerator of the rational expression
 Locating discontinuities of rational functions with technology
 Testing the intervals between the solutions and points of discontinuity
 Evaluating the expression to determine whether values satisfy the inequality
 Analyzing graphs to determine whether values satisfy the inequality
 Using tables to determine whether values satisfy the inequality
Write
THE SOLUTION SET OF THE RATIONAL INEQUALITY IN INTERVAL NOTATION IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Using brackets for closed intervals
 Using parentheses for open intervals
 Using parentheses and the infinity symbol for boundless intervals
 Using the symbol for set union to describe solution sets with more than one interval
Note(s):
 Grade Level(s):
 Algebra II determined the sum, difference, product, and quotient of rational expressions.
 Algebra II formulated and solved rational equations with real solutions.
 Precalculus extends these skills to solve rational inequalities.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
