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- 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)
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- 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)
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P.1 |
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
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P.1A |
Apply mathematics to problems arising in everyday life, society, and the workplace.
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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:
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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.
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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
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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.
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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
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P.1D |
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
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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
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P.1E |
Create and use representations to organize, record, and communicate mathematical ideas.
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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
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P.1F |
Analyze mathematical relationships to connect and communicate mathematical ideas.
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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:
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P.1G |
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
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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
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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:
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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.
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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):
- Grade Level(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
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P.2F |
Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
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Graph
EXPONENTIAL AND LOGARITHMIC 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, 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
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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.
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Graph
FUNCTIONS, INCLUDING EXPONENTIAL AND LOGARITHMIC, FUNCTIONS 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
Including, but not limited to:
- General form of parent function
- Exponential functions: f(x) = 2x, f(x) = ex, f(x) = 10x
- Logarithmic functions: f(x) = log2(x), f(x) = ln(x), f(x) = logx
- Representations with and without technology
- Graphs
- Verbal descriptions
- Algebraic generalizations (including equation and function notation)
- Changes in parameters a, b, c, and d on graphs
- Effects of a on f(x) in af(x)
- a ≠ 0
- |a| > 1, the graph stretches vertically
- 0 < |a| < 1, the graph compresses vertically
- Opposite of a reflects vertically over the horizontal axis (x-axis)
- Effects of d on f(x) in f(x) + d
- d = 0, no vertical shift
- Translation, vertical shift up or down by |d| units
- Effects of c on f(x) in f(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):
- 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, 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
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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.
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Determine, Analyze
THE KEY FEATURES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS SUCH AS DOMAIN, RANGE, 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
- 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
- {x|x
, x ≥ 0}
- {y|y
, y < 1}
- Represented with interval notation
- 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 < x ≤ 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):
- 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:
- VII. Functions
- B1 – Understand and analyze features of a function.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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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.
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Analyze, Describe
END BEHAVIOR OF FUNCTIONS, INCLUDING EXPONENTIAL AND LOGARITHMIC 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) → –∞.
- Interpreting end behavior in real-world 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:
- VII. Functions
- B1 – Understand and analyze features of a function.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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P.2N |
Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.
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Analyze, To Solve
SITUATIONS MODELED BY FUNCTIONS, INCLUDING EXPONENTIAL AND LOGARITHMIC 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.)
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 real-world problems and interpret solutions to those problems.
- 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.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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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:
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P.5G |
Use the properties of logarithms to evaluate or transform logarithmic expressions.
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To Transform, To Evaluate
LOGARITHMIC EXPRESSIONS
Including, but not limited to:
- Evaluating logarithmic expressions
- Changing to exponential notation
- With technology
- Transforming logarithmic expressions
- Numerical expressions
- Algebraic expressions
Note(s):
- Grade Level(s):
- Algebra I simplified expressions using the laws (properties) of exponents, including integral and rational exponents.
- Algebra II rewrote exponential equations to logarithmic equations and vice versa.
- Algebra II formulated and solved exponential and logarithmic equations.
- Precalculus applies the properties of logarithms to transform expressions.
- Various mathematical process standards will be applied to this student expectation as appropriate.
- TxCCRS:
- I. Numeric Reasoning
- B1 – Perform computations with real and complex numbers.
- II. Algebraic Reasoning
- B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
- VII. Functions
- B1 – Understand and analyze features of a function.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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P.5H |
Generate and solve logarithmic equations in mathematical and real-world problems.
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Generate, Solve
LOGARITHMIC EQUATIONS IN MATHEMATICAL AND REAL-WORLD PROBLEMS
Including, but not limited to:
- Solution strategies
- Solving logarithmic equations algebraically
- Simplifying expressions on both sides of an equation by writing them as single logarithms
- Rewriting logarithmic equations in exponential form
- Extraneous solutions
- Solving logarithmic equations with technology
- Various situations
- Mathematical problem situations
- Real-world problem situations
Note(s):
- Grade Level(s):
- Algebra II rewrote exponential equations to logarithmic equations and vice versa.
- Algebra II formulated and solved exponential and logarithmic equations.
- Algebra II determined the resonableness of a solution to a logarithmic equation.
- Precalculus applies the properties of logarithms to simplify expressions and solve equations.
- 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.
- VII. Functions
- B1 – Understand and analyze features of a function.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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P.5I |
Generate and solve exponential equations in mathematical and real-world problems.
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Generate, Solve
EXPONENTIAL EQUATIONS IN MATHEMATICAL AND REAL-WORLD PROBLEMS
Including, but not limited to:
- Various solution strategies
- Solving exponential equations algebraically
- Simplifying expressions on both sides of an equation
- Rewriting exponential equations in logarithmic form
- Solving exponential equations with technology
- Various situations
- Mathematical and real-world problem situations
- Exponential growth
- Exponential decay
- Other exponential behavior
Note(s):
- Grade Level(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 II formulated and solved exponential and logarithmic equations.
- Algebra I and Algebra II analyzed and described the effects of transformations on the parent functions with changes in a, b, c, and d parameters.
- Precalculus extends these skills to generate and solve exponential equations in mathematical and real-world situations.
- 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.
- VII. Functions
- B1 – Understand and analyze features of a function.
- VIII. Problem Solving and Reasoning
- IX. Communication and Representation
- X. Connections
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