TEKS Clarification Document. Mathematics Precalculus

TEKS Clarification Document Mathematics Precalculus 2012 2013

111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12. Source: The provisions of this 111.31 adopted to be effective September 1, 1996, 21 TexReg 7371; amended to be effective August 1, 2006, 30 TexReg 4479 111.35. Precalculus (a) General requirements. The provisions of this section shall be implemented beginning September 1, 1998, and at that time shall supersede 75.63(bb) of this title (relating to Mathematics). Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisites: Algebra II, Geometry. (b) Introduction. (1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model functions and equations and solve real-life problems. (2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problemsolving contexts. 2012, TESCCC 02/22/13 Page 2 of 32

P.1 The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to: P.1A Describe parent functions symbolically and graphically, including f(x) = x n, f(x) = ln x, f(x) = log a x, f(x) = 1/x, f(x) = e x, f(x) = x, f(x) = a x, f(x) = sin x, f(x) = arcsin x, etc. Describe, Define, Translate PARENT FUNCTIONS Parent function relationships Polynomial: f(x) = x n Rational: f(x) = 1/x Power: f(x) = ax b Radical: f(x) = x Absolute value: f(x) = x Exponential: f(x) = b x, f(x) = e x Logarithmic: f(x) = log a x, f(x) = ln x Logistic: f(x) =1/(1 + e x ) Trigonometric: f(x) = sin x, f(x) = cos x, f(x) = tan x, f(x) = arcsin x, or f(x) = sin -1 x, etc. Inverse parent functions Quadratic and square root Exponential and logarithmic Analysis of parent functions with and without technology Symbolically Function notation: f(x) Equation notation: y in terms of x Graphically Domain and range 2012, TESCCC 02/22/13 Page 3 of 32

Symmetry Asymptotes Critical points Numerically Defined values Undefined values Verbal description General shape of the graphs Intervals where the function is increasing or decreasing End behaviors P.1B III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. VII. Functions A1 Recognize whether a relation is a function. VII. Functions B1 Understand and analyze features of a function. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Determine the domain and range of functions using graphs, tables, and symbols. Determine, Define, Describe DOMAIN AND RANGE OF FUNCTIONS Multiple representations of functions with and without technology Graphs Tables Verbal descriptions Algebraic generalizations Notation Set notation Continuous (e.g., {x -3 x < 4}) 2012, TESCCC 02/22/13 Page 4 of 32

Discrete (e.g., x = {2, 4, 6}) Interval notation (e.g., [5, )) Restrictions of the domain determined by type of function Ex: rational functions, denominator 0 Problem situations Domain of the representative function versus domain of the problem situation Discrete versus continuous domain Increasing versus decreasing P.1C III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. VII. Functions B1 Understand and analyze features of a function. Describe symmetry of graphs of even and odd functions. Describe, Define SYMMETRY OF GRAPHS OF EVEN FUNCTIONS AND ODD FUNCTIONS Graphical analysis with and without technology Reflective symmetry with respect to the y-axis (even functions) Rotational symmetry with respect to the origin (odd functions) Symbolic analysis Even functions: f(-x) = f(x) Odd functions: f(-x) = -f(x) Tabular analysis with and without technology Repeating y-values (even functions) Ex: x -2-1 0 1 2 y 4 1 0 1 4 2012, TESCCC 02/22/13 Page 5 of 32

Opposite y-values (odd functions) Ex: x -2-1 0 1 2 y -8-1 0 1 8 P.1D III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. III. Geometric Reasoning B2 Identify symmetries of a plane figure. III. Geometric Reasoning B3 Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures. VII. Functions B1 Understand and analyze features of a function. Recognize and use connections among significant values of a function (zeros, maximum values, minimum values, etc.), points on the graph of a function, and the symbolic representation of a function. Recognize, Use, Define, Describe CONNECTIONS AMONG SIGNIFICANT VALUES, POINTS, AND SYMBOLIC REPRESENTATIONS OF A FUNCTION Significant values with and without technology y-intercepts x-intercepts (zeros) Local extrema Maximum values (where the function changes from increasing to decreasing) Minimum values (where the function changes from decreasing to increasing) Points on a graph with and without technology Symbolic representation with and without technology Determination of y-intercepts from functions Determination of x-intercepts (zeros) from functions Connections between x-intercepts (zeros) to roots and solutions of equations Real solutions 2012, TESCCC 02/22/13 Page 6 of 32

Complex solutions P.1E II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. Investigate the concepts of continuity, end behavior, asymptotes, and limits and connect these characteristics to functions represented graphically and numerically. Investigate, Connect, Describe, Translate CONCEPTS OF CONTINUITY, END BEHAVIOR, ASYMPTOTES, AND LIMITS Graphically with and without technology Continuity Discontinuity Vertical discontinuity Point discontinuity Jump discontinuity Greatest integer function f(x) = [x] Asymptotes Horizontal (x increases without bound, y-values approach a constant) Vertical (x-values approach a constant, y-values increase or decrease without bound) Slant (x increases without bound, y-values approach those of a linear function) End-behavior (x increases without bound, y-values approach those of another function) Piecewise functions Numerically (with tables) with and without technology y-values defined y-values undefined 2012, TESCCC 02/22/13 Page 7 of 32

y-values approach a limit Symbolically Constants in the functions affecting function behavior Coefficients Degrees (exponents) Parts of the functions that limit the domain or range Ex: rational functions, denominator 0 Ex: logarithmic function, argument > 0 III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. P.2 The student interprets the meaning of the symbolic representations of functions and operations on functions to solve meaningful problems. The student is expected to: P.2A Apply basic transformations, including a f(x), f(x) + d, f(x c), f(b x), and compositions with absolute value functions, including f(x), and f( x ), to the parent functions. Apply, Interpret TRANSFORMATIONS AND COMPOSITIONS General forms of parent functions Polynomial: f(x) = x n Rational: f(x) = 1/x Power: f(x) = ax b Radical: f(x) = Absolute value: f(x) = x Exponential: f(x)= b x, f(x) = e x x 2012, TESCCC 02/22/13 Page 8 of 32

Logarithmic: f(x) = log a x, f(x) = ln x Trigonometric: f(x) = sin x, f(x) = cos x, f(x) = tan x, f(x) = arcsin x, or f(x) = sin -1 x, etc. Transformations of parent functions with and without technology a f(x) Vertical stretch ( a > 1) Vertical compression (or shrink) (0 < a < 1) Reflection over the x-axis (a < 0) f(x) + d Vertical translation (shift) up (d > 0) Vertical translation (shift) down (d < 0) f(x c) Horizontal translation (shift) right (c > 0) Horizontal translation (shift) left (c < 0) f(b x) Horizontal stretch (0 < b < 1) Horizontal compression (or shrink) ( b > 1) Reflection over the y-axis (b < 0) Compositions of absolute value with and without technology f(x) Parts of the original graph below the x-axis reflect above the x-axis Parts of the original graph above the x-axis remain the same f( x ) Parts of the original graph to the left of the y-axis delete Parts of the original graph to the right of the y-axis reflect over the y-axis (to the left), and the original part to the right remains Analysis of effects and comparisons with and without technology Graphical Tabular Algebraic 2012, TESCCC 02/22/13 Page 9 of 32

Verbal P.2B III. Geometric Reasoning B1 Identify and apply transformations to figures. VII. Functions B1 Understand and analyze features of a function. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C1 Apply known function models. Perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically. Perform, Describe, Interpret OPERATIONS AND COMPOSITIONS OF FUNCTIONS Symbolic representations of operations with functions (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) Compositions f(g(x)) (f g)(x) Numerical and graphical representations of operations with functions with and without technology Add, subtract, multiply, or divide y-values of two functions at the same x-value Numerical and graphical representations of compositions of functions with and without technology y-values of one function serve as x-values of another Verbal descriptions of operations and composites of functions Substituting given values into appropriate functions with and without technology Find, Describe, Interpret INVERSES OF FUNCTIONS 2012, TESCCC 02/22/13 Page 10 of 32

Symbolic representations of inverses f -1 (x) where f -1 (f(x)) = x Switching the x and y variables then solving for y Numerical representations of inverses with and without technology Switching the x- and y-values in a table or an ordered pair Graphical representations of inverses with and without technology Reflections over the line y = x Verbal descriptions of inverses of functions Substituting given values into appropriate functions with and without technology Solve PROBLEM SITUATIONS Problem situations involving operations, compositions, and inverses of functions Symbolic representations of operations with functions (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) Compositions f(g(x)) (f g)(x) Numerical and graphical representations of operations with functions with and without technology Add, subtract, multiply, or divide y-values of two functions at the same x-value. Numerical and graphical representations of compositions of functions with and without technology y-values of one function serve as x-values of another. 2012, TESCCC 02/22/13 Page 11 of 32

Symbolic representations of inverses f -1 (x), where f -1 (f(x)) = x Switching the x and y variables then solving for y Numerical representations of inverses with and without technology Switching the x- and y-values in a table or an ordered pair Graphical representations of inverses with and without technology Reflections over the line y = x Verbal descriptions effects of operations, compositions, and inverses of functions P.2C II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. VII. Functions B1 Understand and analyze features of a function. Investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties. Investigate, Verify, Interpret IDENTITIES AND PROPERTIES Graphically with and without technology Ex: (exponential): Show (e 2 ) x = e 2x by graphing y 1 = (e 2 ) x and y 2 = e 2x Ex: (logarithmic): Show ln(x) 2 = 2 ln(x) by graphing y 1 = ln(x) 2 and y 2 = 2 ln(x) Ex: (trigonometric): Show sin(x) = cos(x π/2) by graphing y 1 = sin(x) and y 2 = cos(x π/2) Symbolically Substitution and simplification Properties of exponents Properties of logarithms Trigonometric identities Logical reasoning and proofs 2012, TESCCC 02/22/13 Page 12 of 32

II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. P.3 The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to: P.3A Investigate properties of trigonometric and polynomial functions. Investigate, Use, Model PROPERTIES OF TRIGONOMETRIC AND POLYNOMIAL FUNCTIONS Representations of trigonometric and polynomial functions with and without technology Tables Graphs Verbal descriptions Algebraic generalizations Characteristics of trigonometric functions Unit circle (positive and negative rotations) Standard position Reference angles Terminal side Coterminal angles Radian measure versus degree measure Trigonometric ratios defined by x- and y-values on the unit circle Right triangle relationships Correlation of x- and y-values on the unit circle to special right triangle relationships Graphing of trigonometric functions with and without technology Period Amplitude 2012, TESCCC 02/22/13 Page 13 of 32

Critical points Asymptotes Characteristics of polynomial functions with and without technology End behavior (increasing, decreasing) Extreme values (maxima, minima) End-behavior asymptotes Zeros and roots Fundamental Theorem of Algebra Real versus complex roots Algebraic determination of roots to polynomial equations Rational root theorem Synthetic substitution with polynomial functions Zeros of polynomial functions Factors of polynomial functions P.3B III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. IV. Measurement Reasoning C3 Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry. VII. Functions B1 Understand and analyze features of a function. Use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data. Use, Model, Solve FUNCTIONS REPRESENTING REAL-LIFE DATA Data collection activities (hands-on, CBR/CBL, problem situations) Representations of data with and without technology Tables Graphs 2012, TESCCC 02/22/13 Page 14 of 32

Verbal descriptions Algebraic generalizations Characteristics of data with and without technology Independent versus dependent Domain range Increasing/decreasing Continuous/discrete Data analysis with and without technology Scatter plots Determination of appropriate parent function Regression models Reasonableness of solutions mathematically and in context of problem situation P.3C III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. IV. Measurement Reasoning C3 Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry. VII. Functions C1 Apply known function models. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. Use regression to determine the appropriateness of a linear function to model real-life data (including using technology to determine the correlation coefficient). Use, Determine, Model APPROPRIATE LINEAR FUNCTIONS AS REPRESENTATIONS OF REAL-LIFE DATA Data collection activities (hands-on, CBR/CBL, problem situations) Representations of data with and without technology Tables Graphs 2012, TESCCC 02/22/13 Page 15 of 32

Verbal descriptions Algebraic generalizations Data analysis with and without technology Scatter plots Positive correlation (increasing) Negative correlation (decreasing) No correlation Linear regression with technology Correlation coefficient Positive versus negative values of r Goodness of fit as r 2 approaches 1 Reasonableness of solutions mathematically and in context of problem situation P.3D III. Geometric Reasoning C2 Make connections between geometry, statistics, and probability. VI. Statistical Reasoning B1 Determine types of data. VI. Statistical Reasoning B2 Select and apply appropriate visual representations of data. VI. Statistical Reasoning B4 Describe patterns and departure from patterns in a set of data. VI. Statistical Reasoning C2 Analyze data sets using graphs and summary statistics. VI. Statistical Reasoning C3 Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software. VII. Functions C1 Apply known function models. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning B2 Use various types of reasoning. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. IX. Communication and Representation C1 Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, graphs, and words. X. Connections B1 Use multiple representations to demonstrate links between mathematical and real world situations. Use properties of functions to analyze and solve problems and make predictions. Use PROPERTIES OF FUNCTIONS 2012, TESCCC 02/22/13 Page 16 of 32

Representations of functions with and without technology Tables Graphs Verbal descriptions Algebraic generalizations Characteristics of functions with and without technology Domain and range Continuous/discrete Intercepts Zeros Rate of change Maximum/minimum Symmetry Asymptotic behavior End behavior Determination of appropriate parent function with and without technology Analysis of scatter plot Analysis of rates of change Analyze, Solve Ex: constant rate of change (linear) Ex: common ratio (exponential) Ex: increasing then decreasing (quadratic or polynomial) Ex: periodic (trigonometric) PROBLEM SITUATIONS Representations of functions with and without technology Tables 2012, TESCCC 02/22/13 Page 17 of 32

Graphs Verbal descriptions Algebraic generalizations Characteristics of functions with and without technology Domain and range Continuous/discrete Intercepts Zeros Rate of change Maximum/minimum Symmetry Asymptotic behavior End behavior Determination of appropriate parent function with and without technology Analysis of scatter plot Analysis of rates of change Ex: constant rate of change (linear) Ex: common ratio (exponential) Ex: increasing then decreasing (quadratic or polynomial) Ex: periodic (trigonometric) Analysis of problem situation with and without technology Domain and range Maximum and minimum Continuous or discrete Zeros Various real-life problem situations, including Area (quadratic) Volume (cubic) 2012, TESCCC 02/22/13 Page 18 of 32

Proportion (rational) Square root (radical) Periodic (trigonometric) Reasonableness of solutions mathematically and in context of problem situation Make PREDICTIONS FROM FUNCTION MODELS REPRESENTING PROBLEM SITUATIONS Representations of functions with and without technology Tables Graphs Verbal descriptions Algebraic generalizations Determination of appropriate parent function with and without technology Analysis of scatter plot Analysis of rates of change Ex: constant rate of change (linear) Ex: common ratio (exponential) Ex: increasing then decreasing (quadratic or polynomial) Ex: periodic (trigonometric) Analysis of problem situation with and without technology Domain and range Maximum and minimum Continuous or discrete Zeros Various real-life problem situations, including Area (quadratic) Volume (cubic) 2012, TESCCC 02/22/13 Page 19 of 32

Proportion (rational) Square root (radical) Periodic (trigonometric) Reasonableness of solutions mathematically and in context of problem situation P.3E II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. VII. Functions B1 Understand and analyze features of a function. VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed. Model, Solve PHYSICAL PROBLEM SITUATIONS Right triangle problems with and without technology Sine, cosine, and tangent ratios (sides) Inverse trigonometric functions (angles) Oblique triangles Law of Sines (opposite pair known) Law of Cosines (side-angle-side and side-side-side cases) Ambiguous case (side-side-angle) Unit conversions Degrees versus radian measure Dimensional analysis (e.g., ft/sec into mi/hr) Polygon problems Trigonometry to find triangle heights Formulas to find area (Heron s formula) 2012, TESCCC 02/22/13 Page 20 of 32

Sub-dividing polygons into triangles Circle problems Arc length and sector area Degrees versus radian measure Linear and angular velocity Units Degrees Revolutions Radians Gear problems Wheels or gears connected by belts (equivalent linear velocity) Wheels or gears on the same axis (equivalent angular velocity) Reasonableness of solutions mathematically and in context of problem situation Incorporate RADIAN MEASURE Circle problems Arc length and sector area Degrees versus radian measure Linear and angular velocity Units Degrees Revolutions Radians Gear problems Wheels or gears connected by belts (equivalent linear velocity) Wheels or gears on the same axis (equivalent angular velocity) 2012, TESCCC 02/22/13 Page 21 of 32

Reasonableness of solutions mathematically and in context of problem situation III. Geometric Reasoning A3 Recognize and apply right triangle relationships including basic trigonometry. III. Geometric Reasoning C3 Make connections between geometry and measurement. IV. Measurement Reasoning B1 Convert from one measurement system to another. IV. Measurement Reasoning C3 Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. X. Connections A2 Connect mathematics to the study of other disciplines. X. Connections B1 Use multiple representations to demonstrate links between mathematical and real world situations. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. P.4 The student uses sequences and series as well as tools and technology to represent, analyze, and solve real-life problems. The student is expected to: P.4A Represent patterns using arithmetic and geometric sequences and series. Use, Represent, Analyze PATTERNS IN ARITHMETIC AND GEOMETRIC SEQUENCES AND SERIES Types Arithmetic (common difference) Comparison to linear functions Geometric (common ratio) Comparison to exponential functions Comparisons Term value versus term number (position) n versus a n Common difference versus common ratio Recursive versus explicit Sequence versus series a n versus S n 2012, TESCCC 02/22/13 Page 22 of 32

Notation Formulas Terms Arithmetic: a n = a 1 + d(n 1) Geometric: a n = a 1 r n 1 Sums Arithmetic series: S n = n 2 (a 1 + a n ) P.4B 1 r Geometric series: Sn = a1 1 r n a1 Infinite series: S = 1 r Subscripts (e.g., a 1, a n, s n ) Summation with sigma (Σ) Use arithmetic, geometric, and other sequences and series to solve real-life problems. Use, Represent, Analyze ARITHMETIC, GEOMETRIC, AND OTHER SEQUENCES AND SERIES Real-life problem situations Determination of appropriate model Arithmetic versus geometric Sequences versus series Application of appropriate formulas with and without technology Analyze, Represent, Solve PROBLEM SITUATIONS INVOLVING ARITHMETIC, GEOMETRIC, AND OTHER SEQUENCES AND SERIES 2012, TESCCC 02/22/13 Page 23 of 32

Real-life problem situations Determination of appropriate model Arithmetic versus geometric Sequences versus series Application of appropriate formulas with and without technology Reasonableness of solutions mathematically and in context of problem situation P.4C VII. Functions C2 Develop a function to model a situation. VIII. Problem Solving and Reasoning C2 Use a function to model a real world situation. X. Connections B1 Use multiple representations to demonstrate links between mathematical and real world situations. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. Describe limits of sequences and apply their properties to investigate convergent and divergent series. Describe, Apply, Investigate, Analyze PROPERTIES OF LIMITS OF SEQUENCES IN CONVERGENT AND DIVERGENT SERIES Types of sequences Divergent (no limit) Convergent (approaches a number) Comparisons Convergence of a sequence versus convergence of the related series Convergent infinite series Geometric series (-1< r < 1, r 0)) P.4D Apply sequences and series to solve problems including sums and binomial expansion. Apply, Use, Represent, Analyze SEQUENCES AND SERIES IN PROBLEM SITUATIONS 2012, TESCCC 02/22/13 Page 24 of 32

Real-life problem situations Arithmetic sequences and series Geometric sequences and series Summation notation Factorial notation Permutations Combinations Pascal s triangle Binomial expansion Coefficients of terms Degrees of terms Binomial probability Solve PROBLEM SITUATIONS INVOLVING SUMS AND BINOMIAL EXPANSION Real-life problems involving arithmetic and geometric sequences and series Binomial expansions to find specific terms Binomial probability to find the probability of a two-outcome event P.5 The student uses conic sections, their properties, and parametric representations, as well as tools and technology, to model physical situations. The student is expected to: P.5A Use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets. Use, Model CONIC SECTIONS IN PROBLEM SITUATIONS INVOLVING MOTION 2012, TESCCC 02/22/13 Page 25 of 32

Graphical analysis with and without technology Ex: velocity versus location (position) Planetary orbits and satellites Elliptical, parabolic, or hyperbolic paths Foci and vertices Apogee and perigee P.5B III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. III. Geometric Reasoning C3 Make connections between geometry and measurement. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. Use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound. Use, Describe PROPERTIES OF CONIC SECTIONS Geometry of conic sections with and without technology Center Symmetry Vertices Foci Directrices Axes Eccentricity Asymptotes (hyperbola) Determination of geometric representations of conic sections from working or standard form equations Circle: a, b, h, and k on the graph 2 2 2 ( x h) + ( y k) = r, where r is the radius of the circle 2012, TESCCC 02/22/13 Page 26 of 32

2 2 ( x h) ( y k) + = 1, where a = b 2 2 a b Ellipse: a, b, h, and k on the graph 2 2 2 2 ( x h) ( y k) ( y k) ( x h) + = 1 or + = 1, where a b 2 2 2 2 a b a b 2 2 ( x h) ( y k) + = 1, where a b 2 2 a b Hyperbola: a, b, h, and k on the graph 2 2 2 2 ( x h) ( y k) ( y k) ( x h) = 1 or = 1 2 2 2 2 a b a b 2 2 ( x h) ( y k) ± = 1 2 2 a b Parabola: a, p. h, and k on the graph 2 2 ( x h) = 4p( y k) or ( y k) = 4p( x h) y = a( x h) 2 + k or ( ) 2 x = a y h + k, where a = General form equation representing conic section Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 Identification of the type of conic 1 4p Transformation to standard form equations by completing the square Use, Describe, Model CONIC SECTIONS IN PROBLEM SITUATIONS INVOLVING REFLECTIVE PROPERTIES OF LIGHT AND SOUND Physical applications of conic sections with and without technology Ex: lenses, satellite dishes, whisper chambers, etc. Determination of appropriate conic section to represent the situation 2012, TESCCC 02/22/13 Page 27 of 32

Analysis of the properties of the conic model in terms of the problem situation P.5C III. Geometric Reasoning A1 Identify and represent the features of plane and space figures. III. Geometric Reasoning C3 Make connections between geometry and measurement. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. Convert between parametric and rectangular forms of functions and equations to graph them. Convert, Graph, Use, Model PARAMETRIC AND RECTANGULAR FORMS OF FUNCTIONS AND EQUATIONS Determination of geometric representations of conic sections from parametric functions with and without technology Circle: x = h+ rcost y = k + r sint Ellipse: x = h+ acost y = k + bsint Hyperbola: x = h+ asec t x = h+ atant or y = k + btant y = k + bsec t Parabola: 2 x = t x= at ( k) + h or 2 y= at ( h) + k y = t Other rectangular graphs from parametric functions with and without technology: x= ft () y= gt () Solving for t in terms of x: t = f -1 (x) 2012, TESCCC 02/22/13 Page 28 of 32

Substituting the expression for t in terms of x into the y = function: y = g(f -1 (x)) P.5D X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. Use parametric functions to simulate problems involving motion. Use PARAMETRIC FUNCTIONS IN PROBLEMS INVOLVING MOTION Linear motion with and without technology Starting point Direction vector t parameter (time) Projectile motion with and without technology Constants Initial velocity, v 0 Initial height, h 0 Angle of elevation, θ Parabolic functions x = v 0 (cos θ)t y = v 0 (sin θ)t 16t 2 + h 0 or y = v 0 (sin θ)t 4.9t 2 + h 0 Planetary motion with and without technology Parametric functions for appropriate representative conic section model Simulate MOTION IN PROBLEMS USING PARAMETRIC FUNCTIONS 2012, TESCCC 02/22/13 Page 29 of 32

Linear motion with and without technology Starting point Direction vector t parameter (time) Projectile motion with and without technology Constants Initial velocity, v 0 Initial height, h 0 Angle of elevation, θ Parabolic functions x = v 0 (cos θ)t y = v 0 (sin θ)t 16t 2 + h 0 or y = v 0 (sin θ)t 4.9t 2 + h 0 Planetary motion with and without technology Parametric functions for appropriate representative conic section model III. Geometric Reasoning C3 Make connections between geometry and measurement. VII. Functions C2 Develop a function to model a situation. X. Connections A1 Connect and use multiple strands of mathematics in situations and problems. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. P.6 The student uses vectors to model physical situations. The student is expected to: P.6A Use the concept of vectors to model situations defined by magnitude and direction. Use, Model VECTORS AS REPRESENTATIONS OF MAGNITUDE AND DIRECTION IN PROBLEM SITUATIONS Vector vocabulary and notation Vectors as directed distances (v = Magnitude as distance ( v ) RP ) 2012, TESCCC 02/22/13 Page 30 of 32

Direction as rotation angle (θ) Vectors in component form, x, y Situations with vectors Ex: magnitude as speed, direction as rotation angle, compass bearing, etc. Vector formulas x = v cos θ y = v sin θ ( ) θ = tan -1 y x v = 2 2 x + y P.6B III. Geometric Reasoning C3 Make connections between geometry and measurement. X. Connections B1 Use multiple representations to demonstrate links between mathematical and real world situations. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. Analyze and solve vector problems generated by real-life situations. Analyze, Solve, Use, Model VECTOR PROBLEMS GENERATED BY REAL-LIFE SITUATIONS Vector operations Determination of magnitude and direction from components (and vice-versa) Computation with scalar multiples of vectors Determination of resultant vectors by adding components Vector geometry Triangles to model vector addition Vector problems Ex: objects in motion, forces acting on an object in motion 2012, TESCCC 02/22/13 Page 31 of 32

III. Geometric Reasoning C3 Make connections between geometry and measurement. X. Connections B1 Use multiple representations to demonstrate links between mathematical and real world situations. X. Connections B2 Understand and use appropriate mathematical models in the natural, physical, and social sciences. 2012, TESCCC 02/22/13 Page 32 of 32