Unit Standard Description PreCalc clarifications
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PreCalculus Unit Standard Description Pre­Calc clarifications Unit 1 Transformations and Parent Functions Build new functions from existing functions CCSS.Math.Content.HSF­BF.B.3 Identify the effect on the graph of replacing f ( x ) by f ( x ) + k , k f ( x ), f ( kx ), and f ( x + k ) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. CSS.Math.Content.HSF­IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity .★ CSS.Math.Content . F­BF.B.4 Find inverse functions. CSS.Math.Content. F­BF.B.4b Verify by composition that one function is the inverse of another. CSS.Math.Content. F­BF.B.4c Read values of an inverse function from a graph or a table, given that the function has an inverse. CSS.Math.Content. F­BF.B.4d Produce an invertible function from a non­invertible function by restricting the domain. Analyze functions using different representations. CSS.Math.Content. F­IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Unit 2 Polynomials and Rational Functions CCSS.Math.Content.HSF­IF.C.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 Polynomial functions and their graphs are covered in Algebra 2. These concepts can be reviewed in this class and extended to rational functions. page 1 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications CCSS.Math.Content.HSF­IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity .★ Unit 3 Exponential and Logarithms Extend the properties of exponents to rational exponents. CCSS.Math.Content.HSN­RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the 1/33 (1/3)3 1/33 cube root of 5 because we want (5) = 5 to hold, so (5) must equal 5 . CCSS.Math.Content.HSN­RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. CCSS.Math.Content.HSF­BF.B.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Unit 4 Trig Functions Extend the domain of trigonometric functions using the unit circle. CCSS Math Content.HSF­TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. CCSS Math Content.HSF­TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. CCSS.Math.Content.HSF­TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x , π + x , and 2π ­ x in terms of their values for x , where x is any real number. Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 page 2 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications CCSS.Math.Content.HSF­TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions. CCSS.Math.Content.HSF­TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. CCSS.Math.Content.HSF­TF.B.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. CCSS.Math.Content.HSF­TF.B.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. Unit 5 Analytic Trig Unit 6 Trig Applications Prove and apply trigonometric identities. 2 2 CCSS.Math.Content.HSF­TF.C.8 Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. CCSS.Math.Content.HSF­TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. CCSS.Math.Content.HSG­SRT.D.9 Derive the formula for A = 1/2absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. CCSS.Math.Content.HSG­SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 page 3 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications CCSS.Math.Content.HSG­SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non­right triangles (e.g., surveying problems, resultant forces). Unit 7 Vectors Represent and model with vector quantities. CCSS.Math.Content.HSN­VM.A.1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v , | v |, || v ||, v ). CCSS.Math.Content.HSN­VM.A.2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. CCSS.Math.Content.HSN­VM.A.3. Solve problems involving velocity and other quantities that can be represented by vectors. Perform operations on vectors. CCSS.Math.Content.HSN­VM.B.4. Add and subtract vectors. CCSS.Math.Content.HSN­VM.B.4a Add vectors end­to­end, component­wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. CCSS.Math.Content.HSN­VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. CCSS.Math.Content.HSN­VM.B.4c Understand vector subtraction v ­ w as v + (­ w ), where ­ w is the additive inverse of w , with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component­wise. CCSS.Math.Content.HSN­VM.B.5 Multiply a vector by a scalar. CCSS.Math.Content.HSN­VM.B.5a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component­wise, e.g., as c ( vx , vy ) = ( cvx , cvy ). Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 page 4 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications Unit 8 Matrices (optional) CCSS.Math.Content.HSN­VM.B.5b Compute the magnitude of a scalar multiple c v using || c v || = | c | v . Compute the direction of c v knowing that when | c | v ≠ 0, the direction of c v is either along v (for c > 0) or against v (for c < 0). Perform operations on matrices and use matrices in applications. CCSS.Math.Content.HSN­VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. CCSS.Math.Content.HSN­VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when CCSS.Math.Content.HSN­VM.C.8 Add, subtract, and appropriate dimensions. CCSS.Math.Content.HSN­VM.C.9 Understand that, unlike multiplication of numbers,matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. CCSS.Math.Content.HSN­VM.C.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. CCSS.Math.Content.HSN­VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. CCSS.Math.Content.HSN­VM.C.12 Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. Matrices will be covered only if there is time at the end of the year. Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 page 5 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications Unit 9 Conics Translate between the geometric description and the equation for a conic section. MCC9­12.HSG.GPE.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant Unit 10 Polar & Parametric Unit 11 Series & Sequences Ohio.Math.Content.Gr12.PFA.9 Convert points and equations from rectangular and polar form and vice versa. Ohio.Math.Content.Gr12.PFA.9 Graph polar equations by point plotting and recognize special polar graphs. Apply DeMoivre’s Theorem for roots and powers of complex numbers Parametric Equations Graph curves that are represented by sets of parametric equations Rewrite sets of parametric equations as single rectangular equations by eliminating the parameter. Understand the concept of a function and use function notation. CCSS.Math.Content.HSF­IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n­1) for n ≥ 1 . The equations for circles will be covered in Geometry and equations for parabolas will be covered in Algebra. Equations for Ellipses and Hyperbolas will be the focus for Pre­Calc. The depth to which these concepts are covered will be dependent upon time. The majority of this content is covered in Algebra 2. The focus in Pre­Calc will be infinite series. Build a function that models a relationship between two quantities. CCSS.Math.Content.HSF­BF.A.1 Write a function that describes a relationship between two ★ quantities. CCSS.Math.Content.HSF­BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context Archdiocese of Cincinnati & C. Sherman/HCESC/December 2014 page 6 of 7 PreCalculus Unit Standard Description Pre­Calc clarifications CCSS.Math.Content.HSA­APR.5 Know and apply the Binomial Theorem for the expansion of ( x + y )n in powers of x and y for a positive integer n , where x and y are any numbers, with coefficients determined for example by Pascal's Triangle. (NOTE: The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) Unit 12 Limits & Derivatives Ohio.Math.Content.Gr12.Measurement.A.3 Apply informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations; e.g., measurement of some quantities, such as volume of a cone, can be determined by sequences of increasingly accurate approximations. Make a mathematical argument using the concepts of limits. Use the concept of limit to find instantaneous rate of change for a point on a graph as the slope of a tangent at a point. Archdiocese of Cincinnati & C. 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