# Pre-Calculus and Trigonometry Capacity Matrix

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1 Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their Graphs Chapter 3 Sec 1.5 Use interval notation Sec 1.5 Solve and use properties of inequalities Sec 1.6 Solve equations involving Absolute Value Sec 1.6 Solve Inequalities involving Absolute Value Sec 1.7 Verbal descriptions into mathematical expressions Shot in the Dark Sec 1.7 Solve interest problems, uniform motion problems, mixture problems and constant rate job problems Know and use a definition of a function to decide if a given P1.1 relation is a function Perform algebraic operations (including compositions) on functions and apply transformations(translations, P1.2 reflections and rescaling) P1.6 P5.3 P5.3 Sec 3.6 Sec 3.6 Identify and describe discontinuities of a function(greatest integer function) and how these relate to the graph Know and apply the definition and geometric interpretation of the difference quotient Simplify difference quotients and interpret them as rates of change and slopes of secant lines Translate written description of a real world problem into a mathematical model Assign independent and dependent variables Be able to find minimum or maximum value in a real world problem Shot in the Dark Shot in the Dark

2 Linear and Quadratic Chapter 4 P1.1 Know and use a definition of a function to decide if a given relation is a function Perform algebraic operations on functions and apply transformations(translations, reflections, and P1.2 rescaling) Shot in the Dark Polynomials and Rational Chapter 5 P1.8 P3.2 P3.3 Sec 4.4 P4.1 P4.2 P4.3 Explain how the rates of change of functions in different families (ex. Linear functions and quadratics) differ, referring to graphical representations Apply quadratic functions and their graphs in context of motion under gravity and simple optimization problems find a quadratic function to model a given data set or situation Solve applied problems involving the law of demand using the demand equation Given a polynomial function whose roots are known or can be calculated, find the intervals on which the function s value are positive and those where it is negative Solve polynomial equations and inequalities of degree greater than or equal to three. Graph the polynomial functions given in factored form using zeros and their multiplicities, testing the sign-on intervals and analyzing the function s large-scale behavior. Know and apply fundamental facts about polynomials: the remainder theorem, the factor theorem, and the fundamental theorem of algebra x x Shot in the Dark

3 Polynomials and Rational Chapter 5 P5.1 Solve equations and inequalities involving rational functions. Graph rational functions given factored form using zeros, identifying asymptotes, analyzing their behavior for large x values and testing intervals. Given vertical and horizontal asymptotes, find an expression for a rational function with these functions. P5.2 Exponential and Logarithmic Chapter 6 P1.2 P1.3 P1.4 P1.5 P2.1 P2.2 P2.3 Know and use a definition of a function to decide if a given relation is a function. Write an expression for the composition of one given function with another and find domain, range and graph of the composite function. Recognize components when a function is composed of two or more elementary functions. Determine whether a function (given symbolically or graphically) has an inverse and express the inverse if it exists. Know and interpret the function notation for inverses Determine whether two given functions are inverses, using composition. Use the inverse relationship between exponential and logarithmic functions to solve equations and problems. Graph logarithmic functions. Graph translations and reflections of these functions Solve exponential and logarithmic equations. For those that cannot be solved analytically, use graphical methods to find approximate solutions. P2.4 Solve exponential and logarithmic equations when possible. For those that cannot be solved analytically, use graphical methods to find approximate solution.

4 Exponential and Logarithmic functions Chapter 6 P2.5 P3.1 Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay. Solve quadratic-type equations by substitution(eg. e 2x - 4e x+4 = 0) Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find a quadratic function to model given data or situation. P3.3 Trigonometric Chapter 7 P6.7 Define using the unit circle, graph and use all trigonometric functions of any angle. Convert between radian and degree measure. Calculate arc length, and area of a sector in a given circle. Unit Circle P6.1 Graph transformations of the sine and cosine functions (involving changes in amplitude, period, midline and phase changes) and explain the relationship between P6.2 constraints in the formula and transformed graph. Sec 7.7 Graph transformations of tangent functions Prove trigonometric identities and derive some of P6.6 the basic ones. Know the fundamental identities. Unit Circle Find a sinusoidal function to model a given data set or situation and explain how the parameters of the model relate to the data set or situation.

5 Analytic Chapter 8 Miscellaneous P6.3 P6.4 P6.6 P6.5 Know the basic properties of the inverse trigonometric functions: sin -1 x, cos -1 x, tan -1 x, including their domains and ranges. Recognize their graphs. Know basic trigonometric identities for sine cosine and tangent ( Fundamental, sum and difference, co functions, double and half angle formulas) Prove trigonometric identities and derive some of the basic ones ( double angle formulas from sum and difference formulas, half angles formula from double angle formula) Solve trigonometric equation using basic identities and inverse trigonometric functions Prove the addition and subtraction formula for sine, cosine, and tangent and use them to solve problems. CCSS Application of Trigonometric Real world applications of problems involving trigonometry such as the laws of sine and cosine. Polar Coordinates Chapter 10 P9.1 Convert between polar and rectangular coordinates. Graph functions given in polar coordinates P9.2 Write complex numbers in polar form. Know and use De Moivre s Theorem P9.3 Evaluate parametric equations for given values of parameter Convert between parametric and rectangular forms of P9.4 equations Unit Circle Unit Circle

6 Polar Coordinates Chapter 10 P9.5 Graph curves described by parametric equation and find parametric equations for a given graph P9.6 Use parametric equations in applied contexts to model situations and solve problems CCSS Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. Vectors, Matrices, and Perform operations (addition, subtraction, and Systems of Equations multiplication by scalars) on vectors in the plane. Solve P7.1 applied problems using vectors. Chapter 12 P7.2 P7.3 P7.5 P7.6 P7.7 P7.8 Know and apply the algebraic and geometric definitions of the dot product using vectors Know the definitions of matrix addition and multiplication. Add, subtract, and multiply matrices. Multiply a vector by a matrix. Define the inverse of a matrix and compute the inverse of two-by-two and three-by-three matrices when they exist Explain the role of determinants in solving systems of linear equation using matrices and compute determinants of twoby-two and three-by-three matrices. Use Crammer s Rule Write systems of two and three equations in matrix form. Solve such systems using Gaussian elimination or inverse matrices. Represent and solve systems of inequalities in two variable and apply these methods in linear programming situations to solve problems

7 Sequences, Series and Math Induction Chapter 13 Analytical Geometry Chapter 11 P8.1 Know, explain and use sigma and factorial notation Given arithmetic, geometric, or recursively defined sequence, write an expression for the nth term when possible. Write a particular term of a sequence when given P8.2 the nth term. Understand, explain and use the formulas for the sums of P8.3 finite arithmetic and geometric sequences Compute the sums of infinite geometric series. Understand P8.4 and apply the convergence criterion for geometric series. Understand and explain the principle of mathematical induction and prove statements using mathematical P8.5 induction Prove the binomial theorem using mathematical induction. Show its relationship to Pascal s Triangle and to combinations. Use the binomial theorem to find terms in P8.6 the expansion of a binomial to a power greater than 3. Know, explain, and apply the locus definitions of parabolas, ellipses and hyperbolas and recognize conic sections in P9.7 applied situations Identify parabolas, ellipses and hyperbolas from equations, write the equations in standard form, and sketch an P9.8 appropriate graph of the conic section Derive equation for a conic section from given geometric information. Identify key characteristics of a conic section P9.9 form its equation or graph. Identify conic sections whose equations are in polar or P9.10 parametric form.

### Pre-Calculus and Trigonometry Capacity Matrix

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