PS 1 Graphs: Graph equations using rectangular coordinates and graphing utilities, find intercepts, discuss symmetry, graph key equations, solve equations using a graphing utility, work with lines and circles. Use the distance formula and the midpoint formula, graph equations by hand by plotting points and with a graphing utility, use a graphing utility to create tables, find intercepts from a graph, and use a graphing utility to approximate intercepts. Find intercepts algebraically from an equation, test an equation for symmetry with respect to the x-axis, the y-axis and the origin, and know how to graph key equations. Solve equations by using a graphing utility. Calculate and interpret the slope of the line, graph lines given a point and the slope, find the equation of a vertical line, use the point-slope form of a line, identify horizontal lines, find the equation of a line given two points, write the equations of a line in slope-intercept form, identify the slope and y-intercept of a line from its equation, graph lines written in general form using intercepts, and find equations of parallel and perpendicular lines. Write the standard form of the equation of a circle, graph a circle by hand and by using a graphing utility, and work with the general form of the equation of the circle. PS 2 Functions and Their Graphs: Discuss and graph functions, investigate the properties of functions, graph the library of functions and piecewise-defined functions, use graphing techniques and transformations to graph functions, and build functions using mathematical models. Determine whether a relation represents a function, find the value of a function, find the domain of a function defined by an equation and form the sum, difference, product and quotient of two functions. Identify the graph of a function, and obtain information from or about the graph of a function. Determine even and odd functions from a graph, identify even and odd functions from the equation, use a graph to determine where a function is increasing, decreasing, or constant, use a graph to locate local maxima and local minima, use a graphing utility to approximate local maxima and local minima and to determine where a function is increasing or decreasing, and fine the average rate of change of a function. Graph the functions listed in the library of functions and graph piecewise-defined functions. Graph functions using vertical and horizontal shifts, graph functions using compressions and stretches, and graph functions using reflections about the x-axis or y-axis. Build and analyze functions.
PS 3 Linear Functions and Quadratic Functions: Discuss linear functions, their properties and linear models, build liner function from data, discuss quadratic functions and their properties, build quadratic models from verbal descriptions, and solve inequalities involving quadratic functions. Graph linear functions, use average rate of change to identify linear functions, determine whether a linear function is increasing, decreasing, or constant and build linear models from verbal descriptions. Draw and interpret scatter diagrams, distinguish between linear and nonlinear relations, and use a graphing utility to find the line of best fit. Graph a quadratic function using transformations, identify the vertex and axis of symmetry of a quadratic function, graph a quadratic function using its vertex, axis and intercepts, and find the maximum or minimum value of a quadratic function. Build quadratic models from verbal descriptions and from data. Solve inequalities involving quadratic functions PS 4 Polynomial and Rational Functions: Discuss polynomial functions and models, examine the properties of rational functions, graph rational functions, solve polynomial and rational inequalities, find the real zeros of a polynomial function, find complex zeros and use the fundamental theorem of algebra. Identify polynomial functions and their degree, graph polynomial functions using transformations, identify the real zeros of a polynomial function and their multiplicity, analyze the graph of a polynomial function, and build cubic models from data. Find the domain of a rational function, find the vertical asymptotes of a rational function, and find the horizontal or oblique asymptotes of a rational function. Analyze the graph of a rational function, and solve applied problems involving rational functions. Solve polynomial inequalities algebraically and graphically, solve rational inequalities algebraically and graphically. Use the remainder and factor theorems, use the rational zeros theorem to list the potential ration zeros of a polynomial function, find the real zeros of a polynomial function, solve polynomial equations, use the theorem for bounds on zeros, and use the intermediate value theorem. Use the conjugate pairs theorem, find a polynomial function with specified zeros, and find the complex zeros of a polynomial function.
PS 5 Exponential and Logarithmic Functions: Form composite functions, determine one-to-one functions and inverse functions, evaluate exponential functions, solve logarithmic functions, discuss properties of logarithms, solve logarithmic and exponential equations, apply financial models, discuss exponential growth and decay models, Newton s Law and logistic growth and decay models, and build exponential, logarithmic and logistic functions from data. LT 7 LT 8 LT 9 Form a composite function, and find the domain of a composite function. Determine whether a function is one-to-one, determine the inverse of a function defined by a map or a set of ordered pairs, obtain the graph of the inverse function from the graph of the function, and find the inverse of a function defined by an equation. Evaluate exponential functions, graph exponential functions, define the number e, and solve exponential equations. Change exponential statements to logarithmic statements and logarithmic statements to exponential statements, evaluate logarithmic expressions, determine the domain of a logarithmic function, graph logarithmic functions, and solve logarithmic equations. Work with the properties of logarithms, write a logarithmic expression as a sum or difference of logarithms, write a logarithmic expression as a single logarithm, evaluate logarithms whose base is neither 10 nor e, and graph logarithmic functions whose base is neither 10 nor e. Solve logarithmic equations, solve exponential equations, solve logarithmic and exponential equations using a graphing utility. Determine the future value of al ump sum of money, calculate effective rates of return, determine the present value of a lump sum of money, and determine the rate of interest or time required to double a lump sum of money. Find equations of populations that obey the law of uninhibited growth, find equations of populations that obey the law of decay, use Newton s Law of Cooling, and use logistic models. Build an exponential model from data, build a logarithmic model from data, and build a logistic model from data.
PS 6 Systems of Equations and Inequalities: Solve systems of linear equations using substitution and/or eliminations, matrices, determinants, use matrix algebra, decompose using partial fraction decomposition, solve systems of nonlinear equations, solve systems of inequalities, solve linear programming problems. LT 7 LT 8 Solve systems of equations by substitution and/or elimination, identify inconsistent systems of equations containing two variables, express the solution of a system of dependent equations containing two variables, solve systems of three equations containing three variables, identify inconsistent systems of equations containing three variables, and express the solution of a system of dependent equations containing three variables. Write the augmented matrix of a system of linear equations, write the system of equations from the augmented matrix, perform row operations on a matrix and solve a system of linear equations using matrices. Evaluate 2 x 2 determinants, use Cramer s Rule to solve a system of two equations containing two variables, evaluate 3 x 3 determinants, Use Cramer s Rule to solve a system of three equations containing three variables, know properties of determinants. Find the sum and difference of tow matrices, find scalar multiples of a matrix, find the product of two matrices, find the inverse of a matrix and solve a system of linear equations using an inverse matrix. Decompose P/Q where Q has only non-repeated linear factors, repeated linear factors, a non-repeated irreducible quadratic factor and /or a repeated irreducible quadratic factor. Solve a system of nonlinear equations using substitution and/or elimination. Graph an inequality by hand and using a graphing utility, and graph a system of inequalities. Set up and solve a linear programming problem.
PS 7 Sequences; Inductions; The Binomial Theorem: Discuss sequences, arithmetic sequences, geometric sequences, geometric series, prove statements using mathematical induction, and use the binomial theorem. Write the first several terms of a sequence, write the terms of a sequence defined by a recursive formula, use summations notation, find the sum of a sequence algebraically and using a graphing utility, and solve annuity and amortization problems. Determine if a sequence is arithmetic, find a formula for an arithmetic sequence, and find the sum of an arithmetic sequence. Determine if a sequence is geometric, find a formula for a geometric sequence, find the sum of a geometric sequence and determine whether a geometric series converges or diverges. Prove statements using mathematical induction. Evaluate " $ # n j % ' &, and use the binomial theorem.