Math 1050 course objectives
Students will be able to:
• solve quadrating equations by factoring, using the square root method, by completing the square,
or by using the Quadratic Formula
• rewrite square roots of negative numbers as complex numbers
• perform arithmetic operations on complex numbers and rewrite the result in standard form a + bi
• solve rational equations
• solve equations involving radicals
• solve equations quadratic in form using substitution
• solve equations by factoring
• identify inequalities with intervals (their solution sets)
• find the union and intersection of intervals
• apply the method of test points and test intervals of solving different types of inequalities (linear,
quadratic, polynomial, rational)
• identify that a relation is a function (in particular, using the vertical line test)
• identify domains and ranges of relations (functions in particular), in particular from the graph
• interpret different ways of defining a relation (a function in particular)
• interpret function notation
• evaluate functions
• find domains of functions (the three conditions)
• solve applied problems using functions
• verify if a function is even, odd, or neither
• identify (from the graph) intervals on which the function is increasing, decreasing, or constant
• identify (from the graph) relative maximum and minimum values
• find the average rate of change of a function
√ √
• draw and recognize the shapes of basic functions’ graphs (y = x, x2 , x3 , x, 3 x, x1 , |x|)
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• use the Diagram of Transformations to graph transformations of basic functions or identify a
transformation of a function from the graph
• graph piecewise-defined functions
• form and evaluate arithmetic combinations of functions and state their domain
• form and evaluate compositions of functions
• state the domain of a composite function
• decompose a function
• use the concept of composite functions in applications
• identify if a function is one-to-one, in particular using the horizontal line test
• verify if two functions are inverses of each other
• graph inverse functions
• determine domains and ranges for a pair of inverse functions
• find the formula for the inverse function (if the inverse function exists)
• use the concept of inverse functions in applications
End of Exam 1 material
• transform the formula defining a quadratic function to vertex (standard) form
• identify the coordinates of the vertex of the parabola from vertex form
• graph a quadratic function in vertex form (using the diagram of transformations)
• identify the quadratic function from the graph
• solve optimization problems by finding the appropriate coordinate(s) of the vertex of the quadratic
function modeling the quantity to be optimized
• identify the leading term (in particular, the leading coefficient and degree) of a polynomial from
expanded or factored form
• determine the end-behavior of polynomial functions
• identify the multiplicity of a zero of a polynomial
• use the multiplicity of a real zero to determine if the graph crosses or touches the x-axis at the
zero
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• find the y-intercept of the graph of a polynomial function
• factor polynomials (over the reals, as a product of linear and irreducible over the reals quadratic
factors; and completely over the complexes, as a product of linear factors corresponding to all,
including complex, zeros)
• identify zeros and their multiplicities from the factored form of a polynomial
• write the factor corresponding to the given zero
• find a polynomial with given zeros (with multiplicities)
• divide polynomials using long and synthetic (where applicable) division
• identify the quotient and remainder of the division of polynomials
• use synthetic division to: evaluate polynomials, divide polynomials, verify zeros of polynomials,
and perform partial factorization
• find real zeros of polynomials and their corresponding factors
• use the Remainder and Factor Theorems
• determine and use the list of possible rational zeros
• find the possible number of positive and negative zeros of polynomials using Descartes Rule of
Signs
• use the Conjugate Pairs Theorem to find zeros of polynomials
• multiply the factors corresponding to the pair of complex conjugate zeros
• solve polynomial inequalities (by graphing/sketching)
• graph polynomial functions by hand
• find the domain of a rational function
• identify vertical asymptotes and holes of a rational function, if any
• identify the horizontal asymptote of a rational function, if applicable
• find the equation of the slant (oblique) asymptote of a rational function, if applicable
• identify the y- and x-intercepts of a rational function, if any
• graph rational functions by hand (with the aid of a table of points, if needed)
• determine the limiting, or “long run” behavior of a rational function in applications
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• graph a simple exponential function or a transformation of such (using the diagram of transformations)
• recognize graphs of simple exponential functions or transformations of such and identify the function from the graph
• identify the domain and range of an exponential function
• evaluate exponential functions
• evaluate expressions involving the number e
• use (in particular, evaluate) formulas involving exponential functions
• convert between the exponential and equivalent logarithmic form
• find the domain of an expression involving logarithms.
• graph a simple logarithmic function or a transformation of such (using the diagram of transformations)
• recognize graphs of simple logarithmic functions or transformations of such and identify the function from the graph
• identify the domain and range of a logarithmic function
• evaluate “simple” logarithms without using a calculator
• use formulas involving logarithmic functions
• expand (rewrite as a sum or difference of possibly constant multiples of logarithms) and condense
(rewrite as a single logarithm) logarithmic expressions using the rules of logarithms
• evaluate (or approximate) logarithms using the Change of Base formula (and a calculator)
• solve exponential and logarithmic equations (using the one-to-one property, or rewriting in equivalent exponential/logarithmic form)
• solve exponential equations quadratic in type that require a substitution
• recognize equations that don’t require a solution process
• use initial values to find exact models (especially for the exponential growth/decay models)
• use a (provided) formula to answer application questions
• recognize if the application question is about the argument or value, and write the appropriate
equation
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• interpret half-life
• use logarithmic scales
• use a calculator to approximate answers
• determine the limiting value (or carrying capacity) in the logistic model
• solve for a specific unknown in exponential and logarithmic models
End of Exam 2 material
• solve systems of linear equations using substitution, elimination or any matrix method (Gaussian
elimination with back-substitution, Gauss-Jordan elimination, or using the inverse matrix)
• identify the order of a matrix
• write the augmented matrix corresponding to a system of linear equations
• write a system of linear equations as a matrix equation
• find a row-echelon form or the reduced row-echelon form of a matrix using elementary row operations
• label elementary row operations
• recognize if a matrix is in a row-echelon form or the reduced row-echelon form
• find the reduced row-echelon form of a matrix using a calculator
• interpret the reduced row-echelon form of an augmented matrix corresponding to a system of linear
equations to identify the solution set of the system
• write the infinite solution set using parameters for free variables
• add, subtract, or multiply matrices by a scalar
• multiply matrices, if possible
• determine if a square matrix is invertible and find the inverse matrix using elementary row operations or a calculator
• perform partial fraction decomposition on proper fractions
• graph solution sets to systems of linear or nonlinear inequalities in two variables
• determine the consumer and producer surplus provided the demand and supply equations
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• determine extreme values of linear functions of two variables over polygonal regions using linear
programming
• solve systems of nonlinear equations algebraically
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Students will know :
• the defining property of i: i2 = −1
• the “old” methods of factoring
• the zero-product property
• the square root property
• the definition of a relation and a function
• function notation
• the definition of an even and odd function
• the Diagram of Transformations
• the shapes and “standard” guide points for the graphs of basic functions
• the definitions of arithmetic combinations of functions
• the definition of a composite function
• the conditions for the domain of a composite function
• that only one-to-one functions have inverse functions
• the verification equations for inverse functions
• the relationship between the graphs of inverse functions
• the general and vertex (standard) forms of a quadratic function
• the formulas for h and k (the coordinates of the vertex of the parabola)
• the terminology connected with polynomials (the degree, leading coefficient, leading term, constant
term, multiplicity of a zero)
• the relationship between the zeros of a polynomial and their corresponding factors
• long division of polynomials
• synthetic division and its applications
• the Remainder and Factor Theorems
• the list of possible rational zeros
• different “levels” of factoring (over the reals; over the complexes)
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• the Fundamental Theorem of Algebra and its corollaries
• the complex conjugate zeros property
• the list of possibilities for a horizontal asymptote of rational functions
• how the graph of a rational function behaves with respect to vertical, slant, and horizontal asymptotes
• the definition and properties of simple exponential functions f (x) = bx and their graphs
• the natural base e
• the continuously compounded interest formula
• the definition and properties of simple logarithmic functions f (x) = logb (x) and their graphs
• the three conditions for the domain of functions (no zeros in denominators, no negatives under
even index radicals, and strictly positive arguments of logarithms)
• the “inverse” relationship between exponential and logarithmic functions with the same base
• the definition of the common and natural logarithm
• the Change of Base formula
• the one-to-one property for exponential and logarithmic functions
• the exponential growth/decay model
• matrix methods for solving systems of linear equations
• matrix algebra rules
• the cases of partial fraction decomposition
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