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MATH 179 List of Topics [DRAFT--February 23, 2005]

Items that are not covered in Math 141 (or are significantly expanded) are in blue.

I.  Introductory Review

          A. Interval notation

          B. Using calculator to simplify expressions and evaluate functions

          C. Properties of functions, function notation, evaluating and solving functions

          D. Modeling linear relationships (from two points or slope-intercept applications)

          E. Interpreting linear functions and their graphs (input variable, output variable, rule, slope
                  and intercepts)

          F. Solving linear formulas for a specified variable

          G. Solving linear inequalities and functions algebraically and graphically

          H. Algebra of functionssum, difference, product, quotient

          I. Solving quadratic equations -- factoring, completing the square, quadratic formula
                  (including role of discriminant), and graphically

          J. Graphing parabolas -- finding vertex and intercepts algebraically and graphically

          K. Simplifying radical expressions and solving radical equations (review rational exponents)

          L. Simplifying rational expressions and solving rational equations (review negative

          M. Solving linear absolute value equations algebraically and graphically

II.  Functions

          A. Algebra of functions -- algebraically and graphically

          B. Recognize linear, polynomial, radical, rational, exponential, logarithmic, and greatest
functions and their graphs

          C. Compose functions and recognize composition

          D. Expand understanding of domain and range with composed functions (polynomial and
                    rational expressions inside radicals or logs)

          E. Graph functions that have been transformedshifted, stretched (horizontally or
                    vertically), or reflected

          F. Write the equation of a given function that has been shifted, stretched or reflected -- from
                    a description or a graph

          G. Finding inverse functions when only piecewise one-to-one

          H. Finding difference quotient for linear, polynomial and rational functions

          I. Finding equation of secant line

          J. Writing formulas using composition of functions

III.  Graphing

          A. Read, interpolate and extrapolate graphical data

          B. Graph the greatest integer function and recognize stretches, shifts and reflections

          C. Graph conic sections

                   1. circles -- find center and radius (use distance and midpoint formulas)

                   2. parabolas -- find focus, directrix, vertex, focal chord

                   3. ellipses -- find center, foci, vertices and major and minor axes

                   4. hyperbolas -- find vertices, center, foci, and equations of asymptotes

          D. Find equation of secant line and recognize it's role

          E. Interpreting slope, intercepts, and asymptotes of a graph (in context of units and

          F. Graph linear inequalities in two variables and their intersection and union

          G. Recognize "shape" of linear, absolute value, quadratic, cubic, root, greatest integer,
 exponential, and logarithmic functions and their graphs

          H. Graph functions (see previous) that have been transformedshifted, stretched, or

          I. Determine the equation of a rational function from its graph

          J. Graph rational functions with holes, and horizontal, vertical, and oblique asymptotes

          K. Graph the inverse of a function

          L. Graph polynomial functions and find domain, range, and local maximum or minimum

          M. Graph piece-wise defined functions and write the piece-wise function of a given graph

IV.  Graphing Calculator Skills

          A. Using graph to solve application problems

          B. Using intersection of graphs to solve equation in one variable polynomial, rational,
                    radical, exponential or logarithmic equations

          C. Using intersection of graphs to solve system of two linear equations

          D. Finding maximum and minimum values

          E. Finding zeros (roots) and y-intercept

          F. Doing matrix operations (algebra, row operations, solving systems of linear equations)

          G. Using quadratic formula and synthetic division programs to find irrational roots of
                    polynomial functions

         H. Data analysisgenerating scatter plots and regression equations and graphs and
                    evaluating the appropriateness of the regression equationlinear, quadratic, cubic,
                    power, and logarithmic regressions

          I. Solve inequalities graphically including composed functions (roots with polynomial and
                    radicals "inside")

          J. Graphing relations (circles and sideways parabolas)

V.  Modeling (and Solving) Applications

          A. Applications of composed functions

          B. Interpret slope, intercepts and asymptotes in context of problemunits and causality of

          C. Applications of linear inequalities

          D. Systems of linear equations (up to three variable linear equations)

          E. Quadratic relationships (area and projectiles)

          F. Finding maximum and minimum with quadratic applications

          G. Applications of rational relationships -- formulas and equating rates or time

          H. Applications involving direct variation, joint variation and inverse variation

          I. Applications of inequalities in two variables (feasible space) Linear Programming ?

          J. Applications involving roots

                   1. Pythagorean theorem (triangles)

                   2. Distance formula (points certain distance from other point)

                   3. Solving formulas for a variable

          K. Polynomial functions and inequalities -- cost functions, etc.

          L. Applications of exponential functions -- compound interest

          M. Applications of logarithmic functions and formulas

          N. Applications of non-real complex numbers (???)

          O. Applications of conic sections

VI.  Linear Equations and Formulas

          A. Recognize parallel or perpendicular equations

          B. Find equation of line parallel (or perpendicular) to given line through a point

          C. Find equation of secant line (after finding difference quotient)

          D. Extend concept of "slope" to non-linear functions -- increasing, decreasing, constant

VII.  Linear Inequalities

A. Solve, graph and interpret linear inequalities in one variable

          B. Solve, graph and interpret union and intersection of linear inequalities in two variables

          C. Solve constrained maximization and minimization problems graphically (linear

          D. Review use of "and," "or," "inclusive," and "exclusive" terminology

          E. Solve nonlinear systems of inequalities graphically

VIII.  Absolute Value Equations

          A. Solve absolute value equations of degree higher than one algebraically and graphically

          B. Solve absolute value inequalities of degree higher than one algebraically and graphically

          C. Solve equations with two absolute values algebraically and graphically

          D. Solve inequalities with two absolute values algebraically (sign chart) and graphically

IX.  Systems of Equations (Review and expand applications)

          A. Solve two variable linear systems by graphing, substitution and elimination

          B. Determine whether lines intersect, are parallel, or coincide

          C. Solve three variable linear systems using row operations and matrices

          D. Identify inconsistent and dependent systems of equations

          E. Solve nonlinear systems of equations (using substitution)

X.  Complex Numbers

          A. Define complex unit

          B. Add, subtract, multiply and divide (using conjugate to simplify) complex numbers

          C. Raise i to integer powers and simplify

XI.  Polynomial Expressions and Equations

          A. Recognize the connection between zeros of a polynomial equation and the factors

          B. Use long division and synthetic division to divide two polynomials

          C. Find rational, irrational, and non-real zeros (roots) of polynomials

          D. Extensions of factoring (to solving non-polynomials equations with integer, rational, and
                    variable exponents)

          E. Extensions of quadratic formula and taking roots (to solving equations with irrational
                   coefficients and solving formulas for a given variable)

          F. Recognize relationship of discriminant to # of real zeros and x-intercepts

          G. Use formula -b/(2a) to find the vertex of a parabola

          H. Calculate difference quotient for polynomial functions

          I. Properties of 3rd degree and higher polynomial equations, functions and expressions

                 1. relationship of factored equation, zeros, multiplicity of zeros, and graph

                 2. relate domain and range, local extrema and end behavior to degree of polynomial

                 3. use sign chart to find solution set for polynomial inequalities

          J. Apply theorems -- intermediate value, remainder, factor, Descartes' rule of signs,
                   conjugate zeros, fundamental theorem, number of zeros, rational zeros, and complex
                   roots theorems

XII.  Roots and Radical Expressions and Equations

          A. Solve radical equations with up to two radicals graphically and algebraically

          B. Re-write radical function to determine horizontal and/or vertical shift, stretch, and

          C. Find domain of radical functions -- including ones with rational expressions as radicand

          D. Solve radical inequalities algebraically (sign chart) and graphically

          E. Solve radical formulas for a designated variable

          F. Find the inverse of a radical function

XIII.  Rational Expressions and Equations

          A. Review simplifying rational expressions

          B. Review dividing rational expressions (simplifying complex fractions)

          C. Find domain and range of rational functions

          D. Find inverse of one-to-one rational functions

          E. Re-write functions to determine any shift, stretch or compression

          F. Find equation from graph of rational function

          G. Find intercepts and asymptotes of rational functions (including oblique asymptotes)

          H. Solve rational inequalities algebraically (sign chart) and graphically

          I. Model and solve problems with variation (direct, inverse, joint)

          J. Find difference quotient with rational function

XIV.  Exponential Expressions and Equations

          A. Review exponential functions

          B. Identify the domain and range and restrictions on base

          C. Relate equation/graph to shifting, stretching, and reflecting

          D. Solve exponential equations algebraically and graphically

          E. Solve exponential inequalities algebraically and graphically

          F. Find the inverse of an exponential function

          G. Solve exponential equations that are quadratic in form (using substitution)

XV.   Logarithmic Expressions and Equations

          A. Review logarithmic expressions, equations and functions

          B. Switch between exponential and logarithmic forms of an equation

          C. Use properties of logarithms

          D. Identify the domain and range and restrictions on base

          E. Find the inverse of a logarithmic function

          F. Solve logarithmic equations algebraically

          G. Relate equation/graph to shifting, stretching, and reflecting

          H. Solve logarithmic inequalities algebraically and graphically

XVI. Conic Sections

          A. Review circle -- graphing, standard form, completing the square, finding equation from
                    two points on diameter

          B. Interpreting/obtaining quadratic equations in conic form

          C. Graphing directrix, vertex, focus, and focal chord of parabola

          D. Completing the square to find the equation of a parabola

          E. Graph ellipses -- including vertices, major and minor axes, and foci

          F. Find equation of ellipse given information such as vertices, foci, major and minor axes

          G. Complete the square to find the standard equation of ellipse

          H. Graph ellipses with center not at origin

          I. Graph hyperbolas -- including vertices, asymptotes, and foci

          J. Find equation of hyperbola given information such as vertices, foci, or equation of

          K. Complete the square to find the standard equation of hyperbola

          L. Graph hyperbolas with center not at origin

          M. Identify type of conic from second degree polynomial

XVII. Parametric Equations

          A. Graphing parametric equations and their rectangular equivalents

          B. Find alternative forms of parametric equations

XVIII. Sequences and Series -- an Introduction

XIX. Binomial Theorem

XX. Mathematical Induction -- Introduction to Proofs (odd and even functions, square root property, in , why x >6 and x >0 is x >6)