| 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
exponents)
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
integer
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
causality)
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
reflected
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
values
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
problem
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
programming)??
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
compression
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
asymptote
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)
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