Advanced Math Syllabus
Textbook:
Precalculus: Graphical, Numerical, Algebraic (Eighth Edition)
Demana, Waits, Foley, and Kennedy
This class is beyond what the Nebraska State Standards list as necessary content. The
standards are all met before a student enrolls in this class. The student should expect a
challenge and are preparing to take Calculus.
PreRequisite Chapter Key Concepts
 Properties, Theorems and Formulas
o Trichotomy Property
o Properties of Algebra
o Properties of Equality
o Properties of Inequalities
o Distance Formula
o Midpoint Formula (Coordinate Plane)
o Quadratic Formula
o Equations of a line
o Equations of a Circle
 Procedures
o Completing the Square
o Solving Quadratic Equations Algebraically
o Agreement About Approximate Solutions
PreRequisite Chapter Lesson Objectives
 The student will convert between decimals and fractions
 The student will write inequalities
 The student will apply the basic properties of algebra
 The student will work with exponents and scientific notation
 The student will graph points
 The student will find distances and midpoints on a number line and in a
coordinate plane
 The student will write equations of circles in standard form
 The student will solve linear equations and inequalities in one variable
 The student will use slope and y-intercept to graph lines
 The student will use slope and y-intercept to write equations of lines in two
variables
 The student will solve equations involving quadratic, absolute value, and
fractional exponents by finding x-intercepts or intersections on graphs, using
algebraic techniques, or by using numerical techniques
 The student will add, subtract, multiply and divide complex numbers
 The student will find complex zeros of quadratic functions
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The student will solve inequalities involving absolute value, quadratic
polynomials, and expressions involving fractions
Chapter 1 Key Concepts – Functions and Graphs
 Properties, Theorems and Formulas
o The Zero Factor Property
o The Vertical Line Test
o Tests for an Even Function
o Tests for an Odd Function
o Horizontal Line Test
o Inverse Reflection Principle
o Inverse Composition Rule
o Translations of Graphs
o Reflections of Graphs Across Axes
o Stretches and Shrinks of Graphs
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Procedures
o Root, Zero, x-intercept
o Problem Solving
o Agreement about Domain
o Inverse Notation
o Finding the Inverse of a Function
Chapter 1 Lesson Objectives – Functions and Graphs
 The student will use numerical, algebraic, and graphical models to solve problems
and move from one model to another
 The student will represent functions numerically, algebraically, and graphically
 The student will determine the domain and range for functions
 The student will analyze function characteristics such as extrema, symmetry, and
asymptotes
 The student will recognize graphs of 12 basic functions and determine domains of
those functions
 The student will build new functions by adding, subtracting, multiplying, dividing
and composing functions
 The student will find inverses of functions and relations
 The student will algebraically and graphically represent translations, reflections,
stretches, and shrinks of functions
 The student will model real world problems using the basic functions
Chapter 2 Key Concepts – Polynomial, Power, and Rational Functions
 Properties, Theorems and Formulas
o Local Extrema
o Zeros of Polynomial Functions
o Zeros of Odd and Even Multiplicity
o Fundamental Connections for Polynomial Functions
o Upper and Lower Bound Test for Real Zeros
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o Fundamental Polynomial Connections in the Complex Case
o Factors of a Polynomial with Real Coefficients
o Polynomial Function of Odd Degree
o Remainder Theorem
o Factor Theorem
o Rational Zeros Theorem
o Fundamental Theorem of Algebra
o Linear Factorization Theorem
o Complex Conjugate Zeros Theorem
o Vertex Form of a Quadratic Function
o Vertical Free-Fall Motion
o Leading Term Test for Polynomial End Behavior
o Division Algorithm for Polynomials
o Graph of a Rational Function
Procedures
o Regression Analysis - website
o Polynomial Long Division
o Synthetic Division
o Solving Inequalities Using Sign Charts
Chapter 2 Lesson Objectives – Polynomial, Power, and Rational Functions
 The student will recognize and graph linear and quadratic functions, and use these
functions to model situations and solve problems
 The student will sketch power functions in the form of f(x) = kxa (where k and a
are rational numbers.)
 The student will graph polynomial functions, predict their end behavior, and find
their real zeros using a grapher or an algebraic method
 The student will divide polynomials using long division or synthetic division
 The student will apply the Remainder Theorem, Factor Theorem, and Rational
Zeros Theorem
 The student will find upper and lower bounds for zeros of polynomials
 The student will factor polynomials with real coefficients using factors with
complex coefficients
 The student will describe the graphs of rational functions, identify horizontal and
vertical asymptotes, and predict the end behavior of rational functions
 The student will solve equations involving fractions using both algebraic and
graphical techniques and to identify extraneous solutions
 The student will solve inequalities involving polynomials and rational functions
by using both algebraic and graphical techniques
Chapter 3 Key Concepts – Exponential, Logistic, and Logarithmic Functions
 Properties, Theorems and Formulas
o Basic Properties of Logarithms
o Basic Properties of Common Logarithms
o Basic Properties of Natural Logarithms
o Properties of Logarithms
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One-to-One Properties
Exponential Growth and Decay
Exponential Functions f(x) = bx
Exponential Functions and the Base e
Exponential Population Model
Changing Between Logarithmic and Exponential Form
Change-of-Base Formula for Logarithms
Logarithmic Functions f(x) = logbx, with b>1
Interest Compounded Annually
Interest Compounded k Times per Year
Interest Compounded Continuously
Chapter 3 Lesson Objectives – Exponential, Logistic, and Logarithmic Functions
 The student will evaluate exponential expressions and identify and graph
exponential and logistic functions
 The student will use exponential growth, decay, and regression to model real-life
problems
 The student will convert equations between logarithmic form and exponential
form
 The student will evaluate common and natural logarithms
 The student will graph common and natural logarithmic functions
 The student will apply the properties of logarithms to evaluate expressions and
graph functions, and be able to re-express data
 The student will apply the properties of logarithms to solve exponential and
logarithmic equations algebraically and solve application problems using these
equations
 The student will use exponential functions and equations to solve business and
finance applications related to compound interest
Chapter 4 Key Concepts – Trigonometric Functions
 Properties, Theorems, and Formulas
o Arc Length
o Right Triangle Trig Ratios
o Trig Functions of Real Numbers
o Inverse Sine Function
o Inverse Cosine Function
o Inverse Tangent Function
o Special Angles
o Sinusoids
 Procedures
o Angle Measure Conversion
Chapter 4 Lesson Objectives – Trigonometric Functions
 The student will convert between radians and degrees, find arc lengths, convert to
nautical miles, and solve problems involving angular speed
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The student will define the 6 trig functions using the lengths of the sides of a right
triangle
The student will solve problems involving the trig functions of real numbers and
the properties of the sine and cosine as periodic functions
The student will generate the graphs of the sine and cosine functions and explore
various transformations of these graphs
The student will generate the graphs for the tangent, cotangent, secant, and
cosecant functions and to explore various transformations of these graphs
The student will graph sums, differences, and other combinations of trig and
algebraic functions
The student will relate the concept of inverse functions to trig functions
The student will apply the concepts of trig to solve real-world problems
Chapter 5 Key Concepts – Analytic Trigonometry
 Properties, Theorems and Formulas
o Reciprocal Identities
o Quotient Identities
o Pythagorean Identities
o Cofunction Identities
o Odd-Even Identities
o Sum/Difference Identities
o Double-Angle Identities
o Power-Reducing Identities
o Half-Angle Identities
o Law of Sines
o Law of Cosines
o Triangle Area
o Heron’s Formula
 Procedures
o Strategies for Proving an Identity
Chapter 5 Lesson Objectives – Trigonometric Functions
 The student will use the fundamental identities to simplify trig expressions and
solve trig equations
 The student will decide whether an equation is an identity and to confirm
identities analytically
 The student will apply the identities for the cosine, sine, and tangent of a
difference or sum
 The student will apply the double-angle identities, power-reducing identities, and
half-angle identities
 The student will understand the proof of the Law of Sines and use the
computational applications of the Law of Sines to solve a variety of problems
 The student will apply the Law of Cosines to solve acute and obtuse triangles and
to determine the area of a triangle in terms of the measures of the sides and angles
Chapter 6 Key Concepts – Applications of Trigonometry

Properties, Theorems and Formulas
o Polar Coordinates
o Coordinate Conversion Equations
Chapter 6 Lesson Objectives – Applications of Trigonometry
 The student will convert points and equations from polar to rectangular
coordinates and vice versa
 The student will graph polar equations and determine the maximum r-value and
the symmetry of a graph