COURSE: PRECALCULUS ACCELERATED

TEXT: PRECALCULUS: GRAPHING AND DATA ANALYSIS (PRENTICE HALL)

Exam Information
 Both the midyear exam and the final exam consist of two parts.
o Section #1: Calculators are allowed
o Section #2: No calculators
 Topics in red are assessed on the midyear exam.
 Topics in green are assessed on the final exam.


Midyear Exam

I. Complex numbers
Students should be able to:
a. Apply basic operations on complex numbers.
b. Express complex numbers in proper form.
c. Find the argument of a complex number.
d. Find the modulus or magnitude.
e. Graph complex numbers.

II. Parametric equations

Students should be able to:
a. Understand the concept of a parameter.
b. Identify a pair of parametric equations.
c. Sketch the curve represented by a pair of parametric equations.
d. Convert a pair of parametric equations into Cartesian form.
e. Determine any point(s) of collision for a pair of parametric equations.

III. Trigonometry introduction

Students should be able to:
a. Identify the components of an angle in standard position.
b. Identify coterminal, primary, and reference angles.
c. Understand positive and negative direction.
d. Understand arc length and radians.   L
r
e. Measure angles in radians and degrees and be able to convert.
f. Solve problems involving circular motion.
g. Understand the development of circular and trigonometric functions.
h. Understand the connection between circular and trigonometric functions.
i. Determine the values of trig functions by reference angles and calculators.
j. Identify and evaluate special right triangles.

IV. Graphing Trigonometric Functions

Students should be able to:
a. Understand and explain various properties - domain, range, period, odd, even, etc.
b. Graph basic trigonometric functions.
c. Graph transformations of the basic functions.
d. Write expressions for trig functions.
e. Use circular functions to model periodic phenomena.
f. Evaluate expressions involving inverse trig functions.

Precalculus Accelerated Course Outline – page 1 revised Spring 2010

V. Trigonometric Identities
Students should be able to:
a. Apply the fundamental identities - ratios, Pythagorean, reciprocal.
b. Understand and explain the concept of co-functions.
c. Understand and explain the concept of an identity.
d. Understand the development of various identities - sum, difference, double, one-half.

VI. Trigonometric Proofs

Students should be able to:
a. Apply identities to the verification of statements.

VII. Trigonometric Inverses

Students should be able to:
a. Understand and explain principal values and principle valued functions.
b. Calculate inverse trig function values by use of reference triangles or calculators.
c. Apply identities to numerical problems involving angle of elevation, depression, bearing and course.

VIII. Trigonometric Equations

Students should be able to:
a. Understand and explain the various algebraic methods for solving equations.
b. Solve equations involving trigonometric functions by either algebraic methods and/or identities.
c. Check for extraneous roots and domain restrictions.
d. Solve problems involving right triangles and their applications.

IX. Law of Sines and Cosines: ambiguous case, area formula

Students should be able to:
a. Derive and apply the law of sines and cosines for oblique triangles.
b. Solve problems and/or applications involving oblique triangles using Law of Sines or Law of Cosines.
c. Calculate the area of a triangle.
d. Understand and explain the ambiguous case.

X. Polar Coordinates: converting, multiplying, dividing, and powers.

Students should be able to:
a. Understand and use the complex plane to plot complex numbers in standard form.
b. Transform complex numbers in standard form to complex form.
c. Find powers and quotients of complex numbers in polar form.
d. Use DeMoivre's theorem.
e. Find roots of complex numbers.
f. Find symmetry in polar relations.
g. Graph polar relations.

Precalculus Accelerated Course Outline – page 2 revised Spring 2010

Final Exam

XI. Sets
Students should be able to:
a. Know the various number sets
b. Identify if a set is bounded above, bounded below, bounded, or unbounded.
c. Determine the continuity of a set.

XII. Relations
Students should be able to:
a. Graph equations by hand and by using a graphing utility.
b. Determine the domain and range of a relation.
c. Determine the intercepts.
d. Determine if a relation is one-to-one
e. Determine the symmetry of a relation.
f. Derive the inverse of a relation.
g. Graph relations by hand and by using a graphing utility.

XIII. Functions
Students should be able to:
a. Understand and explain the concept of a function.
b. Obtain information from/about the graph of a function.
c. Identify and explain the properties of functions.
d. Evaluate composite functions.
e. Identify parent graphs and transformations.
f. Determine/prove intervals in which a function is increasing or decreasing.
g. Apply Blonzo’s Theorem and the Intermediate Value Theorem to determine the existence of roots and
continuity.
h. Analyze a piecewise defined function.
i. Graph by using transformations.
j. Be able to perform operations on functions.
k. Construct mathematical models.
l. Solve systems of equations by substitution, elimination, and determinants.
m. Solve systems of nonlinear equations using substitution and elimination.
n. Solve quadratic equations or inequalities using the appropriate method.
o. Apply the discriminant of a quadratic equation.
p. Graph a quadratic function by hand and by using a graphing utility.
q. Solve applied problems using quadratic methods.

XIV. Solving systems of equations/word problems

Students should be able to:
a. Solve equations and inequalities, involving real and complex numbers, algebraically and by using a
graphing utility.
b. Solve applied problems using mathematical modeling.

XV. Rational Functions

Students should be able to:
a. Determine the domain and range of a rational function.
b. Identify the symmetry of a rational function.
c. Determine the vertical, horizontal, and oblique asymptotes of a rational function.
d. Analyze the graph of a rational function.

Precalculus Accelerated Course Outline – page 3 revised Spring 2010

XVI. Polynomials
Students should be able to:
a. Identify polynomials and their degree.
b. Apply the theory of polynomials (division algorithm, synthetic division, remainder theorem, factor
theorem, and rational root theorem).
c. Determine the roots and factors of a polynomial function.
d. Use Descartes' Rule of Signs to determine the possible zeros of a polynomial function.
e. Apply the power rule of derivatives to find turning points and points of inflections.
f. Graph a polynomial function using all pertinent data.

XVII.Absolute Value functions/equations

Students should be able to:
a. Solve absolute value equations and inequalities.
b. Graph absolute value functions and relations.

XVIII. Logarithmic functions

Students should be able to:
a. Solve exponential equations.
b. Identify and explain the characteristics of an exponential function.
c. Graph exponential equations.
d. Derive and define the number e.
e. Convert exponential expressions to logarithmic form and vice versa.
f. Evaluate logarithmic functions.
g. Determine the domain of a logarithmic function.
h. Graph logarithmic functions.
i. Apply properties of logarithms.
j. Evaluate and graph logarithmic functions.
k. Apply exponential and logarithmic functions to solve compound interest problems and growth and
decay problems.

XIX. Limits
Students should be able to:
a. Find a limit using tables
b. Find a limit using a graph
c. Find a limit of a sum, difference, product, or a quotient.
d. Find a limit of a polynomial
e. Find a limit of a power or a root.
f. Find a limit of an average rate of change.
g. Find the one-sided limit of a function.
h. Determine whether a function is continuous.
i. Find an equation of the tangent line to the graph of a function.

XX. Derivatives
Students should be able to:
a. Know various definitions of derivatives.
b. Use correct notation.
c. Find the derivative of a function.
d. Find instantaneous rates of change.
e. Find the speed of a particle.
f. Understand the relationship between f and f’.
g. Graph a derivative from given data.
h. Apply one sided derivatives.

Precalculus Accelerated Course Outline – page 4 revised Spring 2010

 i. Identify when f’(a) may fail to exist.
j. Differentiability implies local linearity.
k. Calculate derivatives at x = a using a graphing utility.
l. Understand that differentiability implies continuity.

Precalculus Accelerated Course Outline – page 5 revised Spring 2010