Download Math 1431 - College of DuPage

College of DuPage
FY Fall/17
ACTIVE COURSE FILE
Curricular Area: Mathematics
Course Number: 1431
Title: Precalculus I
Semester Credit Hours: 5 Lecture Hours: 5
Lab Hours: 0 Clinical Hours: 0
This course is not an IAI approved general education course.
Changes from the present course must be accompanied by a yellow
Course Revision or Deletion Form.
Course description to appear in catalog:
A formal study of algebra with emphasis on concepts needed for calculus.
Topics include, but are not limited to, functions, conic sections, matrices
and determinants, polynomial theory, rational functions, sequences and
series, logarithmic and exponential functions, combinatorial mathematics,
and mathematical induction.
Prerequisite:
Demonstrated geometry competency (level 2), and Math 0482 (or college
equivalent) with a grade of C or better or a qualifying score on the
mathematics placement test or a qualifying A.C.T. math sub-score
A.
General Course Objectives:
Upon successful completion of this course, students should be able to
do the following:
1. Solve equations and inequalities that involve the following:
quadratic, rational, and absolute value expressions
2. Identify, analyze, classify, and graph functions and relations
3. Determine limits of functions numerically and/or graphically
4. Determine, analyze, and graph inverse functions
5. Classify and graph conic sections
6. Determine the equation of a conic section
7. Perform matrix operations
8. Calculate the value of determinants
9. Solve systems of linear equations using various methods
10. Solve systems of non-linear equations
11. Analyze and find the zeros of polynomials
12. Graph polynomial functions
13. Analyze and graph exponential and logarithmic functions
14. Solve exponential and logarithmic equations
15. Solve applications of exponential growth and decay
16. Use the Binomial Expansion Theorem
17. Use sequence and series notation including sigma notation
18. Determine elements and sums of arithmetic and geometric
series
19. Use the Principle of Mathematical Induction
20. Determine the domains and ranges of rational functions
21. Construct the graphs of rational functions indicating
horizontal, vertical, and oblique asymptotes
B.
Topical Outline: (Optional topics are indicated by *.)
This topical outline is not necessarily sequential
1.
Review of and further topics from intermediate algebra
a. Quadratic equations and equations quadratic in form
b. Equations involving absolute value
c. Linear relations
1)
The slope of a line
2)
Graphs of linear relations using slope and intercepts
3)
Parallel and perpendicular lines
4)
Determination of an equation for a line
d. Exponents and radicals
2.
Inequalities
a. Quadratic inequalities
b. Rational inequalities
c. Inequalities involving absolute value
3.
Relations and functions
a. Definitions
b. Function notation
c. Domain and range
d. Algebra of functions
e. Composite of two functions
f.
Graphs of functions
1)
Determination of whether a graph is the graph of a
function
2)
Determination of whether the graph of a relation is
symmetric to the x-axis, y-axis or origin.
3)
Use of symmetry in graphing
4)
Horizontal and vertical translations
g. Odd and even functions
h.
Graphs of some special functions
1)
Functions involving absolute value
Greatest integer function
Square root function
Functions defined by more than one formula
depending on the value of the independent variable
Inverse of a function
1)
Determination a formula for f-l given f in
function notation
2)
Determination the domain and range of f-l
3)
Construction of a graph of a function and its inverse
Introduction to limits*
1)
Graphical approach
2)
Numerical approach
2)
3)
4)
i.
j.
4.
Analytic Geometry
a. The distance and midpoint formulas
b. Parabolas
1)
Construction of a graph of quadratic functions and quadratic
relations whose graphs are parabolas
2) Determination of the coordinates of the vertex and focus, the
equation of the directrix, and the equation of the axis of
symmetry
3)
Construction of the graphs of functions whose graphs are half
of a parabola
c. Circles
1)
Determination of the center-radius form of the
equation of a circle
2)
Determination of the center and radius of a
circle whose equation is given in general form
3)
Construction of the graphs of relations whose graphs are
circles or semi-circles
d. Ellipses
1)
Construction of the graphs of relations whose graphs are
ellipses or half of an ellipse
2)
Determination of the length of the major and
minor axes
3)
Determination of the coordinates of the center, vertices,
and foci
4)
Determination of the eccentricity
e. Hyperbolas
1)
Construction of the graphs of relations whose graphs are
hyperbolas or half of a hyperbola
2)
Determination of the equations of the asymptotes
3)
Determination of the lengths of the transverse
and conjugate axes
4)
Determination of the coordinates of the center, vertices, and
foci
5)
Determination of the eccentricity
f.
Systems of non-linear equations
g. Systems of non-linear inequalities*
5.
Matrices and determinants
a. Definition and dimension
b. Operations with matrices
c.
d.
e.
f.
g.
h.
Addition and subtraction
Scalar multiplication
Matrix multiplication
Gaussian elimination to solve linear systems
Evaluation of determinants by using cofactors
Evaluation of determinants by using row and column
operations to introduce zeros
Cramer's rule to solve a linear system
Gaussian elimination to find the inverse of a
nonsingular matrix
Use of the inverse of the coefficient matrix to
solve a linear system
1)
2)
3)
6.
Theory of polynomials
a. Use of synthetic division to divide a
polynomial by a linear polynomial of the form x-c
b. Use of the Remainder Theorem
c. Use of synthetic division and the Remainder Theorem
to evaluate polynomials
d. Use of the Factor Theorem
e. Determination whether a given number is a zero of a
polynomial function
f.
Use of the Conjugate Pair Theorem
g. Use of Descartes' Rule of Signs
h.
Determination of the integral bounds for zeros of a
polynomial
i.
Use of the Rational Zero Theorem
j.
Approximation of irrational zeros of a polynomial*
k.
Construction of the graphs of polynomial functions
7.
Exponential and logarithmic functions
a. Exponential functions
1)
Definition
2)
Graphs of exponential functions
3)
Exponential equations involving the same base
b. Logarithmic functions
1)
Definition using the concept of function
inversion on the exponential function
2)
Graphs of logarithmic functions
3)
Applications involving common logarithms and
natural logarithms
5)
Properties of logarithms with emphasis on
natural logarithms
5)
Logarithmic equations
6)
Use of natural logarithms in solving exponential
equations involving different bases
7)
The change of base formula
c. Exponential growth and decay
8.
Combinatorial mathematics
a. Use of factorial notation
b. Binomial Theorem
1)
Use of the Binomial Theorem to expand a
binomial
2)
9.
Use of the Binomial Theorem to find a
particular term in the expansion of a binomial
Sequences and series
a. Definitions
b. Determination of the terms of a sequence given a
formula for an or given a recursive definition
c. Expansion of a series given in sigma form, and the
determination of its sum
d. Arithmetic sequences and series
1)
Determination whether a given sequence is arithmetic
2)
Determination of any term of an arithmetic sequence
3)
Determination of a formula for an
4)
Determination of the sum of an arithmetic series
e. Geometric sequences and series
1)
Determination whether a given sequence is geometric
2)
Determination of any term of a geometric sequence
3)
Determination of a formula for an
4)
Determination of the sum of a geometric series
5)
Determination of the sum (if it exists) of an infinite
geometric series
10. Mathematical induction
a. Principle of Mathematical Induction
b. Use of the Principle of Mathematical Induction in proofs
11. Partial Fractions*
a. Partial fraction decomposition with distinct linear factors
b. Partial fraction decomposition with repeated linear factors
c. Partial fraction decomposition with distinct linear and
quadratic factors
d. Partial fraction decomposition with repeated quadratic
factors
12. Rational functions
a. Domain and range
b. Horizontal asymptotes
c. Vertical asymptotes
d. Oblique asymptotes
C.
Methods of Evaluating Students:
Unit tests at appropriate intervals; quizzes, homework, projects, and a
comprehensive final examination, all at the discretion of the instructor.
_______________________________
Initiator
Date
_______________________________
Sponsor
Date
_______________________________
Division Dean
Date
Textbook for Math 1431
Title:
College Algebra, 12th Edition
Author:
Lial, Hornsby, Schneider and Daniels
Publisher:
Pearson, Addison Wesley
Copyright:
2017
The following chapters and sections of the textbook should be covered.*
Chapter R:
All sections (Review as time permits.)
Chapter 1:
Sections 1.1 – 1.5 (Review as time permits.)
Chapter 1:
Sections 1.6 – 1.8
Chapter 2:
All sections (In section 2.6 the topic “continuity” is optional.)
Chapter 3:
Sections 3.1 – 3.5 (Omit section 3.6, xy = k should be
discussed here or in Chapter 6.)
Chapter 4:
All sections
Chapter 5:
Sections 5.1 - 5.3, 5.5, and 5.7 – 5.8. (Section 5.4 is optional. In
5.6, systems of inequalities is optional, but omit linear
programming.)
Chapter 6:
Sections 6.1 – 6.4 (In section 6.4, the concept of geometric
definitions of conic sections is optional.)
Chapter 7:
Sections 7.1 – 7.5
* Omit “modeling” exercises.
Use of Technology in Math 1431
The mathematics faculty recommends to all mathematics instructors that
any technology be allowed and encouraged in any level mathematics
course when it can be used by a student to either
1.
simplify calculations where the mechanics of the problem have
already been mastered or
2.
explore and experiment with concepts and problems that
enrich the understanding of the material that is being taught.
Videos for Math 1431
Instructional videos for Math 1431 are available to be viewed in the Math
Assistance Center and in the Library. They are also available for check out
in the Library.
In all Mathematics courses, students with a documented learning disability
that specifically requires a calculator as determined by Health Services, will
be allowed to use a basic calculator for all test/quiz questions where
arithmetic calculations are not the main objective. The specific disability
must be verified with Health Services before the accommodation can be
made.