Download Precalculus: Goals and Course Outline

Precalculus: Goals and Course Outline
DEPARTMENT: Mathematics
COURSE TITLE: Precalculus
 YEAR
 SEMESTER
DATE: 2007-2008
COURSE NUMBER: 5420
QPA: 4.5
CREDITS: 5
TEXTBOOK: (Title, Author, Publisher, Edition)
Precalculus: Mathematics for Calculus, Stewart, Redlin, Watson, Brooks/Cole, 2002
MATERIALS USED: TI-84 Plus graphing calculator, IBM ThinkPad, supplementary
materials
GOALS:
1.
2.
3.
4.
To provide the rigorous approach students will need for college level math.
To reinforce fundamental concepts learned in previous math courses.
To prepare students for Calculus.
To build new mathematical knowledge through problem solving that involves a variety of
strategies.
5. To master the language of mathematics by expressing mathematical ideas precisely using
correct mathematical notation.
6. To use graphing calculators and modern computer graphics to analyze, solve, visualize,
and clarify mathematical concepts.
CONTENT OF COURSE:
FIRST SEMESTER
Unit 1: Review Topics from Algebra II
1. Operations on monomials and polynomials
2. The real number systems
3. Union and intersection of sets
Unit 2: Relations and Functions
1. Introduction to families of functions
2. Four ways to represent functions
3. Domain and range
4. Rational expressions and complex fractions
5. Arithmetic operations on functions
6. Composition of functions
7. Inverses of functions
8. Vertical and horizontal line tests
Unit 3: Constant and Linear Functions
1. Domain and Range
2. Degree, coefficients, end behavior
3. Graphs of constant and linear functions
4. Slope, distance, and midpoint calculations
5. x- and y- intercept points
6. Point-slope, slope-intercept, and standard forms of equations
7. Parallel and perpendicular lines, perpendicular bisector
8. Systems of linear equations
9. Modeling with linear functions
10. Graphical analysis; determine domain interval where f  x   0 , f  x   0 , f  x   0 .
Unit 4: Quadratic Functions
1. Quadratic function, expression, equation, and the quadratic formula
2. Domain and range
3. Graphing quadratic functions - 5 key parts of a parabola
4. Roots/zeros of quadratic functions
5. Factoring
6. Completing the square
7. Modeling with quadratic functions
8. Intersection of two functions (linear and quadratic; quadratic and quadratic)
Unit 5: Complex Numbers, Quadratic Inequalities, and Circles
1. Complex numbers
2. The discriminant
3. Quadratic inequalities
4. Analysis of the domain and range
5. Circles
SECOND SEMESTER
Unit 6: Trigonometric Functions of Angles (Chapter 6)
1. Right triangle trigonometry and its application
2. 30-60-90 and 45-45-90 triangles
3. Six trigonometric functions
4. Degree measure and radian measure
5. Coterminal, reference, and quadrantal angles
6. “All Students Take Calculus” rule
7. Law of sines and cosines
8. Fundamental trigonometric identities
Unit 7: Trigonometric Functions of Real Numbers (Chapter 5)
1. 17 special angles used to graph the trigonometric functions
2. Terminal point of an arc and its coordinates
3. Graphs of the six trigonometric functions
4. Domain, range, period, amplitude, and phase-shift
Unit 8: Analytic Trigonometry (Chapter 7)
1. Trigonometric identities
2. The addition and subtraction formulas
3. Inverse sine and cosine functions
4. Analyzing the graphs of the inverse sine and cosine functions
Unit 9: Exponential and Logarithmic Functions (Chapter 4)
1. Graphs of exponential and logarithmic functions
2. Inverse of an exponential function and logarithmic function
3. Evaluating logarithmic expressions
4. Change of base formula
5. Expanding and condensing logarithmic expressions
6. Exponential and logarithmic equations
7. Applications of logarithms in problem solving
Unit 10: Polynomial Functions of Degree 3 or Higher (Chapter 3)
1. Graphs of polynomial functions
2. Local maxima and minima, and end behavior
3. Synthetic division
4. Factoring higher degree polynomials
5. Remainder and factor theorems
6. Rational zeros theorem
7. Finding zeros of polynomials
Precalculus Unit Proficiencies
Students will be able to…
FIRST SEMESTER
Unit 1: Review Topics from Algebra II
1. Add, subtract, multiply, and divide monomials and polynomials.
2. Identify rational, irrational, natural, whole numbers, and integers.
3. Find and graph intersection and union of sets.
4. Write sets in set-builder notation and interval notation.
Unit 2: Relations and Functions
1. Determine whether a relation is a function.
2. Describe the four ways to represent a function – verbal, table of values, graphic, and
algebraic.
3. Identify families of functions.
4. Find the domain and range of a given function, including restrictions on radical and
rational functions.
5. Graph piecewise defined functions.
6. Add and subtract rational expressions with unlike denominators and simplify complex
fractions.
7. Add, subtract, multiply, divide, and evaluate functions.
8. Compose functions and apply to real-life problems.
9. Differentiate the purposes of vertical and horizontal line tests
10. Identify a function as one-to-one and its implications on inverse functions.
11. Find the inverse of a function and verify two functions are inverses by using the property
of inverse functions.
12. Graph a function and its inverse (reflections across the identity).
Unit 3: Constant and Linear Functions
1. Classify a polynomial function by its degree and write its coefficients.
2. Graph a linear function, analyze the graph and separate the domain into intervals that
makes a function positive, negative, and zero, noticing the intercepts.
3. Describe the end behavior of a graph, using function and infinity notations.
4. Given two end points of a line segment, find the slope, distance, and midpoint.
5. Distinguish the point-slope form, slope-intercept form, and standard form.
6. Write equations of lines given different conditions, including parallel/perpendicular lines
and perpendicular bisector.
7. Solve systems of linear equations by graphing, substitution, and elimination.
8. Use a calculator to confirm the solution(intersecting point) of a linear system.
9. Apply linear systems in solving real life problems.
Unit 4: Quadratic Functions
1. Distinguish between a quadratic function, expression, equation, and the quadratic
formula.
2. Identify the domain and range of a quadratic function.
3. Find x- and y-intercepts, axis of symmetry, vertex, and point symmetric to y-intercept
of a quadratic function, and graph the function.
4. Describe the end behavior of the function.
5. Factor polynomials, including sum and difference of two cubes and factor by
grouping.
6. Find zeros of a quadratic function by factoring, completing the square, using
the quadratic formula.
7. Solve maximum/minimum problems by modeling a quadratic function.
8. Utilize the maximum, minimum, and zero features of the calculator in problem solving.
9. Find intersecting point(s) of a quadratic function and a linear function by solving
the system.
10. Find intersecting point(s) of two quadratic functions.
Unit 5: Complex Numbers, Quadratic Inequalities, and Circles
1. Add, subtract, multiply, divide complex numbers, then simplify them - including FOIL,
powers of i , and rationalizing the denominator.
2. Determine the nature of the roots by the value of the discriminant, stating the number
and type of solutions.
3. Solve and graph quadratic inequalities in one variable.
4. Determine the part of the domain that makes the value of the function positive, zero,
or negative.
5. Graph circles given equations in standard form.
6. Write an equation of a circle in standard form given the center and radius.
7. Convert an equation of a circle from general form to standard form.
SECOND SEMESTER
Unit 6: Trigonometric Functions of Angles(Chapter 6)
1. Find the trigonometric ratios of 30-60-90 triangle and 45-45-90 triangle without a
calculator.
2. Find the exact value of a trigonometric function of an angle associated with the special
triangles and quadrantal angles.
3. Identify quadrantal angles and their trigonometric values.
4. Find trigonometric functions of any angle.
5. Solve right triangles and apply these skills in problem solving.
6. Find the other five trigonometric ratios if given one trigonometric ratio.
7. Convert degree measure into radian measure and vice versa.
8. Find coterminal angles and reference angle of a given angle.
9. Identify positive and negative trigonometric values in all quadrants.
10. Apply the Law of Sines and Law of Cosines in solving triangles and word problems.
11. Find the six trigonometric values of an angle if a point is known on the terminal side.
12. Express a trigonometric function in terms of another by manipulating the fundamental
identities.
Unit 7: Trigonometric Functions of Real Numbers (Chapter 5)
1. Determine whether a point is on the unit circle.
2. Find the point on the unit circle if given one of the coordinates.
3. Find the x and y-coordinates of the 17 major points (0,  / 6,  / 4,  / 3,  / 2 .......... 2 )
on the unit circle and relate them to the six trigonometric ratios.
4. Identify the quadrant which contains the terminal point of an arc whose length is given in
terms of  .
5. Find the terminal point ( x , y ) on the unit circle determined by a real number.
6. Find the reference number and the terminal point determined by a real number.
7. Find all trigonometric functions from one given terminal point.
8. Find the exact value of the trigonometric function of a given number associated with
the 17 major points on the unit circle.
9. Evaluate trigonometric functions with or without a calculator.
10. Write from memory the reciprocal and Pythagorean identities.
11. Graph the six trigonometric functions by incorporating the amplitude, period and phase
shift.
Unit 8: Analytic Trigonometry (Chapter 7)
1. Simplify trigonometric expressions and verify trigonometric identities.
2. Find the exact value of a trigonometric expression by using the addition and subtraction
formulas.
3. Prove identities by using the addition and subtraction formulas.
4. Analyze and graph the inverse sine and cosine functions.
5. Evaluate the inverse sine and cosine functions.
Unit 9: Exponential and Logarithmic Functions (Chapter 4)
1. Identify whether the function represents exponential growth or decay.
2. Graph exponential and logarithmic functions by incorporating the domain, range,
intercepts, asymptote, and translations.
3. Find the inverse of an exponential function and logarithmic function.
4. Convert exponential form into logarithmic form and vice versa.
5. Evaluate logarithmic expressions with or without a calculator.
6. Apply the change of base formula to calculate logarithmic expressions.
7. Expand and condense logarithmic expressions by using the properties of logarithms.
8. Solve exponential and logarithmic equations.
9. Solve compound interest problems by applying logarithms.
Unit 10: Polynomial Functions of Degree 3 or Higher (Chapter 3)
1. Graph and analyze polynomial functions by incorporating intercepts, end behavior,
local maxima and minima, and translations.
2. Apply synthetic division to divide polynomials.
3. Apply the remainder theorem to evaluate polynomials.
4. Make the connection between the remainder theorem and factor theorem.
5. List the possible rational zeros of a polynomial function by using the rational zeros
theorem.
6. Find all zeros of a polynomial function by utilizing the remainder, factor, and rational
zeros theorems.
7. Find all zeros by factoring if possible.
8. Write a polynomial function if given its zeros.