Download College Algebra and Trigonometry with Applications

College Algebra and Trigonometry
with Applications
Course Outcome Summary
10-804-197
Information
Credits
5
Contact Hours
90
02/23/2004
Development Date
Types of Instruction
Type of Instruction
Contact Hours
Credits
Classroom Presentation
90
5
Totals
90
5
Competencies, Linked Exit Learning Outcomes, and
Performance Standards
1.
Apply mathematical problem solving skills
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
1.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
1.b.
by participating actively in class discussions and activities.
Performance will be successful when:
1.a.
1.b.
you show work in a clear and logical manner.
you verify solutions.
1.c.
you verify that the solution is within the stated range and reflect appropriate accuracy or
precision.
you label solutions with appropriate units.
1.d.
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College Algebra and Trigonometry with Applications
Outcome Summary
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Learning objectives
What you will learn as you master the competency:
a.
Review the method of verifying the solution to equations
2.
b.
Predict that the solution to an equation is within a range
c.
Review the determination of appropriate units for solutions to equations
d.
Describe what is necessary to present clear and logical mathematical work
Use the fundamental concepts of algebra
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
2.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
2.b.
by participating actively in class discussions and activities.
Performance will be successful when:
2.a.
you apply properties of real numbers.
2.b.
you locate real numbers on the number line.
2.c.
2.d.
you evaluate expressions using absolute value.
you relate absolute value to distances on the number line.
2.e.
you represent absolute value expressions without the use of the absolute value.
2.f.
you model a problem situation algebraically.
2.g.
you apply positive integer exponents, negative integer exponents, and rational number
exponents to evaluation and simplification of expressions.
2.h.
you relate roots to rational number exponents.
2.i.
2.j.
you define mononomial, binomial, trinomial, and polynomial.
you perform algebraic operations on polynomials.
2.k.
you apply the binomial theorem to expand powers of binomials.
2.l.
you factor polynomials.
Learning objectives
What you will learn as you master the competency:
a.
Identify properties of real numbers
b.
Describe the evaluation of absolute numbers
c.
Define inequality symbols
d.
Identify the location of numbers on a number line
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3.
e.
Recognize algebraic expressions with one or more algebraic terms
f.
g.
Review the properties of exponents
Evaluate algebraic expressions
h.
Describe the concept of factoring polynomials
i.
Describe the expansion of powers of binomial
Analyze the features of a graph of a given function or relation
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
Competence will be demonstrated:
3.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
3.b.
by participating actively in class discussions and activities.
Performance will be successful when:
3.a.
you use the Cartesian coordinate system.
3.b.
you find distances in the Cartesian coordinate system.
3.c.
you use function notation correctly.
3.d.
3.e.
you graph a relation or function in X and Y.
you differentiate between symmetry about the X-axis, the Y-axis, and the origin.
3.f.
you determine the intercept of a graph.
3.g.
you apply the vertical line test to distinguish between a relation and a function.
3.h.
you define a relation or function.
3.i.
you determine the domain and range of a relation or function.
3.j.
you define a 1-1 function.
3.k.
you apply the horizontal line test for 1-1.
3.l.
3.m.
you determine the inverse function of a 1-1 function.
you specify the relation between the domain and the range of a function and its inverse.
Learning objectives
What you will learn as you master the competency:
a.
Describe the Cartesian coordinate system
b.
Describe the process of finding the distance between two points on a coordinate grid
c.
Define functional notation
d.
Distinguish between a function and relation
e.
Define the concept of symmetry around an axis
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4.
f.
Describe the vertical line test
g.
h.
Distinguish between the domain and range of a function
Define the inverse of a function
i.
Explain the intercepts of a graph
Solve linear and quadratic equations and inequalities
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
4.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
4.b.
by participating actively in class discussions and activities.
Performance will be successful when:
4.a.
4.b.
you define a linear function, a linear equation, and a linear inequality.
you define a solution of an equation.
4.c.
you relate the solution of an equation to the X-intercept of a graph of the function and to
the root (or zero) of the function.
4.d.
you distinguish between an identity and a conditional equation.
4.e.
you define equivalent equations and inequalities.
4.f.
you solve linear equations and inequalities (with or without absolute values)
algebraically.
4.g.
you solve linear equations and inequalities (with or without absolute values) graphically.
4.h.
4.i.
you define quadratic equations.
you solve quadratic equations using: graphical methods, factoring, completing the
square, and the quadratic formula.
4.j.
4.k.
you solve 2nd or higher order equations or inequalities algebraically or graphically.
you model a verbal problem with an algebraic representation and a graph.
4.l.
you solve the algebraic representation of a verbal problem.
4.m.
you interpret the algebraic answer to a verbal problem in terms of the original problem.
4.n.
you use the equivalent interval notation, absolute value notation, and/or inequality
notation when writing solutions of inequalities.
Learning objectives
What you will learn as you master the competency:
a.
Describe the concept of linearity
b.
Relate the concept of the solution of an equation to the X intercept (or zero) of the
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equation
5.
c.
Review the concept of solving linear equations and inequalities both algebraically and
graphically
d.
Describe the methods of solving quadratic equations
e.
Distinguish between linear and quadratic equations
f.
Relate the verbal expression of an applied problem to it's mathematical representation
g.
Describe the proper notation when solving inequalities
h.
Distinguish between an identity and conditional equation
Analyze the properties of linear and quadratic functions
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
5.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
5.b.
by participating actively in class discussions and activities.
Performance will be successful when:
5.a.
you determine slop, X-intercept, and Y-intercept.
5.b.
you relate slope to parallel and perpendicular lines.
5.c.
you use point-slope, slope-intercept, and standard forms of the equation of a line.
5.d.
you convert between point-slope, slope-intercept, and standard forms of the equation of
a line.
5.e.
5.f.
you use the midpoint formula.
you find the equation of a line that is parallel to or perpendicular to a given line through
a specified point.
5.g.
5.h.
you find the equation of the perpendicular bisector of a given line segment.
you prove geometric theorems algebraically.
5.i.
you graph quadratic functions.
5.j.
you apply the concepts of vertically stretching or shrinking or shrinking, reflecting, or
shifting a graph of a quadratic to transform the graph.
5.k.
you determine the sum, difference, product, and quotient of two functions.
5.l.
you determine the composition of two functions.
5.m.
you note that function composition is non-commutative.
5.n.
you apply geometric transformations as compositions.
5.o.
you determine the vortex, axis of symmetry, and direction from the standard form of a
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quadratic function.
5.p.
5.q.
you change the form of a quadratic function to a standard form.
you find zeros of quadratic functions algebraically and graphically.
Learning objectives
What you will learn as you master the competency:
6.
a.
b.
Define the slope, X and Y intercepts of a line
Describe the forms of an equation of a line
c.
Describe the concept of parallel and perpendicular lines through their mathematical
expressions
d.
Describe the properties of functions
e.
Explain transformation of functions
f.
Review the concepts associated with quadratic equations
Use theories of equations to find the zeros of a polynomial function
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
6.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
6.b.
by participating actively in class discussions and activities.
Performance will be successful when:
6.a.
you define the degree of a polynomial function.
6.b.
you identify local maxima and minima of a polynomial function using graphical and
analytical methods.
6.c.
you determine whether a function is increasing or decreasing on an interval.
6.d.
you apply the concepts of continuity, discontinuity, the intermediate value theorem, and
the bisection method of finding a zero to determine the features of a graph.
6.e.
you determine the behavior of a polynomial function at the end points of its domain.
6.f.
you apply transformations to polynomial functions.
6.g.
you apply the remainder theorem, the factor theorem, the rational roots theorem,
synthetic division, the upper and lower bounds theorem, and Descartes' law of signs to
find zeros of a polynomial function.
6.h.
you use the fundamental theorem of algebra to determine the number of roots of a
polynomial equation.
you use graphical methods to estimate the roots of a polynomial equation.
6.i.
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Outcome Summary
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Learning objectives
What you will learn as you master the competency:
a.
Identify the degree of a polynomial
7.
b.
Describe the methods of finding the zeros of polynomial functions
c.
Define the properties of polynomial functions of higher degree
d.
e.
Explain graphical aspects of polynomial functions
Explain analytical aspects of polynomial functions
Determine complex solutions to polynomial equations
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
7.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
7.b.
by participating actively in class discussions and activities.
Performance will be successful when:
7.a.
you define the imaginary unit i.
7.b.
7.c.
you define complex numbers and complex conjugates.
you perform operations with complex numbers including addition, subtraction,
multiplication, and division.
7.d.
you apply the fundamental theorem of algebra and the linear factorization theorem to
determine solutions to polynomial equations.
7.e.
you find complex (non-real) solutions to polynomial equations.
Learning objectives
What you will learn as you master the competency:
8.
a.
Define the concept of a complex number including it's representation and conjugate
b.
Explain the process of performing arithmetic operations on complex numbers
c.
Describe the method of finding imaginary solutions to polynomial equations
d.
Describe the application of fundamental theorem of algebra to determine the solution to
polynomial equations
e.
Describe the application of the linear factorization theorem to determine the solution to
polynomial equations
Perform computations with rational functions
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Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
8.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
8.b.
by participating actively in class discussions and activities.
Performance will be successful when:
8.a.
you define a rational expression.
8.b.
you recognize a rational expression.
8.c.
you simplify rational expressions.
8.d.
8.e.
you add, subtract, multiply and divide rational expressions.
you simplify complex fractions.
8.f.
you define a rational function and the domain of a rational function.
8.g.
you find vertical, horizontal, and oblique asymptotes.
8.h.
you discuss the properties of f(x) = 1/x, graph it and apply transformations to rational
functions.
8.i.
you determine the behavior of rational functions as the absolute value of the variables
becomes large.
8.j.
you identify removable discontinuities.
8.k.
you determine the end behavior for rational functions where the degree of the numerator
does not exceed the degree of the denominator.
8.l.
you graph rational functions.
8.m.
you solve rational functions algebraically and graphically.
8.n.
you discuss extraneous solutions.
Learning objectives
What you will learn as you master the competency:
a.
b.
Define a rational expression
Describe the process of simplifying a rational expression
c.
Explain the performance of arithmetic operations on rational expressions
d.
Describe the process of finding asymptotes
e.
Describe the properties and behavior of rational functions and their transformations
f.
Discuss the process of graphing rational functions
g.
Describe the process of solving rational functions graphically
h.
Describe the process of solving rational functions algebraically
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9.
Perform computations with radical functions
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
9.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
9.b.
by participating actively in class discussions and activities.
Performance will be successful when:
9.a.
you define radical, index, and radicand.
9.b.
you define principal square root, cube root, nth root.
9.c.
9.d.
you determine the domain of radical expression.
you solve radical equations algebraically and check for extraneous roots.
Learning objectives
What you will learn as you master the competency:
10.
a.
b.
Define radical, index, and radicand
Define principal square root, cube root, nth root
c.
Describe the process of determining the domain of radical expressions
d.
Describe the process of solving radical equations algebraically
e.
Explain the process of checking for extraneous roots in solving radical equations
Analyze exponential and logarithmic functions
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
Competence will be demonstrated:
10.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
10.b.
by participating actively in class discussions and activities.
Performance will be successful when:
10.a.
you define base and exponent.
10.b.
you define the exponential function with base 10, base 2, fractional base, an arbitrary
base, and base e.
10.c.
you graph the exponential function with base 10, base 2, fractional base, an arbitrary
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base, and base e.
10.d.
10.e.
you apply transformations to exponential functions.
you apply exponential functions to such problems as exponential growth and decay,
half-life, interest, annuities, and mortgages.
10.f.
you solve exponential equations algebraically and graphically.
10.g.
you define logarithmic functions with base a- also discuss possible values for the base.
10.h.
you define the common logarithmic function and the natural logarithmic function.
10.i.
you discuss the inverse relationship between the logarithmic function and the
exponential function (same base).
10.j.
you determine the domain and range of logarithmic functions.
10.k.
you produce the graph of a logarithmic function (with various bases) considering the
domain and the range.
10.l.
you solve exponential and logarithmic functions algebraically and graphically.
Learning objectives
What you will learn as you master the competency:
11.
a.
Describe exponential functions of any base
b.
Describe the application of exponential and logarithmic functions
c.
d.
Define the common logarithmic function and the natural logarithmic function
Discuss determining the domain and range of logarithmic functions
e.
Discuss the inverse relationship between the logarithmic function and the exponential
function
f.
Explain the solution of exponential and logarithmic functions algebraically and
graphically.
Solve non-linear systems of equations
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
Competence will be demonstrated:
11.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
11.b.
by participating actively in class discussions and activities.
Performance will be successful when:
11.a.
you solve a nonlinear system of equations algebraically.
11.b.
you use graphing calculator to solve non-linear systems of equation.
11.c.
you use nonlinear systems of equation to solve applied problems.
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Learning objectives
What you will learn as you master the competency:
a.
Describe the methods of solving a nonlinear system of equations algebraically
12.
b.
Discuss the use of a graphing calculator to solve non-linear systems of equation
c.
Explain the use of nonlinear systems of equations to solve applied problems
Solve systems of linear equations
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
12.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
12.b.
by participating actively in class discussions and activities.
Performance will be successful when:
12.a. you solve systems of linear equations in three or more variables algebraically.
12.b.
you use the graphing calculator to solve systems of linear equations in three or more
variables.
12.c.
you use systems of three equations to solve applied problems.
Learning objectives
What you will learn as you master the competency:
13.
a.
Review thee solution of systems of linear equations in three or more variables
algebraically
b.
Describe the use of the graphing calculator to solve systems of linear equations in three
or more variables
c.
Describe the use of systems of three equations to solve applied problems
Perform basic operations with matrices
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
13.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
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13.b.
by participating actively in class discussions and activities.
Performance will be successful when:
13.a.
you add matrices.
13.b.
you subtract matrices.
13.c.
13.d.
you multiply a matrix by a scaler.
you multiply matrices when possible.
13.e.
you solve application problems using basic operations with matrices.
13.f.
you perform basic operations with matrices using the graphing calculator.
Learning objectives
What you will learn as you master the competency:
14.
a.
Describe the process of adding and subtracting matrices
b.
Describe the process of multiplying a scalar by a matrix
c.
d.
Discuss the conditions under which it is possible to multiply matrices
Explain the process of multiplying matrices
e.
Explain the use of a graphing calculator to perform matrix operations
f.
Discuss solving application problems using basic operations with matrices.
Use the inverse of a square matrix
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
14.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
14.b.
by participating actively in class discussions and activities.
Performance will be successful when:
14.a. you recognize the identity matrix of a square matrix.
14.b.
you recognize when two square matrices are inverses of one another.
14.c.
you find the inverse of a square matrix, if it exists.
14.d.
you use inverses of matrices to solve systems of equations.
14.e.
you solve application problems involving matrix inverses.
14.f.
you use a graphing calculator to compute matrix inverses.
Learning objectives
What you will learn as you master the competency:
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15.
a.
Recognize the identity matrix of a square matrix
b.
c.
Recognize when two square matrices are inverses of one another
Describe the steps to find the inverse of a square matrix, if it exists
d.
Explain the use inverses of matrices to solve systems of equations
e.
Describe the solution of application problems involving matrix inverses
f.
Describe the use a graphing calculator to compute matrix inverses
Solve systems of equations using matrix equations
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
Competence will be demonstrated:
15.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
15.b.
by participating actively in class discussions and activities.
Performance will be successful when:
15.a.
you identify a matrix equation.
15.b.
you identify a matrix equation.
15.c.
15.d.
you write a system of linear equations as a matrix equation.
you find the solution of a system of linear equations by using inverses of matrices.
15.e.
you solve application problems using systems of linear equations and matrix equations.
Learning objectives
What you will learn as you master the competency:
16.
a.
Identify a matrix equation
b.
Describe the writing of a system of linear equations as a matrix equation.
c.
Describe finding the solution of a system of linear equations by using inverses of
matrices
d.
Discuss the solution of application problems using systems of linear equations and
matrix equations
Solve systems of linear inequalities
Linked Core Abilities
A.
Learn effectively
Performance Standards
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Competence will be demonstrated:
16.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
16.b.
by participating actively in class discussions and activities.
Performance will be successful when:
16.a.
you graph linear inequalities algebraically.
16.b.
you graph systems of linear inequalities algebraically.
Learning objectives
What you will learn as you master the competency:
17.
a.
Explain the process of graphing linear inequalities algebraically
b.
Explain the process of graphing systems of linear inequalities algebraically
Produce the graph of a conic section
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
Competence will be demonstrated:
17.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
17.b.
by participating actively in class discussions and activities.
Performance will be successful when:
17.a. you distinguish between the type of conic sections produced when a quadratic
polynomial in two unknowns is graphed.
17.b. you find the center and the semi-major and semi-minor axes given the polynomial form
of the equation of an ellipse.
17.c.
you find the center, the axes, and the asymptotes given the polynomial form of the
equation of a hyperbola.
17.d.
you find the center, the axis, and the directrix given the polynomial form of the equation
for a parabola.
17.e.
you sketch the graph of a conic section given its equation in polynomial form.
Learning objectives
What you will learn as you master the competency:
a.
b.
Distinguish between the type of conic sections produced when a quadratic polynomial in
two unknowns is graphed
Explain how to find the center and the semi-major and semi-minor axes given the
polynomial form of the equation of an ellipse
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18.
c.
Explain how to find the center, the axes, and the asymptotes given the polynomial form
of the equation of a hyperbola
d.
Explain how to find the center, the axis, and the directrix given the polynomial form of
the equation for a parabola
e.
Describe the process of sketching the graph of a conic section given its equation in
polynomial form
Solve problems involving sequences and series
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
18.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
18.b.
by participating actively in class discussions and activities.
Performance will be successful when:
18.a.
you find terms of sequences given the nth term.
18.b.
you find a general term for a sequence.
18.c.
you convert between sigma notation and other notation for a series.
18.d.
you find the nth term of an arithmetic and geometric sequence.
18.e.
you find the common difference of an arithmetic sequence.
18.f.
you construct an arithmetic and geometric sequence.
18.g.
you find the common ratio of a geometric sequence.
18.h.
you find the sum of the first n terms of an arithmetic and geometric sequence.
18.i.
you find the sum of an infinite geometric series, if is exists.
Learning objectives
What you will learn as you master the competency:
a.
Describe the process of finding terms of sequences given the nth term
b.
Describe the process of finding a general term for a sequence
c.
Explain sigma notation and other notations and the conversion between them
d.
Describe the process of finding the nth term of an arithmetic and geometric sequence
e.
Describe the process of finding the common difference of an arithmetic sequence
f.
Explain the construction of an arithmetic and geometric sequence
g.
Describe the process of finding the common ratio of a geometric sequence
h.
Describe the process of finding the sum of the first n terms of an arithmetic and
geometric sequence
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i.
19.
Describe the process of finding the sum of an infinite geometric series, if is exists
Use the Binomial Theorem
Linked Core Abilities
A.
Learn effectively
Performance Standards
Competence will be demonstrated:
19.a.
by submitting all in-class and take home assignments with passing grades according to
the grading scale in the syllabus.
19.b.
by participating actively in class discussions and activities.
Performance will be successful when:
19.a. you expand a power of a binomial using Pascal's triangle or factorial notation.
19.b.
you find a specific term of binomial expansion.
Learning objectives
What you will learn as you master the competency:
a.
Recognize Pascal's triangle
20.
b.
Explain the process of expanding a power of a binomial using Pascal's triangle or
factorial notation
c.
Define binomial expansion
d.
Describe the process of finding a specific term of binomial expansion.
Define the trigonometric functions
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
20.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
20.a.
you graph angles in standard position.
20.b.
20.c.
you solve problems involving special right triangles.
you use trigonometric functions defined by using a point on the terminal side of an
angle.
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20.d.
you use the trigonometric functions defined on the basis of a right triangle.
20.e.
20.f.
you prove simple trigonometric identities.
you use degree-measures of angles including decimals as well as minutes and seconds.
Learning objectives
What you will learn as you master the competency:
21.
a.
b.
Define angles and their measure
Identify the terminal points of an angle on a coordinate grid
c.
Identify sides of a right triangle in relation to an angle
d.
Find sides of a right triangle using the Pythagorean Theorem
e.
f.
Define trigonometric functions using the sides of a right triangle
Find values of trigonometric functions using the terminal point of a right triangle
g.
Identify the angles on a right triangle to determine trigonometric identities
Evaluate the trigonometric functions
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
21.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
21.a.
you identify exact values of trigonometric functions where appropriate.
21.b.
you find approximate values of trigonometric functions using a calculator.
21.c.
you use inverse trigonometric calculator functions to find measures of acute angles.
21.d.
you solve right triangles using trigonometric functions and other properties of such
triangles.
you use vectors applying a geometric approach.
21.e.
Learning objectives
What you will learn as you master the competency:
a.
Find values of trigonometric functions using length of sides of a right triangle
b.
Explain the steps to determine the values of trigonometric functions using a calculator
c.
Explain the steps to determine the measures of acute angles using inverse functions of a
calculator
d.
Find all sides and angles of a right triangle
e.
Identify components of vectors using a geometric approach
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22.
Use radian measures of angles
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
22.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
22.a.
you use the concept of a reference angle.
22.b.
22.c.
you relate radian measures to real numbers.
you convert between radian measures and degree measures of angles.
22.d.
you find exact and approximate values of angles described in radians.
22.e.
you use trigonometric functions defined for the unit circle.
22.f.
22.g.
you use the concept arc length and area of a sector.
you use the concepts of linear velocity and angular velocity.
22.h.
you convert between linear velocity and angular velocity.
Learning objectives
What you will learn as you master the competency:
a.
Determine the reference angle given an angle of any degree
23.
b.
Describe radians in relation to real numbers
c.
Find arc length of a circle using radians
d.
Convert between radian and degree measures of angles
e.
Find the area of a sector of a circle
f.
Find the linear speed of an object traveling in circular motion
g.
Convert between linear and angular velocity
h.
Find the exact value of the trigonometric functions using a point on the unit circle
Interpret graphs of trigonometric functions
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
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23.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
23.a.
you use basic graphs of trigonometric functions.
23.b.
you identify domains and ranges of trigonometric functions using correct notation.
23.c.
you identify undefined and restricted values for each basic trigonometric function.
23.d.
you use concept of amplitude, period, and phase shift related to the graphs of
trigonometric functions.
Learning objectives
What you will learn as you master the competency:
24.
a.
Identify the components of graph values of trigonometric functions
b.
Determine the domain and ranges of trigonometric functions
c.
Determine the cases where division by zero renders a trigonometric function to be
undefined
d.
Determine the values to which basic trigonometric functions are restricted
e.
Determine amplitude, period and phase shift in sinusoidal functions
Graph equations of basic and modified trigonomertric functions
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
24.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
24.a.
24.b.
you sketch graphs of one period of basic trigonometric functions clearly and accurately.
you graph equations of trigonometric functions with variations in amplitude and/or
period and with/without phase shifts.
24.c.
you find equations from graphs of trigonometric functions.
Learning objectives
What you will learn as you master the competency:
a.
Identify the steps necessary to graph one period of trigonometric functions
b.
Explain the steps necessary to graph trigonometric functions with variations of
amplitude, period and phase shifts
c.
Explain the steps necessary to derive trigonometric functions from their graphs
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25.
Define the inverse of a trigonometric functions
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
25.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
25.a.
you relate the inverse of trigonometric functions to the functions themselves.
25.b.
25.c.
you identify domains and ranges of inverse trigonometric functions.
you sketch graphs of the inverse of trigonometric functions.
25.d.
you evaluate the inverse of trigonometric functions for given values of the domain.
25.e.
you find inverse trigonometric function values, with and without a calculator.
Learning objectives
What you will learn as you master the competency:
26.
a.
Outline the relationship of trigonometric functions to the inverse of trigonometric
functions
b.
c.
Describe the relationship between domains and ranges of trigonometric functions and
their inverse
Describe the steps necessary to sketch the graphs of inverse trigonometric functions
d.
Identify the inverse of trigonometric functions given the values of the domain
e.
Relate the steps necessary to find the inverse of trigonometric functions with and
without a calculator
Derive trigonometric identities
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
26.a. By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
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26.a.
you use the reciprocal, ratio, Pythagorean and other established identities to derive new
identities.
26.b.
you use the techniques of algebra to prove new identities.
Learning objectives
What you will learn as you master the competency:
27.
a.
Describe the processes involved in using established mathematical identities to derive
new ones
b.
Relate the use of algebra to prove new identities
Apply new trigonometric identities
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
27.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
27.a.
you use sum and difference formulas to evaluate trigonometric functions of angles.
27.b.
you use sum and difference formulas to prove other identities.
27.c.
you use double-angle and half-angle formulas to evaluate trigonometric functions of
angles.
27.d.
you use double-angle and half-angle formulas to prove other identities.
27.e.
you use sum-to-product and product-to-sum formulas to prove other identities.
Learning objectives
What you will learn as you master the competency:
28.
a.
Explain how sum and difference formulas are used in evaluating trigonometric functions
of angles
b.
Explain how sum and difference formulas are used to prove identities
c.
Explain how double-angle and half-angle formulas are used to evaluate trigonometric
functions
d.
Explain how double-angle and half-angle formulas are used to prove identities
e.
Explain the use of sum-to-product and product-to-sum formulas to prove identities
Solve trigonometric equations
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Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
28.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
28.a.
you use algebraic techniques and trigonometric identities to solve equations involving
trigonometric expressions of a single angle.
28.b.
you use algebraic techniques and trigonometric identities to solve equations involving
trigonometric functions of multiple angles.
Learning objectives
What you will learn as you master the competency:
29.
a.
Relate the use of algebraic techniques and trigonometric identities in solving
trigonometric equations of a single angle
b.
Relate the use of algebraic techniques and trigonometric identities in solving
trigonometric equations involving multiple angles
Perform operations with parametric equations
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
29.a. By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
29.a.
you eliminate the parameter from a pair of parametric equations.
29.b.
you sketch the graphs which result from eliminating the parameter from a pair of
parametric equations.
Learning objectives
What you will learn as you master the competency:
a.
Define the meaning of parametric equation
b.
Identify the parameters in a parametric equation
c.
Describe the steps necessary to eliminate the parameter in a pair of parametric equations
d.
Relate the steps involved in drawing the graphs that result from eliminating the
parameter from a pair of parametric equations
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30.
Solve oblique triangles
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
30.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
30.a.
you use the Law of Sines, the Law of Cosines, and properties of triangles to solve
oblique triangles.
30.b.
you calculate areas of oblique triangles using formulas that involve trigonometric
functions.
30.c.
you calculate areas of oblique triangles for which the lengths of all sides are known.
30.d.
you solve application problems by solving oblique triangles.
Learning objectives
What you will learn as you master the competency:
a.
Define the Law of Sines, the Law of Cosines
b.
Describe the steps in calculating the areas of oblique triangles using formulas involving
trigonometric functions
c.
Describe the steps in calculating the areas of oblique triangles for which all sides are
known
Apply the solutions to oblique triangles to applied problems
d.
31.
Perform vector operations
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
31.a. By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
31.a.
you define a vector algebraically.
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31.b.
you define unit vectors i and j.
31.c.
31.d.
you identify the horizontal and vertical components of a vector.
you find the magnitude of a vector.
31.e.
you add and subtract algebraic vectors.
31.f.
you perform scalar multiplication of vectors.
31.g.
31.h.
you obtain dot product of vectors.
you determine if two vectors are perpendicular or not.
31.i.
you find the angle between two vectors.
Learning objectives
What you will learn as you master the competency:
a.
Describe the concept of an algebraic vector.
32.
b.
Describe the unit vectors commonly referred to as i and j
c.
Explain the horizontal and vertical components of a vector
d.
e.
Explain the magnitude of a vector
Describe the process of adding and subtracting algebraic vectors
f.
Describe the process of scalar multiplication of vectors
g.
Describe the process of obtaining dot products of vectors
h.
i.
Relate the conditions necessary for two vectors to be perpendicular
Describe the process of determining the angles between two vectors
Use trigonometry in vector applications
Linked Core Abilities
B.
Apply relevant technologies
Performance Standards
You will demonstrate your competence:
32.a. By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
32.a.
you find the resultant of the combination of vectors.
32.b.
you apply trigonometry to vector applications such as physics or engineering.
Learning objectives
What you will learn as you master the competency:
a.
Define the x and y components of vectors
b.
Describe the steps necessary to find the resultant of the combination of vectors
c.
Relate the application of trigonometry to vectors in such fields as physics and
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engineering
33.
Peform operations with complex numbers in trigonometry
Linked Core Abilities
A.
Learn effectively
Performance Standards
You will demonstrate your competence:
33.a.
By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
33.a.
you transform complex numbers from rectangular form to trigonometric form and vice
versa.
33.b.
you graph complex numbers in trigonometric form and rectangular form.
33.c.
33.d.
you find products and quotients of complex numbers written in trigonometric form.
you find powers and roots of complex numbers written in trigonometric form.
33.e.
you summarize DeMoivre's theorem.
33.f.
you use DeMoivre's theorem to find powers and roots of complex numbers.
Learning objectives
What you will learn as you master the competency:
34.
a.
Define the concept of complex numbers
b.
Describe the process of transforming complex numbers between rectangular form and
trigonometric form
c.
Explain the process of graphing complex numbers in trigonometric and rectangular form
d.
Relate the process of determining products and quotients of complex numbers in
trigonometric form
e.
Relate the process of determining powers and roots of complex numbers in
trigonometric form
f.
Explain DeMoivre's theorem
g.
Describe the steps in using DeMoivre's theorem to find powers and roots of complex
numbers
Relate the polar coordinate system to the rectangular coordinate system
Linked Core Abilities
A.
Learn effectively
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Performance Standards
You will demonstrate your competence:
34.a. By a satisfactory score on all tests, quizzes, and/or graded assignments incorporating this
competency.
Your performance will be successful when:
34.a.
you represent a point on the plane in both rectangular and polar coordinates.
34.b.
you convert an equation in polar coordinates to rectangular coordinates and vice versa.
34.c.
you graph equations in polar form.
Learning objectives
What you will learn as you master the competency:
a.
Describe the polar coordinate system
b.
c.
Describe how a point on a plane is represented in the rectangular and polar system
Explain the process of converting an equation between the polar and rectangular system
d.
Describe the process of graphing equations in polar form
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Current Date: 08/19/05D:\582735904.doc