Download Mapping for Instruction - First Nine Weeks

Roanoke County Public Schools
College Algebra
Curriculum Guide
Revised, 2012
College Algebra Curriculum Guide
2012
Mathematics Curriculum Guide
Revised 2012. Available at www.rcs.k12.va.us.
Roanoke County Public Schools does not discriminate with regard to race, color, age, national origin, gender, or handicapping condition in an
educational and/or employment policy or practice. Questions and/or complaints should be addressed to the Deputy Superintendent/Title IX
Coordinator at (540) 562-3900 ext. 10121 or the Director of Pupil Personnel Services/504 Coordinator at (540) 562-3900 ext. 10181.
Acknowledgements
The following people have made tremendous contributions to the completion of this curriculum guide and all are appreciated.
Nancy Hoffman
Cave Spring High School
Tamara Miniclier
Cave Spring High School
Jim Wolfe
Northside High School
Victor Maciel
William Byrd High School
Roanoke County Public Schools Administration
Dr. Lorraine Lange
Superintendent
Cecil Snead
Director of Secondary Instruction
Rebecca Eastwood
Director of Elementary Instruction
Linda Bowden
Mathematics Coordinator
College Algebra Curriculum Guide
2012
Preface
This curriculum guide is written for the teachers to assist them in using the textbooks/resources in a most effective way. This guide will assist the mathematics
teacher in preparing students for the challenges of the twenty-first century. As established by the National Council of Teachers of Mathematics Principles and
Standards for School Mathematics, educational goals for students are changing. Students should have many and varied experiences in their mathematical
training to help them learn to value mathematics, become confident in their ability to do mathematics, become problem solvers, and learn to communicate and
reason mathematically. This guide, along with the available textbook resources, other professional literature, alternative assessment methods, and varied
instruction in-service activities will assist the mathematics teacher in continuing to integrate these student goals into the curriculum.
Table of Contents
Introduction/General Comments ............................................................................................................................................. i
Textbook/Resources Overview ................................................................................................................................................ i
Sequence of Instruction and Pacing Suggestions ................................................................................................................... ii
Mapping for Instruction - First Nine Weeks ............................................................................................................................ 1
Mapping for Instruction - Second Nine Weeks……………………………………………………………………………………………….4
Mapping for Instruction - Third Nine Weeks………………………………………………………………………………………………….8
Mapping for Instruction - Fourth Nine Weekds…………………………………………………………………………………………….11
List of Standards……………………………………………………………………………………………………………………………………14
Supplemental Resources ......................................................................................................................................................... 4
College Algebra Curriculum Guide
2012
Introduction/General Comments
College Algebra is a college preparatory course that extends topics from Algebra II and introduces additional ones. One-quarter of the course deals with
trigonometric topics. The remainder of the course concentrates on the study of functions, matrices, and conics. This course is designed for college bound
students in preparation for the more in-depth course of Precalculus. To support the current trends from institutes of higher learning regarding calculator usage,
methods of calculation in this course will include mental math, paper-pencil, computer, and scientific and graphics calculators. In certain topics, use of the
graphics calculator will be limited.
The teacher should consider the diverse mathematical backgrounds, experiences, and ages of his/her students when preparing lessons and answering questions.
Textbook/Resources Overview
Course Title: College Algebra (3150)
Course Text: Algebra and Trigonometry, 4th edition
Publisher: Pearson
Author: Robert Blitzer
Teacher Supplements:
Instructor's Edition
Instructor's Solution Manual
Student Solution Manual
TestGen CD-ROM
Video Lecture DVDs
Test Item File (download)
PowerPoint Slides (download)
Mini Lecture Notes (download)
Student Supplements:
Math XL Tutorials on CD-ROM
i
College Algebra Curriculum Guide
2012
Sequence of Instruction and Pacing Suggestions
First Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
MPE.1, MPE.18, MPE.24
Chapter P / Sections 1 – 6 / Prerequisites: Fundamental Concepts of Algebra I
10 blocks
MPE.12, MPE.18, MPE.25,
MPE.26
Chapter 1 / Sections 2 – 7 / Equations and Inequalities
10 blocks
Project
2.5 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
First Nine Weeks Total
22.5 blocks
Second Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
MPE.13, MPE.14, MPE.15,
MPE.16, MPE.19
Chapter 2 / Sections 1, 3, 5, 2 / Functions and Graphs
8.5 blocks
MPE.13
Chapter 3 / Section 1 / Quadratic Functions
3 blocks
MPE.2, MPE.16
Chapter 4 / Sections 1 – 5 / Exponential and Logarithmic Functions
8.5 blocks
Project
2.5 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for 45
days in the middle schools.
ii
Second Nine Weeks Total
22.5 blocks
College Algebra Curriculum Guide
2012
Sequence of Instruction and Pacing Suggestions
Third Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
MPE.5, MPE.11, MPE.20,
MPE.27
Chapter 5 / Sections 1 – 6 / Trigonometric Functions
11 blocks
MPE.5
Chapter 5 / Sections 2, 8 / Solving Right Triangles and Applications
2.5 blocks
Chapter 7 / Sections 1, 2 / Solving Triangles and Applications
3.5 blocks
Chapter 6 / Section 1 / Analytic Trigonometry
3.5 blocks
Project
2 blocks
MPE.36
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
Third Nine Weeks Total
22.5 blocks
Fourth Nine Weeks
SOL
Chapter/Sections/Topic
*Time Frame
Chapter 8 / Section 1 – 2 / Systems of Equations
5 blocks
Chapter 9 / Sections 1 – 5 / Matrices and Determinants
7.5 blocks
MPE.21, MPE.29
Chapter 2 / Section 8 / Distance and Midpoint Formulas
1 block
MPE.29
Chapter 10 / Sections 1 – 3 / Conic Sections and Analytic Geometry
7 blocks
Project
2 blocks
*Time Frame is based on 95 minutes of instruction per block.
Math 6, 7 and 8, Pre- Algebra, Algebra I and Geometry require one 95 minute block per day for
45 days in the middle schools.
iii
Fourth Nine Weeks Total
22.5 blocks
Mapping for Instruction - First Nine Weeks
Chapter P: Prerequisites: Fundamental Concepts of Algebra I
SOLs
Textbook Chapters/Sections/Topics
MPE.1
Chapter P/Section 1/Algebraic
Solve practical problems involving
Expressions, Mathematical Models,
rational numbers (including numbers and Real Numbers
in scientific notation), percents, ratios
and proportions.
MPE.24
Describe orally and in writing the
relationships betweenthe subsets of
the real number systems.
Supporting Materials
Notes: Real Numbers
WS: Sets and Set Notation
WS: Simplifying Expressions
MPE.1
Chapter P/Section 2/Exponents and
Solve practical problems involving
Scientific Notation
rational numbers (including numbers
in scientific notation), percents, ratios
and proportions.
Comments
Properties may be
deemphasized according to
teacher preference.
1 block
Scientific Notation may be
omitted.
1 block
MPE.18
Chapter P/Section 3/Radicals and
Given rational, radical, or polynomial Rational Exponents
expressions
b) add, subtract, multiply, divide, and
simplify radical expressions
containing rational numbers and
variables, and expressions containing
rational exponents;
c) write radical expressions as
expressions containing rational
exponents and vice versa;
2 blocks
Chapter P/Section 4/Polynomials
MPE.18
Given rational, radical, or polynomial
expressions
d) factor polynomials completely.
Chapter P/Section 5/Factoring
Polynomials
Time Frame
0.5 block
Factoring Overview
1
Factoring with fractional
exponents does not need to be
included.
1 block
Chapter P: Prerequisites: Fundamental Concepts of Algebra I
SOLs
MPE.18
Given rational, radical, or polynomial
expressions
a) add, subtract, multiply, divide, and
simplify rational algebraic
expressions
Textbook Chapters/Sections/Topics
Supporting Materials
Comments
Chapter P/Section 6/Rational
Expressions
Time Frame
2 blocks
In this guide, “review and
assessment” blocks are divided
by chapter. The instructor my
place these blocks during the
chapter as to his/her
preference.
Review/Assessment
2.5 blocks
Chapter 1: Equations and Inequalities
SOLs
Textbook Chapters/Sections/Topics
MPE.12
Transfer between and analyze
multiple representations of functions,
including algebraic formulas, graphs,
tables, and words. Select and use
appropriate representations for
analysis, interpretation, and
prediction.
Chapter 1/Section 1/Graphs and
Graphing Utilities
MPE.26
Solve, algebraically and graphically,
(c) equations containing rational
algebraic expressions; and Use
graphing calculators for solving and
for confirming the algebraic solutions.
Chapter 1/Section 2/Linear Equations
and Rational Equations
Supporting Materials
Comments
Time Frame
OMIT THIS SECTION
1 block
2
Chapter 1: Equations and Inequalities
SOLs
Textbook Chapters/Sections/Topics
MPE.26
Solve, algebraically and graphically,
(c) equations containing rational
algebraic expressions; and Use
graphing calculators for solving and
for confirming the algebraic solutions.
Chapter 1/Section 3/Models and
Applications
MPE.25
Perform operations on complex
numbers, express the results in
simplest form using patterns of the
powers of i, and identify field
properties that are valid for the
complex numbers.
Chapter 1/Section 4/Complex
Numbers
Supporting Materials
Comments
WS: Word Problems; D = RT
WS: Word Problems; Interest
and Mixture
Time Frame
1.5 blocks
1 block
MPE.26
Chapter 1/Section 5/Quadratic
Solve, algebraically and graphically,
Equations
(b) quadratic equations over the set of
complex numbers; Use graphing
calculators for solving and for
confirming the algebraic solutions.
Completing the square may be
reserved until working with
conic sections.
1.5 blocks
MPE18.
Given rational, radical, or polynomial
expressions,
(d) factor polynomials completely
Chapter 1/Section 6/Other Types of
Equations
The topics that should be
covered in this section are
radical equations and absoute
value equations.
0.5 blocks
MPE26
Solve, algebraically and graphically,
(a) absolute value equations and
inequalities Use graphing calculators
for solving and for confirming the
algebraic solutions
Chapter 1/Section 7/Linear
Inequalities and Absolute Value
Inequalities
Quadratic Inequalities from
section 3.6 should be included
at this point.
1.5 blocks
Review/Assessment
3 blocks
3
Capstone Project 1
Title
Math or Magic?
Applicable Chapters
Weblink
Chapter P
https://sites.google.com/site/mathematicscapstonecourseu
Chapter 1
nits/home/tasks/MathOrMagic.pdf?attredirects=0
Additional Projects may be used from the following link:
https://sites.google.com/site/mathematicscapstonecourseunits/home
4
Mapping for Instruction - Second Nine Weeks
Chapter 2: Functions and Graphs
SOLs
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
MPE.16
Chapter 2/Section 1/Basics of
Investigate and analyze functions
Functions and Their Graphs
(linear, quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts will include
c) domain and range, including limited
and discontinuous domains and
ranges;
k) finding the values of a functionfor
elements in its domain; and
l) making connections between and
among multiplie representations of
functions including concrete, verbal,
numberic, graphic, and algebraic.
1.5 blocks
MPE.19
Chapter 2/Section 3/Linear Functions
Graph linear equations and linear
and Slope
inequalities in two variables, including
a) determining the slope of a line
when given an equation of the line,
the graph of the line, or two points on
the line; describing slope as rate of
change and determine if it is positive,
negative, zero, or undefined; and
b) writing the equation of a line when
given the graph of the line, two points
on the line, or the slope and a point
on the line.
0.5 blocks
MPE.14
Recognize the general shape of
function (absolute value, square root,
cube root, rational, polynomial,
exponential, and logarithmic) families
Chapter 2/Section 5/Transformations
of Functions
The graphs of exponential and
logarithmic functions should be
included at this point.
5
2.5 blocks
Chapter 2: Functions and Graphs
SOLs
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
and convert between graphic and
symbolic forms of functions. Use a
transformational approach to
graphing. Use graphing calculators as
a tool to investigate the shapes and
behaviors of these functions.
MPE.15
Use knowledge of transformations to
write an equation, given the graph of
a function (linear, quadratic,
exponential, and logarithmic).
MPE.16
Chapter 2/Section 2/More on
Investigate and analyze functions
Functions and Their Graphs
(linear,quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts include
a) continuity;
b) local and absolute maxima and
minima;
c) domain and range, including limited
and discontinuous domains and
ranges;
d) zeros;
e) x- and y-intercepts;
f) intervals in which a function is
increasing or decreasing;
k) finding the values of a function for
elements in its domain; and
l) making connections between and
among multiplie representations of
functions including concrete, verbal,
numberic, graphic, and algebraic.
MPE.13
Investigate and describe the
relationships among solutions of an
equation, zeros of a function, x-
WS Piecewise Functions
Chapter 3/Section 1/Quadratic
Functions
Piecewise functions should be
the focus of this section. The
definition of even and odd
functions may be omitted.
1 block
1.5 blocks
6
Chapter 2: Functions and Graphs
SOLs
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
intercepts of a graph, and factors of a
polynomial expression.
Review/Assessment
4.5 blocks
Chapter 4: Exponential and Logarithmic Functions
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
MPE.16
Chapter 4/Section 1/Exponential
Investigate and analyze functions
Functions
(linear, quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts include
Intensive study of the graph of 1 block
exponential functions and
transformations may be
reserved until the graphing unit
in the fourth quarter.
MPE.16
Chapter 4/Section 2/Logarithmic
Investigate and analyze functions
Functions
(linear, quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts include
Intensive study of the graph of 1 block
logarithmic functions and
transformations may be
reserved until the graphing unit
in the fourth quarter.
MPE.16
Chapter 4/Section 3/Properties of
Investigate and analyze functions
Logarithms
(linear, quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts include
WS: Logarithmic Properties
WS: Properties of Logarithms
1 block
MPE.16
Chapter 4/Section 4/Exponential and
Investigate and analyze functions
Logarithmic Equations
(linear, quadratic, exponential, and
logarithmic families) algebraically and
graphically. Key concepts include
WS: Solve Exponential
Equations Using Logarithms
2 blocks
MPE.2
Collect and analyze data, determine
the equation of the curve of best fit,
make predictions, and solve real-
WS: Word Problems Involving
Exponential Equations
WS: Exponential Decay
Half-Life Data
Chapter 4/Section 5/Exponential
Growth and Decay; Modeling Data
7
The instructor may choose to
omit regression equations,
logistic growth models, and
conversion to base e.
1 block
Chapter 4: Exponential and Logarithmic Functions
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
world problems using mathematical
models. Mathematical models will
include polynomial, exponential, and
logarithmic functions.
Review/Assessment
2.5 blocks
Capstone Project 2
Title
To Speed or Not To
Speed? That is the
question.
Applicable Chapters
Chapter 2
Weblink
https://825d99b1-a-62cb3a1a-ssites.googlegroups.com/site/mathematicscapstonecourseuni
ts/home/tasks/ToSpeedorNottoSpeedTask.pdf?attachauth=A
NoY7crP3KtnRhZ_B9CY46aB2bleLzfWqfLEJ5vPB2t81395_Z
Q53pCY1VvEUE1NnzBQIcmHU1EzHuKUjDX1KxcKpcZ_pJSewgHjHEE6ORSjGFRcPA
89
Additional Projects may be used from the following link:
https://sites.google.com/site/mathematicscapstonecourseunits/home
8
Mapping for Instruction - Third Nine Weeks
Chapter 5: Trigonometric Functions
Textbook
Chapters/Sections/Topics
MPE.11
Use angles, arcs, chords, tangents,
and secants to
(c) find arc lengths and areas of
sectors in circles.
Chapter 5/Section 1/Angle and
Radian Measure
MPE.5
Solve real-world problems involving
right triangles by using the
Pythagorean Theorem and its
converse, properties of special right
triangles, and right triangle
trigonometry.
Chapter 5/Section 2/Right Triangle
Trigonometry (partial)
MPE.20
Chapter 5/Section 3/Trigonometric
Given a point other than the origin on Functions of Any Angle
the terminal side of an angle, use the
definitions of the six trigonometric
functions to find the sine, cosine,
tangent, cotangent, secant, and
cosecant of the angle in standard
position. Relate trigonometric
functions defined on the unit circle to
trigonometric functions defined in right
triangles.
MPE.20
Given a point other than the origin on
the terminal side of an angle, use the
definitions of the six trigonometric
functions to find the sine, cosine,
tangent, cotangent, secant, and
cosecant of the angle in standard
position. Relate trigonometric
functions defined on the unit circle to
Supporting Materials
Comments
Time Frame
1 block
At this point, the only topics to
be covered in this section are
definitions of the six
trigonometric functions, the
reciprocal indentities, and
determining the trigonometric
values of special angles.
Blank Unit Circle
WS: Six Trigonometric
Functions
Chapter 5/Section 4/Trigonometric
Functions of Real Numbers; Periodic
Functions
1 block
1 block
1 block
9
Chapter 5: Trigonometric Functions
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
trigonometric functions defined in right
triangles.
MPE.27
Chapter 5/Section 5/Graphs of Sine
Given one of the six trigonometric
and Cosine Functions
functions in standard form,
(a) state the domain and the range of
the function;
(b) determine the amplitude, period,
phase shift, vertical shift, and
asymptotes;
(c) sketch the graph of the function by
using transformations for at least a
two-period interval; and
(d) investigate the effect of changing
the parameters in a trigonometric
function on the graph of the function.
Graphing Sine and Cosine
Sine and Cosine Graph
Properties
Sine and Cosine Graphs Lab
Analyze Sine and Cosine
Graphs
2 blocks
MPE.27
Chapter 5/Section 6/Graphs of Other
Given one of the six trigonometric
Trigonometric Functions
functions in standard form,
(a) state the domain and the range of
the function;
(b) determine the amplitude, period,
phase shift, vertical shift, and
asymptotes;
(c) sketch the graph of the function by
using transformations for at least a
two-period interval; and
(d) investigate the effect of changing
the parameters in a trigonometric
function on the graph of the function.
Graphs of Other Trigonometric
Functions
1 block
Review/Assessment
2.5 blocks
10
Chapter 5/Chapter 7: Solving Triangles using Trigonometry
Textbook
Chapters/Sections/Topics
MPE.5
Solve real-world problems involving
right triangles by using the
Pythagorean Theorem and its
converse, properties of special right
triangles, and right triangle
trigonometry.
Chapter 5/Section 2/Right Triangle
Trigonometry
MPE.5
Solve real-world problems involving
right triangles by using the
Pythagorean Theorem and its
converse, properties of special right
triangles,and right triangle
trigonometry.
Chapter 5/Section 8/Applications of
Trigonometric Functions
Supporting Materials
Right Triangle Word Problems
Comments
Time Frame
At this point, solving right
triangles should be addressed.
0.5 blocks
Intrsuctors may choose to omit
problems on bearing and
harmonic motion.
1 block
Chapter 7/Section 1/The Law of Sines
1.5 blocks
Chapter 7/Section 2/The Law of
Cosines
1 block
Review/Assessment
2 blocks
Chapter 6: Analytic Trigonometry
Textbook
Chapters/Sections/Topics
MPE.36 Verify basic trigonometric identities
and make substitutions, using the basic identities.
Chapter 6/Section 1/Verifying
Trigonometric Identities
Supporting Materials
Trig Memory Sheet
WS: Simplifying Using Trig
Identities
WS: Proving Identities
Review Assessment
Comments
Instructors may choose to
supplement initial problems in
which students simplify
trigonometirc expressions
without verifying identities.
Problems using the negative
angle identities may be omitted.
Time Frame
3 blocks
1.5 blocks
11
Capstone Project 3
Title
Sam, Kyle, and Kirby – A
Love of Triangles
Applicable Chapters
Weblink
Chapter 7
https://sites.google.com/site/mathematicscapstonecourseu
Chapter 5
nits/home/tasks/LoveOfTriangles.pdf?attredirects=0
Additional Projects may be used from the following link:
https://sites.google.com/site/mathematicscapstonecourseunits/home
12
Mapping for Instruction - Fourth Nine Weeks
Chapter 8: Systems of Equations and Inequalities
Textbook Chapters/Sections/Topics
Supporting Materials
Comments
Chapter 8/Section 1/Systems of
Linear Equations in Two Variables
Time Frame
1 blocks
Chapter 8/Section 2/Systems of
Linear Equations in Three Variables
The instructor may choose to
reserve the majority of solving
three variable word problems
until Chapter 9.
Review/Assessment
2 blocks
1.5 blocks
Chapter 9: Matrices and Determinants
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Chapter 9/Section 1/Matrix Solutions
to Linear Equations
Time Frame
1.5 blocks
Chapter 9/Section 2/Inconsistent and
Dependent Systems and Their
Applications
Non-square matrices do not
need to be addressed.
Chapter 9/Section 3/Matrix
Operations and Their Applications
Matrices are no longer
included in the 2009 SOLs for
Algebra I and Algebra II.
Instructors may need to
supplement introductory
material for matrices.
1 block
Chapter 9/Section 4/Multiplicative
Inverses of Matrices and Matrix
Equations
Depending on the needs and
abilities of students, the
instructor may determine how
much of this section is
performed by hand and how
much is performed using the
1 block
13
0.5 blocks
Chapter 9: Matrices and Determinants
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
calculator.
Chapter 9/Section 5/Determinants
and Cramer’s Rule
The instructor may choose to
revisit the word problems from
section 8.2
Review/Assessment
1 block
2.5 blocks
Chapter 10: Conic Sections and Analytic Geometry
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
MPE.21
Chapter 2/Section 8/Distance and
Given the coordinates of the center of Midpoint Formulas; The Circle
a circle and a point on the circle, write
the equation of the circle.
MPE.29
Investigate and identify the
characteristics of conic section
equations in (h, k) and standard
forms. Use transformations in the
coordinate plane to graph conic
sections.
Time Frame
1 block
MPE.29
Investigate and identify the
characteristics of conic section
equations in (h, k) and standard
forms. Use transformations in the
coordinate plane to graph conic
sections.
Chapter 10/Section 1/The Ellipse
The instructor may address the
foci of an ellipse as time allows.
2 blocks
MPE.29
Investigate and identify the
characteristics of conic section
Chapter 10/Section 2/The Hyperbola
The instructor may address the
foci of the hyperbola as time
allows.
1.5 blocks
14
Chapter 10: Conic Sections and Analytic Geometry
Textbook
Chapters/Sections/Topics
Supporting Materials
Comments
Time Frame
equations in (h, k) and standard
forms. Use transformations in the
coordinate plane to graph conic
sections.
MPE.29
Investigate and identify the
characteristics of conic section
equations in (h, k) and standard
forms. Use transformations in the
coordinate plane to graph conic
sections.
Chapter 10/Section 3/The Parabola
Parabola Table
Standard Form of the
Parabola
Review/Assessment
WS: Conics Review
WS: Types of Conics
The instructor may address the
focus and directrix of the
parabola as time allows.
1.5 blocks
2 blocks
Capstone Project 4
Title
Mix it Up
Applicable Chapters
Chapter 8
Weblink
https://825d99b1-a-62cb3a1a-ssites.googlegroups.com/site/mathematicscapstonecourseuni
ts/home/tasks/MixItUp.pdf?attachauth=ANoY7cpfZefSHUEFG
wcfmYJ5QsB6jjdTbm7IsAucj76pj_O61R_AiFmOYVGQMyaFdi
9v3W_InlaDE9EtFdcUjXYVOMY6zG3T7uKxTbfrVJnE7NI5FKi4
aEm625ifEygj8uv7t
Additional Projects may be used from the following link:
https://sites.google.com/site/mathematicscapstonecourseunits/home
15
List of Standards
http://www.doe.virginia.gov/instruction/mathematics/capstone_course/perf_expectations_math.pdf
4) Verify characteristics of quadrilaterals and use properties of
quadrilaterals to solve real-world problems.
5) Solve real-world problems involving right triangles by using the
Pythagorean Theorem and its converse, properties of special right
triangles, and right triangle trigonometry.
6) Use formulas for surface area and volume of three-dimensional
objects to solve real-world problems.
7) Use similar geometric objects in two- or three-dimensions to
a) compare ratios between side lengths, perimeters, areas, and
volumes;
b) determine how changes in one or more dimensions of an
object affect
area and/or volume of the object;
c) determine how changes in area and/or volume of an object
affect one or
more dimensions of the object; and
d) solve real-world problems about similar geometric objects.
8) Compare distributions of two or more univariate data sets,
analyzing center and spread (within group and between group
variations), clusters and gaps, shapes, outliers, or other unusual
features.
9) Design and conduct an experiment/survey. Key concepts include
a) sample size;
b) sampling technique;
c) controlling sources of bias and experimental error;
d) data collection; and
e) data analysis and reporting.
10)Investigate and apply the properties of arithmetic and geometric
sequences and series to solve real-world problems, including writing
Problem Solving, Decision Making, and Integration
Students will apply algebraic, geometric, and statistical concepts and
the relationships among them to solve problems, model relations, and
make decisions using data and situations within and outside of
mathematics. In accomplishing this goal, students will develop and
enhance a repertoire of skills and strategies for solving a variety of
problem types.
1) Solve practical problems involving rational numbers (including
numbers in
scientific notation), percents, ratios, and proportions.
2) Collect and analyze data, determine the equation of the curve of
best fit, make predictions, and solve real-world problems using
mathematical models.
Mathematical models will include polynomial, exponential, and
logarithmic
functions.
3) Use pictorial representations, including computer software,
constructions, and coordinate methods, to solve problems involving
symmetry and transformation.
This will include
a) investigating and using formulas for finding distance,
midpoint, and slope;
b) applying slope to verify and determine whether lines are
parallel or perpendicular;
c) investigating symmetry and determining whether a figure is
symmetric with respect to a line or a point; and
d) determining whether a figure has been translated, reflected,
rotated, or dilated, using coordinate methods.
16
the first n terms, finding the nth term, and evaluating summation
formulas. Notation will include Σ and an.
11)Use angles, arcs, chords, tangents, and secants to
a) investigate, verify, and apply properties of circles;
b) solve real-world problems involving properties of circles;
and
c) find arc lengths and areas of sectors in circles.
d) zeros;
e) x- and y-intercepts;
f) intervals in which a function is increasing or decreasing;
g) asymptotes;
h) end behavior;
i) inverse of a function;
j) composition of multiple functions;
k) finding the values of a function for elements in its domain;
and
l) making connections between and among multiple
representations of functions including concrete, verbal,
numeric, graphic, and algebraic.
17) Determine optimal values in problem situations by identifying
constraints and using linear programming techniques.
Understanding and Applying Functions
Students will be able to recognize, use, and interpret various functions
and their representations, including verbal descriptions, tables,
equations, and graphs to make predictions and analyze relationships in
solving complex, real-world mathematical problems.
12) Transfer between and analyze multiple representations of
functions, including
algebraic formulas, graphs, tables, and words. Select and use
appropriate representations for analysis, interpretation, and prediction.
13) Investigate and describe the relationships among solutions of an
equation, zeros of a function, x-intercepts of a graph, and factors of a
polynomial expression.
14) Recognize the general shape of function (absolute value, square
root, cube root, rational, polynomial, exponential, and logarithmic)
families and convert between graphic and symbolic forms of functions.
Use a transformational approach to graphing. Use graphing calculators
as a tool to investigate the shapes and behaviors of these functions.
15) Use knowledge of transformations to write an equation, given the
graph of a function (linear, quadratic, exponential, and logarithmic).
16) Investigate and analyze functions (linear, quadratic, exponential,
and logarithmic families) algebraically and graphically. Key concepts
include
a) continuity;
b) local and absolute maxima and minima;
c) domain and range, including limited and discontinuous
domains and ranges;
Procedure and Calculation
Students will be able to perform and justify steps in mathematical
procedures and calculations and graph and solve a range of equations
types. Students will reason from a variety of representations such as
graphs, tables, and charts and will use displays of univariate data to
identify and interpret patterns. Students will be able to calculate
probabilities and analyze distributions of data to make decisions.
18) Given rational, radical, or polynomial expressions,
a) add, subtract, multiply, divide, and simplify rational
algebraic expressions;
b) add, subtract, multiply, divide, and simplify radical
expressions containing rational numbers and variables, and
expressions containing rational exponents;
c) write radical expressions as expressions containing rational
exponents and vice versa; and
d) factor polynomials completely.
19) Graph linear equations and linear inequalities in two variables,
including
2
a) determining the slope of a line when given an equation of
the line, the graph of the line, or two points on the line;
describing slope as rate of change and determine if it is
positive, negative, zero, or undefined; and
b) writing the equation of a line when given the graph of the
line, two points on the line, or the slope and a point on the line.
20) Given a point other than the origin on the terminal side of an
angle, use the definitions of the six trigonometric functions to find the
sine, cosine, tangent, cotangent, secant, and cosecant of the angle in
standard position. Relate trigonometric functions defined on the unit
circle to trigonometric functions defined in right triangles.
21) Given the coordinates of the center of a circle and a point on the
circle, write the equation of the circle.
22) Analyze graphical displays of univariate data, including dotplots,
stemplots, and histograms, to identify and describe patterns and
departures from patterns, using central tendency, spread, clusters,
gaps, and outliers. Use appropriate technology to create graphical
displays.
23) Analyze the normal distribution. Key concepts include
a) characteristics of normally distributed data;
b) percentiles;
c) normalizing data, using z-scores; and
d) area under the standard normal curve and probability.
24) Describe orally and in writing the relationships between the
subsets of the real number system.
25) Perform operations on complex numbers, express the results in
simplest form using patterns of the powers of i, and identify field
properties that are valid for the complex numbers.
26) Solve, algebraically and graphically,
a) absolute value equations and inequalities;
b) quadratic equations over the set of complex numbers;
c) equations containing rational algebraic expressions; and
d) equations containing radical expressions. Use graphing
calculators for solving and for confirming the algebraic
solutions.
27) Given one of the six trigonometric functions in standard form,
a) state the domain and the range of the function;
b) determine the amplitude, period, phase shift, vertical shift,
and asymptotes;
c) sketch the graph of the function by using transformations for
at least a two-period interval; and
d) investigate the effect of changing the parameters in a
trigonometric function on the graph of the function.
28) Find, without the aid of a calculator, the values of the
trigonometric functions of the special angles and their related angles as
found in the unit circle. This includes converting angle measures from
radians to degrees and vice versa.
29) Investigate and identify the characteristics of conic section
equations in (h, k) and standard forms. Use transformations in the
coordinate plane to graph conic sections.
30) Using two-way tables, analyze categorical data to describe patterns
and departure from patterns and to find marginal frequency and
relative frequencies, including conditional frequencies.
31) Calculate probabilities. Key concepts include
a) conditional probability;
b) dependent and independent events;
c) addition and multiplication rules;
d) counting techniques (permutations and combinations); and
e) Law of Large Numbers.
Verification and Proof
Students will recognize verification and proof as fundamental aspects
of mathematical reasoning. Students will integrate and apply inductive
and deductive reasoning skills to make, test, and evaluate
mathematical statements. This applies equally through simple
mathematical calculations, in geometric applications, and more
abstract statistical and algebraic processes. Students will use logical
reasoning to analyze an argument and to determine whether
conclusions are valid.
3
32) Use the relationships between angles formed by two lines cut by a
transversal to
a) determine whether two lines are parallel;
b) verify the parallelism, using algebraic and coordinate
methods as well as deductive proofs; and
c) solve real-world problems involving angles formed when
parallel lines are cut by a transversal.
33) Given information in the form of a figure or statement, prove two
triangles are congruent, using algebraic and coordinate methods as
well as deductive proofs.
34) Given information in the form of a figure or statement, prove two
triangles are similar, using algebraic and coordinate methods as well as
deductive proofs.
35) Construct and justify the constructions of
a) a line segment congruent to a given line segment;
b) the perpendicular bisector of a line segment;
c) a perpendicular to a given line from a point not on the line;
d) a perpendicular to a given line at a given point on the line;
e) the bisector of a given angle,
f) an angle congruent to a given angle; and
g) a line parallel to a given line through a point not on the
given line.
36) Verify basic trigonometric identities and make substitutions, using
the basic identities.
Supplemental Resources
4
College Algebra:
Real Numbers and Their Properties
I.
Name________________
Date ________________
Sets
A.
Set: a collection of objects where it is possible to determine whether or not a
certain object is in the set.
Example: { *, &, %, <, >, #}
“&” is in the set
“@” is not in the set
Symbol: Often a capital letter is used to name the set (see below)
B.
The individual objects in the set are called members or elements of the set.
Example: The set above has 6 elements. “*” is a member of the set above,
“$” is not an element of the set above.
Symbol:  “is an element of”  “is not an element of”
C.
Subset: If every element of one set is a member of a second set then the set is a
subset of the second.
Example: if A = { 1, 2, 3 } and B = { 0, 1, 2, 3, 4 … } then “A is a subset of B”.
Symbols:
 “is a subset of”
 “have exactly the same elements”
 “is not a subset of”
AB
BA
D.
Operations on sets are union and intersection.
Union: put all elements together in a single set without listing any object twice.
Symbol - 
Intersection: all elements that are in common to both sets.
Symbol: 
Example: If A = { 1, 2, 3 } and B = { 3, 4 }, then
A  B = { 1, 2, 3, 4 }
A B = { 3 }
E.
More about sets.
A finite set ends. Example: { 1, 2, 3 }, { 2, 4, 6, … 122 }
A infinite set does not end. Example: {0, 1, 2, 3, … }
The null set or empty set has no elements. Symbols:  or { }
2
II.
Real Number Sets
A. Natural Numbers are the counting numbers or any number that simplifies to a
counting number, N = { 1, 2, 3, 4, … }
12
Examples:
; 1,234,000;
81
4
B. Whole Numbers contain zero in addition to all the natural numbers.
W = { 0, 1, 2, 3, 4, … }
0 100
Examples: ;
; 121
2
5
C. Integers contain all the whole numbers and their opposites.
Z = { … -3, -2, -1, 0, 1, 2, 3, … }
36 18
Examples: 
;
;  49
9
2
a
D. Rational Numbers are a ratio of two integers or can be written as a ratio of
where
b
a and b are integers and b  0 . Symbol is Q.
E. Irrational Numbers are all real numbers that are not rational. Symbol is I.
Examples: 1) decimal numbers that do not repeat or end;
3.121121112…
2) Square roots of non perfect squares;
2 ; 3 ; 99
3) pi: 
F. Real Numbers is the union of the rational numbers and the irrational numbers. Symbol
is R. Q  I = R
All Natural Numbers are Whole Numbers, all Whole Numbers are Integers, all
Integers are Rational Numbers, and all Rational Numbers are Real Numbers. All
Irrational Numbers are Real Numbers. NO Rational Numbers are Irrational Numbers
and NO Irrational Numbers are Rational Numbers. The numbers that are not real are
call Imaginary numbers.
F.
Complete the chart based on the notes above. Check every set that the given
number is an element of.
Number
-1.25
40
5
.2323…
- 900
N
W
Z
Q
I

3
R
Name _____________________________
College Algebra
Date ________________
Sets and Set Notation
I.
Period _____
Given set A  1,2,3, 4, 5 and set B  2, 4, 6, 8 , find the following:
1. A
B
2. A
4. Is 4  A
3. Is 5  B ?
II.
B
B?
Given set A  0,2,3, 4, 6, 9 , set B  0,2, 4, 6,8,10 and set C  3, 4, 5, 6 , find the following
:
5. A
C
6. A
B
7. C
A
8. C
B
9. A
B C
10. A
B C
11. A
B
12. C
A
14. A
B
13. C
A
C
C
B
4
C
A
C
Name ________________________
College Algebra
Date __________
Simplifying Expressions
I.
Period _________
Simplify each expression.
1. 5  2x  3  7  4  x 
2. 2  a  b  1   3a  4b  3
3. 3 2  x  4   5x  1
4. 2 5  3  y  2    3  2  4  y  2  
5. 4 5a  2  a  3b   7b 
6. 4 2 3  2  x  2   4


7. 6 5  2  x  2y   3  2x  y 
8. 2    4  p    2  5  


9. 2 3  4a  b   2  5a  3b   5 3a  b  4a   b

 

10. 4 7  4  2  5p  3r   6 3  2  3  p  2r 
5
Name ________________________
Date ______________
I.
College Algebra
Period ____
Solve each word problem. Set up variables, write an equation and check your answer. Show all of
your work.
1. At 8:00 am two buses leave a bus station, one traveling south at 45 mph and the other traveling
north at 30 mph. How long will it take them to get 375 miles apart?
2. At 2:00 pm two cars start to meet each other from towns 240 miles apart. The cars meet at 5:00
pm. If the rate of one car is 20 mph faster then the other car, how fast does each car go?
3. Sam left for school at 6:30 am. He drove at 20 mph. His mom left at 7:30 am and drove at 60
mph in order to catch him to give him has homework he had forgotten. How long will it take her
to catch him?
4. Two trains leave Chicago at the same time and travel in opposite directions. One train travels at
40 mph and the other train at 50 mph. How long will it take the trains to get 450 miles apart?
5. Two planes leave the airport at noon, one flying east at a certain speed and the other flying west
at twice the speed of the first plane. If the planes are 2700 miles apart in 3 hours, how fast is
each plane flying?
6. Exactly 12 minutes after the Smiths head north on the highway the Jones set out from the same
point to overtake them. The Smiths travel at a steady speed of 45 mph and the Jones travel at a
steady speed of 54 mph. How long will it take the Jones to catch the Smiths?
6
Name ________________________
College Algebra
Date ____________ Period ____
I.
Solve each word problem. Show your set-up, write an equation, and check you answer. Show all of
your work.
1. Chris invested $200.00 at a certain interest rate and $700.00 at a rate 6% higher. The annual
return on both investments totaled $114.00. Find each interest rate.
2. Sarah invested $300.00 at a certain interest rate and $900.00 at a rate 5% higher. If she earned
$111.00 less in annual interest on the smaller investment, find the interest rates.
3. Leroy invested $400.00 at a certain interest rate and $1200.00 at a rate 3% higher. He earned
$84.00 less in annual interest on the smaller investment. If the return on both investments totaled
$132.00, find each rate.
4. An isosceles trapezoid has three congruent sides. Each of these sides is 5 feet shorter than the
fourth side. If the perimeter of the trapezoid is 33 feet, find the length of the longer side.
5. A radiator contains 8 liters of a 40% antifreeze solution. How many liters of pure antifreeze
must be added to obtain a 50% solution?
6. Some corn costing 60 cents per pound is added to 50 pounds of oats costing 90 cents per pound
to make an animal feed costing 75 cents per pound. How many pounds of corn should be added?
7. A chemist has 60 ml of a 70% acid solution. How much water should be added to produce a
solution that is 40 % acid?
7
Intro Calculus Worksheet
Name ________________________
Piecewise Functions
Graph the following functions. State the domain and range of each.
1.
2.
3.
4.
5.
6.
 x  2,

f ( x)  1,
 x  4,


x,
f ( x)  

5,
x2
x2
x2
Domain: ______________________________
Range: _______________________________
Domain: ______________________________
x0
x0
Range: _______________________________
Domain: ______________________________

x 1
x,
f ( x)  

3x  4, x  2
2 x  2,

f ( x)   x   x  ,
 2
2 x ,
 x2 ,

f ( x )   x,

 x,
 x  ,

f ( x)   x ,
 2
 x  1,
Range: _______________________________
x  3
Domain: ______________________________
3  x 1
Range: _______________________________
x 1
x  2
2 x  4
Domain: ______________________________
Range: _______________________________
x4
x  3
Domain: ______________________________
 2  x 1
Range: _______________________________
x 1
8
y
y
1.
4.
x
y
x
y
2.
5.
x
y
x
y
3.
x
6.
x
9
College Algebra
Using Log Properties
Name _____________________
Date ________________ Pd __
Use the properties of logarithms to expand each of the following expressions.
1. log a xz
2. log 4  
3
3. log 3 a 2
4. log5 x
5. log b x3 y 2
6. log 4 y 2 x
 x2 y 
7. log b 

 z 
 xy 2 
8. log 

 z
9.
x
log5 3
xy
z2
Write each of the following expressions as a single logarithm. Do not rationalize.
10. logb x  logb y
11. log5 m  log5 y
13. logb x  12 logb y
14. logb x  2logb y  logb z
16.
1
3
 2logb x  logb y  logb z 
12. 2log3 x  log3 y
15. 12 log6 x  2log6 y  3log6 z
17. 2log5 x  3log5 y  log5 z 
18.
1
3
logb x  2logb x
Write the left side as a single logarithm and use the exponential form to solve for x.
19. log 2 x  log 2 8  7
21. log5 3  log5 x  2
20. log 4 x  log 4 8  3
10
Name _______________________
College Algebra
Date ______________
Properties of Logarithms
I.
Period ___
Use the properties of logarithms to rewrite each of the following as a sum or difference of multiple
logarithms.
1. loga xy
3. loga
2. loga x 2y 3
x
y2
5. loga xy
II.
4. loga
x 3y 4
z2
6. loga
xy 4
z3
Rewrite each of the following as a single logarithm.
7. 4log10 5  2log10 5
8. 4log10 2  log10 2  1
9. loga x  3loga y  5loga z
10. loga 18x 2  3loga z  loga 6y
11. loga x  2loga y 
1
loga z
2
12. loga x 2y  2loga 5xy 3  loga 10x 2y 2
11
III.
Express each of the following in terms of log10 3 and log10 5 .
13. log10 15
14. log10 27
15. log10 3
16. log10 45
17. log10
5
3
18. log10
19. log10 75
9
25
20. log10 225
12
Name __________________________
College Algebra
Date ______________ Period ____
Solving Exponential Equations Using Logarithms
I.
Solve each of the following. Show all work. Write your answer in calculator ready form and then
give the decimal approximation to the nearest hundredth.
1. 2.9 x  62.3
2. x  log5 41.6
3. x  log3 18.9
4. 5x 2  12.5
5. 3x 1  5x
6. 32x 3  1.77
7. 102x 2  137.5
8. 113x 1  193.5
9. 23x  3x 1
10. 52x  7 x 1
11. 11x  4  52x
12. 7 x 2  53x
13. 72x  5.2x 1
14. 6x 1  4.53x 1
3
15. x 7  4.63
13
Name _______________________
College Algebra
Date ______________
Word Problems Involving Exponential Equations
I.
Period ___
Solve each word problem. Show all work! Make sure to show your calculator ready form.
1. If $1000 amounts to $1250 in 2 years and the interest is compounded semi-annually, what was
the interest rate?
2. If a colony of bacteria starts with 2000 bacteria and increases 15% of its population each day, the
population P after t days is given by P  2000e 0.14t . How long will it take the population to
reach 8000 bacteria?
3. The atmospheric pressure P at an altitude x miles above sea level is given by P  30e 0.198x ,
where P is measured in inches of mercury. Find the altitude of a mountain peak if the pressure
there is 16 inches of mercury.
4. How long will it take an original principal P to triple if it is invested at 12% compounded
monthly?
5. Starting with 100 milligrams of radium, the amount A of radium remaining after x years of
radioactive decay is given by A  100e 0.000411x . How long will it be until half of the radium
remains?
6. How long will it take for an original investment to double if it is compounded continuously at a
rate of 10% ?
7. A colony of bacteria increases by 20% of its population each day. If the colony started with
3000 bacteria and the population P after x days is given by P  3000e 0.18x , how long will it
take for the population to reach 6000 bacteria?
14
College Algebra
Exponential Decay
Name _____________________
Date ________________ Pd __
There are 2 formulas associated with exponential decay.
A  A0ert where A is the amount remaining , A0 is the original amount, r is the annual rate of
growth/decay and t is the number of years. For decay, the rate is negative. (For growth, like in
continuous compounding, the rate would be positive.)
If the decay is by half-life, the following formula may also be used.
 kt
A  A0 2
where A is the amount remaining , A0 is the original amount, k is the half-life of the
substance in years and t is the number of years.
Set up formulas for each of the problem. Show steps. Be sure to answer the question using proper units.
1. Approximately 4,000,000 curies (a measure of the quantity of a radioactive gas) of Hydrogen-3 were
released by nuclear power plants in the late 1980s. How long will it take for this quantity of Hydrogen-3 gas
(half-life = 12 yr) to be reduced to 31,250 curies by decay?
2. Another nuclear waste, Plutonum-239, has a half-life of 24,000 yr. A rule of thumb is that radioactive
wastes are virtually harmless after 10 half-lives. How long must 1 gram of Plutonium be securely stored before
t s virtually harmless? How much of the Plutonium will remain at that time?
3. Some radioactive waste products of nuclear wastes have half-lives of about 30 yr. If a stockpile of 120 m3 of
these nuclear wastes has accumulated at a given time, how much will be present 30 yr later? How much will be
present 300 yr later?
4. A 50-gram sample of radium decays to 5 grams in approximately 5615 yr. What is the half-life of this
substance?
5. After 100 yr of storage, the nuclear wastes of a nuclear plant have diminished to 247 m3. The half-life of the
waste is 40 yr. Find the original amount of the waste that was stored.
6. Another example of exponential decay is the decrease in atmospheric pressure with increasing height above

h
sea level. In the formula P  P0 2 4795 , where P is the atmospheric pressure (millimeters of mercury) at height
h above sea level (meters). If the atmospheric pressure is 42 mm at a height of 20 km, what is the pressure P0 at
sea level?
7. A typical nuclear power plant produces about 10 lb of Krypton-85 per year. The half-life of Krypton-85 is
11 yr. How long must the Krypton be contained so that only 0.1 lb will remain?
8. Approximate the age of a bone that how contains 84 g of carbon-14 if it originally contained 192 g of that
isotope. The half-life of carbon-14 is 5730 yr.
9. The half-life of radium (Ra226) is 1620 years.
a) Find the constant r in the formula P  P0 e rt
b) Rewrite the equation using the values for r and t.
c) Suppose a 20-gram sample of radium (Ra226) is sealed in a box. Find the mass of the radium after
5000 years. Use P  P0 e rt
d) Repeat c) using A  A0 2
 kt
15
Cesium-137 is a radioactive element with a half-life of thirty years.
Uranium-239 has a half-life of about 23 minutes
Neptunium-239 has a half-life of about 2.4 days
Thorium-233 has a half-life of about 22 minutes
Protactinium-233 has a half-life of about 27 days
bismuth-210, which has a half-life of 5 days
Polonium-210 has a half-life of 138.39 days
plutonium-244, has a half-life of about 82,000,000 years
polonium-209, has a half-life of 102 years
astatine-210, has a half-life of 8.1 hours
radon-222, has a half-life of about 3.8 days
radium-226, has a half-life of about 1600 years
americium-243, has a half-life of about 7,370 years
Americium-241, with a half-life of 432.2 years
curium-247, has a half-life of about 15,600,000 years
berkelium-247, has a half-life of about 1,380 years
californium-251, has a half-life of about 898 years
einsteinium-252, has a half-life of about 471.7 days
16
College Algebra
Coordinates on the Unit Circle
Name _______________
Place the degree and radian measure for each angle in the small box on its terminal
side.
Label the points of intersection with the proper coordinates.
y
x
MEMORIZE
17
College Algebra
Name _____________________
Date _____________ Pd ____
Complete the following using good old Soh Cah Toa with sides x, y, and r in standard position..
1.
sin   =
4.
csc   =
2.
sin  
or
5.
cos   =
sec   =
cos  
3.
or
6.
tan   =
cot   =
tan  
=
or
Since the value of r is _____ in a Unit Circle, these become:
7.
sin   =
8.
cos   =
9.
tan   =
10.
csc   =
11.
sec   =
12.
cot   =
Notice since no matter what value is used for r
tan   
sin  
y
, so tan   
x
cos  
cot   
cos  
x
, so cot   
sin  
y
Complete the following chart for the given angles.
Ordered pair
r
sin  
cos  
13. 00
1
14. 
1

2
1
16. 900
1
15.
tan  
cot  
and
sec  
csc  
Write the exact value for all trig functions of an angle  , if its terminal side passes through  3,5 .
(Hint: First draw a reference angle and then find r using the Pythagorean Theorem.).Be sure to rationalize.
17.
sin   =
18.
cos   =
19.
tan   =
20.
csc   =
21.
sec   =
22.
cot   =
Write the exact value for all trig functions of an angle  , if its terminal side passes through  7 ,2  .
(Hint: First draw a reference angle and then find r using the Pythagorean Theorem.) Be sure to rationalize.
18
23.
sin   =
24.
cos   =
25.
tan   =
26.
csc   =
27.
sec   =
28.
cot   =
5
and  is in Quadrant II
13
(Sketch the reference triangle with a radius of 1)
Write the exact values of the trig functions if cos  
29.
sin   =
30.
cos   =
31.
tan   =
32.
csc   =
33.
sec   =
34.
cot   =
20
and  is in Quadrant IV
29
(Sketch the reference triangle with a radius of 1)
Write the exact values of the trig functions if sin  
35.
sin   =
36.
cos   =
37.
tan   =
38.
csc   =
39.
sec   =
40.
cot   =
19
College Algebra
Name ________________
Date ___________ Pd ___
Sine and cosine Graphs
Comparing the sin and cosine graphs.
1. In terms of , how often does the sine graph repeat? ________ ,
cosine graph repeat? ________.
2. The domain of the sine graph is _________________.
3. The domain of the cosine graph is _______________.
4. The range of the sine graph is _______________.
5. The range of the cosine graph is _______________.
6. How far to the right would you have to slide the cosine graph to have it fit exactly on the
sine graph? __________
7. When the domain is between 0 and /2,you are in Quadrant ____ and
the sine is _____ ( + or - ?) and
the cosine is _____ ( + or - ?)
8. When the domain is between _______ and _______, you are in Quadrant II.
The sine is _____ ( + or - ?) in this quadrant and
the cosine is _____ ( + or - ?) in this quadrant.
9. When the domain is between _______ and _______, you are in Quadrant III.
The sine is _____ ( + or - ?) in Quadrant III and
the cosine is _____ ( + or - ?) in Quadrant III.
10. When the domain is between _______ and _______ , you are in Quadrant IV.
The sine is _____ ( + or - ?) in Quadrant IV and
the cosine is _____ ( + or - ?) in Quadrant IV.
11. Give the coordinates of a least 4 points where the basic sine and cosine graphs intersect:
__________
__________
__________
__________
20
College Algebra
Transformations of Sine and Cosine Graphs
Name ____________________
Date _____________ Pd _____
PURPOSE: This worksheet is to be used to discover what effects changing numbers in trig functions equations
has on its graphs.
General form of the equation: y  A sin  Bx  C   D ,
y  sin  x  , notice A  1 , B  1 , C  0 , and D  0
Basic equation:
All results will be compared to the graph of the basic y = sin x graph,
also referred to as a sine wave, a sinusoidal wave or a sinusoid.
y
1


x
–1
DIRECTIONS: Use your graphing calculator to graph the following over the indicated intervals. Make sure
your calculator is set in radians. When you do each graph, go under ZOOM 7 before you sketch your graph.
Mark on the y-axis the largest y used and the smallest y used in the graph.
1. A) What effect does changing A to a value
between 0 and 1 have on the graph?
B) What effect does changing A to a value
greater than 1 have?
__________________________________
__________________________________
1.5
y
1
1
0.5
0.5
–0.5
C)
1.5


x
y

–0.5
–1
–1
–1.5
–1.5
y  0.5 sin  x 
y  1.5 sin  x 
A is called the amplitude of the graph and is found by the formula
M is the maximum value of y and m is the minimum value of y.
What is the amplitude of the basic sine graph?
What is the amplitude of the graph in # 1. A) above?
What is the amplitude of the graph in # 1. B) above?

M m
2
__________
__________
__________
CONCLUSION: Changing the A value changes the ______________________ of the graph.
21
x
2. A) What effect does changing the B value to a number between 0 and 1 have?
______________________________________________________
1.5
y
1
y  sin  .5x 
0.5

–0.5



x
–1
–1.5
B) What effect does changing the value of B to a number greater than 1 have?
______________________________________________________
1.5
y
1
y  sin  2 x 
0.5

–0.5



x
–1
–1.5
C) The period is defined to be the length over x that it takes the graph to complete one entire cycle.
Notice that our basic graph has a period of .
What is the period of the graph in # 2 A above?
__________
What is the period of the graph in # 2 B above?
__________
Using what you have just observed, what would be the period of:
__________
y  sin  .25x 
__________
y  sin  6 x 
CONCLUSION:
Changing B, changes the __________________ of the graph.
3. A) What effect does changing C to a positive multiple of pi () have on the graph?
______________________________________________________
1.5


y  sin  x  
4

y
1
0.5
 



–0.5

 


x
–1
–1.5
B) What effect does changing C to a negative multiple of pi () have on the graph?
______________________________________________________
1.5


y  sin  x  
2

y
1
0.5
 

–0.5
–1
–1.5
22



 


 

x
C
B
It shows how far the basic graph has been slid to the left (negative shift) or right (positive shift)
C) C is called the phase shift (horizontal shift) of the graph.
What is the phase shift of graph # 3 A?
What is the phase shift of graph # 3 B?
__________
__________
Using what you just observed, what would be the phase shift of :
___________
y  sin  3x   



If B is not 1, then you must solve what is in
y  sin  x  
__________
9

   0 to find the phase shift.
23
4. A) What effect does changing D to a positive number have on our graph?
______________________________________________________
y
2
1.5
y  sin  x   1
1
0.5
–0.5
–1
–1.5
Does this change the amplitude?

M m
=
2

x
= _______
__________
B) What effect does changing D to a negative number have on our graph?
______________________________________________________
1.5
y
1
y  sin  x   0.5
0.5
–0.5


x
–1
–1.5
Does this change the amplitude?
M m
=
2
= _______
__________
C) Changing d shifts the graph up (+) or down ( - ), vertical shift has no special name.
What is the vertical shift of # 4 A? __________
What is the vertical shift of # 4 B? __________
5. Using what you have learned, analyze the graph of the following equation by filling in the blanks. (Do not
graph)


y  2 sin  6 x    2.5
Period __________
Amplitude
__________
2

Phase shift __________
Vertical shift __________
6. A) What effect would changing A to a (-1) have on the graph?
______________________________________________________
24
1.5
y
1
y   sin  x 
0.5
–0.5


x
–1
–1.5
B) What effect would changing B to a (-1) have on the graph?
______________________________________________________
1.5
y
1
y  sin   x 
0.5
–0.5


x
–1
–1.5
7. Write an equation that satisfies the following description. The graph
repeats twice as often as normal
it has been shifted to the right /2
it has a maximum value of 4 and a minimum value of -1
Equation: ____________________________________
Now, graph the basic sine graph and your graph on the same calculator screen and then see if it seems to fit the
above description. If not, try one more time.
25
College Algebra
Analyze Sin/Cos Graphs
Name ________________
Date __________ Pd ____
This worksheet is designed to have you use what you have learned to analyze sine and cosine
graphs by looking at specific information about then, either their equations or their graphs. The
general form of sine may be written y  A sin  Bx  C   D and the general form of cosine may be
written y  A cos  Bx  C   D . Study the examples and then complete the chart.
Equation
Amplitude
Period
Phase Shift
Vertical Shift
A
2
B
Set ( ) = O &
solve
D
y  sin x
1 1
2
 2
1
None
None
y  cos x
1 1
2
 2
1
None
None
y  2 cos 3  x     4
2 2
2
3
(x   )  0
x  
-4
2
y   sin  4 x   
3
2 2

3
3
2 

4
2
y  5 sin  3x     1
y  cos 2 x  6
y  4 sin 5  x   
y   cos  2 x     1
1
1

1
y  sin  x   
3
8 6
8
26
4x    0
x

4
None
The maximum and minimum values are given. Find the amplitude.
M m
. To find the vertical shift, you subtract the amplitude from M or add the
Amplitude 
2
amplitude to m.
Maximum point
Minimum point
Amplitude
Vertical Shift
6.5
4.5
1.4
4.6
10
4
7
7
Complete the following chart for each given graph.
Graph
Minimum
Maximum
point
point
y
4
2
|

|

x
–2
–4
y
2
|

|

|

|

x
y
2
1
|
|
 
|

|

x
|

–1
–2
y
2
1
|

|

|

|

x
|

–1
–2
27
Amplitude
Period
Vertical
shift
College Algebra
Graphs csc, sec , tan , cot .
I.
Name ____________________
Date _____________ Pd _____
Draw y  sin  x  using a faint line. Sin  x   0 at what values? _____________
Sketch dotted vertical lines through the x values where sin  x   0 .
Graph y  csc x over the given interval.
1. What do you type to enter this in your calculator? __________________
1.5
y
2. Period
1
0.5
 
–0.5




 


  

x
–1
3. Amplitude
4. Domain
5. Range
–1.5
.
6. What are the values of x that should have a dotted line through them?
________________________
1
 _____________
7. Use the calculator to find csc ( ) of one of these values. i.e. csc ( ) 
sin ( )
8. What do the dotted lines indicate? ___________________________________________
9. What are these dotted lines called? ___________________________________________
10.Describe to someone what they could do to a sine graph to get a cosecant graph.
___________________________________________________________________
Based on what was done in WS 5.3B and 5.3C, write equations for the following. Then enter your
answers in the graphing calculator to see if they fit the description
11. Shift the cosecant graph

to the right.
________________________________
2
12. Make the cosecant graph repeat twice as often. _________________________________
13. Raise the entire cosecant graph 2 units.
II.
_________________________________
Draw y  cos  x  using a faint line. Cos  x   0 at what values? _____________
Sketch dotted vertical lines through the x values where cos  x   0 .
Graph y  sec  x  over the given interval.
14. What do you type to enter this in your calculator?
1.5
__________________
y
15. Period
1
0.5
 
–0.5




 


  

x
16. Amplitude
17. Domain
–1
–1.5
18. Range
19. Name three values for x for which the secant graph is undefined.
28
_______, _______, _______
III.
What is the x, y, r definition of tangent? _______
Visualize the Unit Circle. At what values of an angle are the coordinates  0,1 or  0, 1 ?
___ ___ degrees or ___ ___ radians. Sketch vertical dotted lines through these values.
Graph y  tan  x  over the given interval.
1.5
y
20. Period
1
21. Amplitude
0.5
 
–0.5




 



  
x
–1
22. Domain
23. Range
–1.5
24. Name four values of x for which the tangent graph is undefined.
25. Write an equation that would shift the tangent graph
_____, _____, _____, _____

to the left.
2
(Test your answer in the calculator and adjust if needed)
_______________________
IV.
What is the x, y, r definition of cotangent? _______
Visualize the Unit Circle. At what values of an angle are the coordinates 1, 0  or  1, 0  ?
__________ degrees or __________ radians. Sketch vertical dotted lines through these values.
Graph y  cot  x  over the given interval.
26. What do you type to enter this in your calculator?
1.5
__________________
y
27. Period
1
28. Amplitude
0.5
 
–0.5




 



  
x
–1
29. Domain
30. Range
–1.5
31. Name four values of x for which the cotangent graph is undefined.
_____, _____, _____, _____
32. Compare the tangent and cotangent graphs.
What is the same?
__________________________________________________
What is different?
___________________________________________________
33. Graph the equation y  2cot  x  in your calculator.
Be sure to put ( ) around  2 tan  x   when you divide, otherwise it divides by 2 then multiplies by tan  x 
What does it change about the basic y  cot  x  graph?
______________________
34. Graph the equation y  cot  2 x  in your calculator.
What does it change about the basic y  cot x graph?
29
______________________
College Algebra
Name ________________
Right Triangle Word Problems
Date ___________ Pd ___
Directions: Work on each problem with your partner. Each person will turn in an answer sheet. Staple the one
you wish me to grade on top. You are required to do any 10 problems, if you do more they will count as extra
credit. If you start a problem and then decide
For each problem:
a) Draw and label a sketch
b) Give the trig equation used (must have an angle and = mark to be a trig equation)
c) Circle final answers (including units)
d) Round sides to tenths and angles to nearest minute
1. From a point on the ground 208 ft. from the
base of the Gateway Arch (in St. Louis), the angle
of elevation to the top is 72˚. Find the height of the
Arch.
2. The angle of depression from the top of a pine
tree to the base of an oak tree is 66˚.If the trees are
30 feet apart, how tall is the pine tree?
3. Find the height of the Sears Tower in Chicago if
there is a 51˚ angle of elevation to the top from a
point on the ground 1177 ft. from its base.
4. A sunken pirate ship on the ocean floor is
sighted at an angle of depression of 42˚ from a
sailboat. After sailing 50 meters, the sailboat is
directly above the sunken ship. How deep is the
ocean at this point?
5. At a certain time of the day, the angle of
elevation to the sun is 25˚. Find the height of a tree
that casts a shadow of 137 feet.
6. The three sides of an isosceles triangle are 6 cm,
6 cm, and 8 cm. Find the measurement of the two
base angles.
7. A 20 foot ladder rests against a vertical wall with
its foot on level ground 11 feet from the bottom of
the wall. What is the angle the ladder makes with
the wall?
8. An airplane rises into the sky in a straight line
that makes a 9˚ angle with the horizon. How many
feet off the ground is the plane after traveling an air
distance of 800 ft.? (The hypotenuse of your
triangle is 800 feet.)
30
9. A lovesick executive looks out his office
window which is 45 feet above the ground. At an
angle of depression of 35˚, he sees his fiancé on the
sidewalk talking to another man. What is the actual
distance between the man and the pair (line of sight)?
10. The famous fountain is Geneva, Switzerland, is
reported to be 400 feet tall. Find the angle of
elevation of the top of the fountain from a point 115
feet from its base.
11. A hot air balloon hovers 718 feet above one end
of a bridge that spans the Mississippi River at New
Orleans. The angle of depression from the balloon
to the other end of the bridge is 24˚. How long is
the bridge?
12. A lifeboat is 75 m below a helicopter. The pilot
holds its coordinates and radios its position back to
base as the crew prepares for the rescue operation.
The lifeboat is now seen at an angle of depression
of 26˚ to their East. How far has the boat drifted?
13. Find the radius of a circle in which a chord of
20 inches (AB) has a central angle (ACB) of 38˚.
14. Each diagonal of a rectangle has a length of 12
cm. The two acute angles formed when the
diagonals cross measure 40˚. Find the length and
width of the rectangle.
C
A
B
31
Trig Memory Sheet
I.
Reciprocal Identities. Works with or without squares.
Variations
II.
III.
1.
1. sin  
1
csc 
a) csc  
1
sin 
b) sin  csc  1
2. cos  
1
sec 
a) sec  
1
cos 
b) cos sec  1
3. tan  
1
cot 
a) cot  
1
tan 
b) tan  cot   1
Quotient Identities. Works with or without square.
sin 
tan


1.
cos 
sin 2 
a) tan  
cos 2 
cos 
cot


2.
sin 
cos 2 
a) cot  
sin 2 
2
2
Pythagorean Identities. Work only with squares.
sin 2   cos 2   1
a) sin
2
b) cos
  1  cos 2 
2
  1  sin 2 
2.
1  tan 2   sec 2 
a)
tan 2   sec 2   1
3.
1  cot 2   csc2 
a)
cot 2   csc2   1
32
College Algebra
Name ____________________
Date _______________ Pd ___
Simplifying Identities
Basic Identities
Use identities to simplify each expression.
2.
1  sec 
sin   tan 
csc y  cot y
4.
1  cot 2 
cot  csc 
sin 4   2sin 2  cos 2   cos 4 
6.
1
1

1  sin x 1  sin x
8.
cos   tan 2   1
1.
csc 
cot   tan 
3.
5.
2
(Hint: LCD)
2
Factor: x  2 xy  y
2
Hint:
now: x  2 x y  y
4
7.
2
2
2
sec  cos   cos 2 
4
Continue to back 
33
9.
sec   cos   sin  tan  
11.
cos x
1  sin 2 x
10.
12.
sin 2  csc  sec 
 sin   cos  
2
1
sin 
13.
sec t  1sec t 1
14.
1  cos 2 y
1  sin y 1  sin y 
15.
sin  1  cot 2  
16.
sin x 
17.
cot  cos  sin 
18.
cos  1  tan 2  
34
cos 2
sin x
College Algebra
Name ________________
Date __________ Pd. ____
Proving Trig Identities
1. Know the fundamental Identities and all variations of them.
2. Choose the more complicated side and reduce it to the simpler side.
3. Always keep the goal in mind.
4. If there are fractions, you may need to
a. combine two or more fractions into one
b. separate one fraction into two or more
c. multiply numerator and denominator by the conjugate, or
e. eliminate complex fractions.
5. If there are trig polynomials, you may need to factor or FOIL.
6. If all else fails, change everything to sines and cosines and use algebraic manipulations to get the
resulting expression.
GOOD LUCK! YOU CANDO THIS!! I KNOW YOU CAN!!! MANY BEFORE YOU HAVE BEEN
SUCCESSFUL!!!!
1.
1
tan 2 B
1 
cos B
sec B  1
2.
tan2 B  1 sec B

tan B
sin B
3.
cot x
1

 sin x
sec x sin x
4.
cos 2 B  sin 2 B  cos 4 B  sin 4 B
5.
csc2 x 
6.
1
7.
sin 2 x 
1  tan2 x
tan2 x
cos y  csc y
cot y
sec2 x  1
sec2 x
35
8.
9.
1  cot A tan A  1

1  cot A tan A  1
2 csc2 A  1 
1  cos A
sin2 A
2
19.
sin x  csc x  sec x   1  tan x
20.
csc2   sec2 
 tan   cot 
csc sec
21.
sin A cos A sec A


cos A sin A sin A
10.
cos A  cos3 A  cos A sin 2 A
11.
sin B  tan B
 sin B
1  sec B
22.
sin B
1  cos B

 2 csc B
1  cos B
sin B
12.
1  sec A 1  cos A

sec A  1 cos A  1
23.
1
13.
csc2 x 1  cos2 x   1
24.
cos B
1

cos B  sin B 1  tan B
14.
tan B sin B  cos B  sec B
25.
1  tan2 B
 1  sec2 B
2
1  cot B
15.
cos x sec x
 cot x
tan x
26.
1  csc A
 cot A  cos A
sec A
27.
cos A  sin A
 cos A
1  tan A
28.
tan x sin x  cos x  sec x
29.
sec2 x 1  cos2 x   tan 2 x
16.
17.
18.
1  2 cos x sec x  tan y cot y
cos y csc y
1
cot y
tan A 
1
sec A

tan A sin A
36
1
tan 2 A

cos A sec A  1
College Algebra
Standard Form of the Parabola
Name ________________
Date __________ Pd. ____
Write an equation for each parabola described.
y  x2
Assume the width is the same as
1. Vertex (3, 4) opens down
in each case.
2. Vertex (-2, 7) opens left
_________________
3. Domain
[1,  )
_________________
axis: y = 5
4. Range
_________________
5. Vertex (6,0)
( , 5)
axis: x = 3
_________________
opens up_________________, down _________________
opens right _________________, left_________________
Sketch the graph using a chart of values with at least 5 values used. Give the important facts (vertex,
axis equation, domain, range).
x  2( y  1)2  2
1
y  ( x  1) 2
3
x
y
5
x
y
Y
y
6
5
4
4
3
3
2
2
1
1
–3
–2
–1
1
2
3
4
x
–1
–2
–1
–1
–2
–3
–2
–4
37
1
2
3
4
5
6
7 x
Use completing the square to put each of the following parabolas in standard form and then
analyze the equation.
1.
y  x 2  12 x  29
2.
x  y 2  8 y  21
3.
x   y2  6 y  5
4.
y  2 x 2  20 x  41
5.
y
6.
x
1 2
x  2x  5
3
# Vertex
Direction
1
2
3
4
5
6
College Algebra
Width
Axis eq.
1 2
y  4y  9
4
Domain
Range
Name _____________________
38
Review of Parabolas, Circles, Ellipses
Date ______________ Pd ____
Complete the chart for each parabola.
Vertex
Direction
1. y  2 x 2  3
2. x  4  y  2 2
Axis Eq
Br,Na ,St
Domain
Range
1 2
x
3
3.
y
4.
y    x  1  5
5.
x
6.
y   x  2  3
7.
x  3y2
8.
x   y  1  6
2
2
2
 y  4  1
3
2
2
Write an equation for each parabola described. Assume the width is the same as y  x 2
9. Range  , 2 , axis equation: x  7 . ____________________________
10. Vertex  2,1 , opens up.
____________________________
11. Vertex  3, 4  , opens right.
____________________________
12. Domain  , 8 , axis equation: y  1 ____________________________
Rewrite the following parabola in analyze form and then complete the given information.
13. y  x 2  10 x  4 Vertex ________ Direction ________
axis of equation ________
Domain _______________ Range ______________
14. y  x 2  14 x  54
Vertex ________ Direction ________
axis of equation _______
Domain _______________ Range ______________
Find the distance between each pair of points.
15.
3,5
and  7,10 
16.
 1,4
39
and
 5,8
Complete the chart for each circle.
Equation
Center Radius Four points that graph goes through
17.  x  2 2   y  32  36
18.
19.
x 2  y 2  144
5,4
 2,5 
20.
8
 2,18 ,
Use completing the square to put the following circles in analyze form.
21. x2  y 2  8x  4 y  4  0
21. x 2  y 2  10 x  24  0
center ________, radius _______
Complete the chart for each ellipse.
Equation
Center
2
2
23.
x
y

1
9
25
2
2
24.
 x  2   y  1  1
16
36
2
25.
 x  3  y  2 2  1


49
center ________, radius _______
Major axis Eq
Domain
Range
Sketch the graph of each of the following. Complete the chart with 5 key points.
2
2
x2
y2
26. x    y  2   1
27.  x  1  y 2  16
28.

1
9
4
x y
x y
x y
40
College Algebra
WS Types of Conics
Name __________________
Date ____________ Pd ___
Directions: Each of the following equations would graph one of the following. Pick the correct type
of graph. Some equations need to be transformed to be able to tell what they graph.
A) line B) point C) circle D) null set E) ellipse H) hyperbola P) parabola
_____ 12. x  y
_____ 1. x 2  y 2  144
_____ 2.
 x  2    y  3
2
2
 25
_____ 13. 9 y 2  4 x 2  36
_____ 3. y  2 x 2  3x  4
_____ 14.
x2
y2
 1
4
9
_____ 4. x  3 y 2  5 y  6
_____ 15.
x2
y2
 1
4
9
_____ 5. 2 x  3 y  6
_____ 16.
x2 y 2

1
4
4
x2 y 2

1
25 25
_____ 17.
x2 y 2

 1
4
4
x2 y 2

1
25 36
_____ 18. 9 x 2  36 y 2  36
_____ 6.
_____ 7.
_____ 8.
_____ 9.
 x  2
2
9
2
2
 y  4

2
16
_____ 19. y 2  4 y  x  4
1
16
 x  2
16
 y  4

_____ 20. x 2  2 x  x 2  y  6
1
_____ 21. x 2  9  y 2
_____ 10. x   y  2   0
2
_____ 11. x 2 
2
_____ 22. 4  x  3  3  y  2   0
y2
 2
4
2
41
2
College Algebra
Conics Project
Name ________________________
Date ________________ Pd _____
This will be a quiz grade. The project is due December 6 but will be returned to make minor
corrections before a final grade is given.
Graph a picture, word or abstract on an 8.5 x 11 piece of graph paper.
On a separate piece of paper, each shape must have an equation with restricted domain or range to
match the part of the shape used. These equations, domains, ranges must allow another person to
recreate your drawing (minus any color or shading). No free hand sketches are allowed.
The project must include at least 2 each of parabolas, circles, and ellipses, and lines (with defined
nonzero slope). There must be at least one hyperbola and one non conic. If only part of a graph is
needed, use domain/range to crop the graph. The non conic could be absolute value, the square root
function or a higher degree function.
For functions, give the DOMAIN used.
Functions include nonvertical lines, y = parabolas and top or bottom semicircles.
It is much easier to use a whole circle or ellipse, but if you only want the top or bottom half, solve for
y, use the + or – result as needed, give DOMAIN
For vertical hyperbolas, restrict the DOMAIN.
Any time you have a x = , give the RANGE
For x = parabolas, give the Range
Again, a whole circle or ellipse is easier, but if you only want the left or right half, solve for x, use the
+ or – result as needed, give the RANGE.
For horizontal hyperbolas, restrict the RANGE
My graphing utility will not eliminate a branch of a hyperbola. Either incorporate both branches into
your picture, or position it so one branch is off the visible coordinate axes. This can be done by
placement of the center and the length of the transverse axis.
42
College Algebra
Part of Picture
Conics Project
Name ________________________
Date ________________ Pd _____
Equation
Domain or Range
Shape
Hyperbola
Ellipse
Ellipse
Parabola
Parabola
Circle
Circle
Line
Line
Other
43
Name _____________________________
Equation
Vertex
Direction
y  a  x  h  k
 h,k 
a  0, up
a  0, down
 h,k 
a  0, right
a  0, left
2
1
y   x  2
2
2
y   x  4
2
3
y   x  3  4
4
y   x  5  4
5
y  2  x  3  2
6
y  3  x  2   1
7
y
8
Domain
Range
xh
B if a  1
N if a  1
  , 
All Real
up   k ,  
down    ,k 
yk
B if a  1
N if a  1
right   h, 
left    ,h 
All Real
  , 
2
2
2
2
1
2
 x  1  3
2
2
2
y   x  2 1
3
x  a y  k  h
2
x  y2  2
9
10
11
x   y  1
12
x   y  3
13
x   y  2 1
14
x   y  4  2
15
x  2  y  3
16
Axis Equation Broader / Standard /
Narrower
x   y2
2
2
2
2
x
2
2
2
 y  3  2
3
44
45
46