1 Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to solve simple radical equations. In working with symmetry of graphs, students will apply the concepts of similarity and transformations of geometric figures inherent in the Grade 7 standards for geometry. The task offers students an in-depth discussion of even and odd symmetry of graphs of functions and transformations of graphs by reflection in the coordinate axes. These topics and the discussion of solving rational equations and quadratic equations of the form x² c = 0, c 0, reinforce topics from geometry, especially the topics of symmetry and transformation of geometric figures, similar triangles, and the Pythagorean Theorem. Key Standards MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither. MM1A3. Students will solve simple equations. a. Solve quadratic equations in the form ax² + bx + c = 0 where a = 1 by using factorization and finding square roots where applicable. d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1. Possible Materials o Miras o Patty paper o Rulers/ Straightedges o TI Navigator System o Graphing calculators
2 Task In middle school, students learned about line and rotational symmetry. Students can be asked to remember that a figure has line symmetry if there is a line that divides the figure into two parts that are mirror images of each other and that a figure has rotational symmetry if, when rotated by an angle of 180 degrees or less about its center, the figure aligns with itself. A figure can also have point symmetry. A figure is symmetric about a single point if when rotated about that point 180 degrees it aligns with itself. So, rotational symmetry of 180 degrees is also symmetry about the center. Sample Questions/Solutions 1.We all see many company logos everyday. These logos often have symmetry. For each logo shown below, identify and explain any symmetries you see. Solution: Except for the Logo 4, Logo 14 and Logo 16, each logo above has line symmetry with respect to the vertical that cuts the figure in half. If we omit the year 1785, the remaining figure in Logo 14 has symmetry with respect to the vertical that divides the figure in half. Logo 4 has rotational symmetry of 180 ; this symmetry is also described as point symmetry about the center of the design. Logos 9, 10, and 17 have rotational symmetry of 120 in the both the clockwise and counterclockwise directions. Logos 9 and 10 also have line symmetry about the lines obtained by rotating the vertical line through the center 120 clockwise and counterclockwise. Logo 11 has rotational symmetry of 72 and 144 in both the clockwise and counterclockwise directions. If we ignore the colors, Logo 16 has rotational symmetry of 72 and 144 in both the clockwise and counterclockwise directions. Logo 2 has rotational symmetry of 20, 40, 60, 80, 100, 120, 140, 160, and 180 in both the clockwise and counterclockwise directions. The logo also has line symmetry across any line obtained by rotating the vertical line through the center through a multiple of 10. These lines are the diagonal lines that go from one of the outer points through the center to another outer point and the lines half way between any two of these diagonals.
3 2. A textile company called Uniform Universe has been hired to manufacture some military uniforms. To complete the order, they need embroidered patches with the military insignia for a sergeant in the United States Army. To save on costs, Uniform Universe subcontracted a portion of their work to a foreign company. The machines that embroider the insignia design require a mathematical description. The foreign company incorrectly used the design at the right, which is the insignia for a British Sergeant. They sent the following description for the portion of the design to be embroidered in black. a. Match the lines in the design to the functions indicated in the table. Comments: Students may use graphing calculators or other technology to reproduce the design. In order to restrict the domain in a graph drawn by technology, divide the expression for the function by the Boolean expression ( 7 x) ( x 7). When the Boolean expression is false, we have division by 0 and no graph is drawn. When the Boolean expression is true, the graph is drawn. Solution: The graphs match the functions listed in the table in order from top to bottom. The absolute value graph is shifted down 1 unit, 3 units, 5 units, and 7 units respectively. b. When the design is stitched on a machine, wide black stitching is centered along the lines given by the equations above. The British Sergeant s insignia has a light-colored embroidery between the lines of black. Write a description for the lines on which the light-colored stitching will be centered. Use a table format similar to that shown above.
4 3. Jessica, a manager at Uniform Universe, immediately noticed the design error when she saw some of the prototype uniforms. The sergeant s insignia was upside down from the correct insignia for a U.S. sergeant, which is shown at the right. Jessica checked the description that had been sent by the foreign contractor. She immediately realized how to fix the insignia. So, she ed the foreign supplier to point out the mistake and to inform the company that the error could be corrected the by reflecting each of the functions in the x-axis. Ankit, an employee at the foreign textile company, ed Jessica back and included the graph below to verify that Uniform Universe would be satisfied with the new formulas. a. What type of symmetry does the incorrect insignia have? Line symmetry b. If it is symmetric about a point, line, or lines, write the associated coordinates of the point or equation(s) for the lines of symmetry. The line of symmetry is the y-axis, which is the vertical line with equaiton x = 0. c. Does the corrected insignia have the same symmetry? The correct insignia is symmetric with respect to the y-axis, that is, the vertical line x = 0. d. Write the mathematical description of the design for the U.S. sergeant insignia, as shown in the graph below. Verify that your mathematical description yields the graphs shown.
5 The correct insignia is described by reflecting each of the equations through the x-axis. To reflect a graph through the x-axis, we set the y-coordinate equal to the negative of the previous y-coordinate. So, to reflect y = x 7 through the x-axis, we form the equation y = ( x 7) = x + 7. We can see that the graphs are the graph of y = x shifted up 1 unit, 3 units, 5 units, and 7 units, respectively. e. Let f denote any one of the functions graphed in the British sergeant s insigniz or the U.S. sergeant s insignia. Compare f(1) and f( 1), f(2.5) and f( 2.5), f(3.7) and f( 3.7). If x is a number such that 0 x 7, how do f(x) and f( x) compare? On each of the graphs, f(1) = f( 1), f(2.5) = f( 2.5), f(3.7) = f( 3.7). If x is a number such that 0 x 7, then f(x) = f( x). f. Let a be a constant other than the number 0 and let g denote the function whose formula is given by g(x) = ax 2. You studied the shapes of these graphs in Unit 1. Look back at some examples for particular choices of a. What type of symmetry do these graphs have? If x is a positive number, how do g(x) and g( x) compare? Graphs of some of the functions of the form g(x) = ax 2 considered in Unit 1, Painted Cubes Learning Task, are shown below. Each of the graphs has line symmetry with respect to the y-axis. For any positive number x, the value of g(x) is the same as the value of g( x).
6 4. We call a function f an even function if, for any number x in the domain of f, x is also in the domain and f( x) = f(x). a. Suppose f is an even function and the point (3, 5) is on the graph of f. What other point do you know must be on the graph of f? Explain. Comments: Since (3, 5) is on the graph, 3 is in the domain and f(3) = 5. Then, by the definition of an even function, 3 is also in the domain and f( 3) = f(3) = 5. Solution: ( 3, 5) is also on the graph of f b. Suppose f is an even function and the point ( 2, 4) is on the graph of f. What other point do you know must be on the graph of f? Explain. Comments: Since ( 2, 4) is on the graph, 2 is in the domain and f( 2) = 4. Then, by the definition of an even function, ( 2) = 2 is also in the domain and f(2) = f( 2) = 4. Solution: (2, 4) is also on the graph of f c. If (a, b) is a point on the graph of an even function f, what other point is also on the graph of f? Comments: Since ( a, b) is on the graph, a is in the domain and f(a) = b. Then, by the definition of an even function, a is also in the domain and f( a) = f( a) = b. Solution: ( a, b) d. What symmetry does the graph of an even function have? Explain why.
7 Solution: The graph of an even function has line symmetry with respect to the y-axis. For any number b, the points (x, b) and ( x, b) are at the same height on the grid and are equidistant from the y-axis; thus, they represent line symmetry with respect to the y-axis. Since f(x) and f( x) are the same number, the points (x, f(x)) and ( x, f( x)) are of the form (x, b) and ( x, b), and thus, are symmetric with respect to the y-axis. See the diagram above. e. Consider the function k, which is an even function. Part of the graph of k is shown at the right. Using the information that k is an even function, complete the graph for the rest of the domain. Comments: Since the graph of an even function is symmetric with respect to the y- axis, we reflect the part of the graph shown in the y-axis to obtain the rest of the graph, as shown.
8 5. When Jessica s supervisor, Malcom, saw the revised design, he told Jessica that he did not think that the U.S. Army would be satisfied. He pointed out that, while the revision did turn the design right-side up, it did not account for the slight curve in the lines in the real U.S. sergeant s insignia. He suggested that a square root function might be a better choice than an absolute value function and told Jessica to work with the foreign contractor to get a more accurate design. Jessica ed Ankit to let him know that the design needed to be revised again to have lines with a curve similar to the picture from the U.S. Army website, as shown above, and suggested that he try the square root function. a. Ankit graphed the square root function, y = x, and decided to limit the domain to 0 x 4. Set up a grid using a scale of ½ -inch for each unit, and graph the square root function on this limited domain. For accuracy, plot points for the following domain values: : 0, 1/4, 1, 25/16, 9/4, 49/16, 4. Comments: The graph, using the scale of ½ -inch for each unit, is shown in blue below, after part b. b. Ankit saw that his graph of the square root function (on the domain 0 x 4 ) looked like the curve that forms the lower right edge of the British sergeant s insignia. He knew that he could reflect the graph through the x-axis to turn the curve over for the U.S. sergeant s insignia, but first he needed to reflect the graph through the y-axis in order to get the full curve to form the lower edge of the British sergeant s insignia. He knew that he needed to input a negative number and then get the square root of the corresponding positive number, so he tried graphing the function y = x on the domain -4 x 0. Reproduce Ankit s graph using the scale of ½-inch for each unit and using the opposites of the domain values in part a as some accurate points on the graph: 0, 1/4, 1, -25/16, -9/4, -49/16, 4. Comments: This graph is shown below in red on the same axes with the graph for part a.
9 c. To see an additional example of reflecting a graph through the axes, graph each of the following and compare the graphs. d. Given an equation of the form y = f(x), there is a specific process for creating a second equation whose graph is obtained by reflecting the graph of y = f(x) through the x-axis, and there is different process for creating an equation whose graph is obtained by reflecting the graph of of y = f(x) through the y-axis. Expalin these processes in your own words. Comments: This item begins a process of helping students understand the similarities and differences in the two types of reflection. Reflecting the graph of a function through the x-axis is the simpler idea; it was introduced in Unit 1. Understanding reflection through the y-axis requires more sophisticated understanding of function notation and, hence, was delayed until Unit 5. One of the difficulties in helping students at this level understand the distinction is that, for three of the six basic functions studied in Mathematics I, the two types of reflection produce the same result. In particular, for the function f(x) = x Given an equation of the form y = f(x).: to reflect through the x-axis, form the equation y = -f(x).. to reflect through the y-axis, form the equation y = f(-x)..
10 e. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part a in the x-axis and then shifting it up 8 units. Call this function f1, read f sub one. f. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part b in the x-axis and then shifting it up 8 units. Call this function f2, read f sub two. When graphed on the same axes, the graphs of the functions f1 and f2 give the curve shown above. Solution: f 2 (x) = - -x + 8
11 g. Ankit completed his mathematical definition for the U.S. Army sergeant s insignia and sent it to Jessica along with the graph shown at the right. Based on the graph and your work above, write Ankit s specifications for black embroidery of the insignia as shown in the graph. Supporting Lessons Available :
Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
Lesson 17 Name Date Lesson 17: Graphing the Logarithm Function Exit Ticket Graph the function () = log () without using a calculator, and identify its key features. Lesson 17: Graphing the Logarithm Function
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
The straight line 2.7 Introduction Probably the most important function and graph that you will use are those associated with the straight line. A large number of relationships between engineering variables
Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable
Section 3.1 Quadratic Functions in Vertex Form Pre-Calculus 20 Chapter 3 Notes Using a table of values, graph y = x 2 y x y=x 2-2 4-2 4 x Using a table of values, graph y = -1x 2 (or y = -x 2 ) y x y=-x
Key Terms for This Session Session 7 Symmetry Previously Introduced coordinates reflection rotation translation New in This Session: frieze pattern glide reflection reflection or line symmetry rotation
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
Title: Graphing Quadratic Equations in Standard Form Class: Math 100 or 107 Author: Sharareh Masooman Instructions to tutor: Read instructions under Activity and follow all steps for each problem exactly
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary axis of symmetry standard form minimum value maximum
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
TEKS: a(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete,
Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)
Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
Concepts Patterns Coordinate geometry Rotation and reflection transformations of the plane Materials Student activity sheet Transformational Geometry in the Coordinate Plane Rotations and Reflections Construction
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
1.4 Exponential and logarithm graphs. Example 1. Recall that b = 2 a if and only if a = log 2 (b) That tells us that the functions f(x) = 2 x and g(x) = log 2 (x) are inverse functions. It also tells us
Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
This is Solving Quadratic Equations and Graphing Parabolas, chapter 9 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
1.5 ANALYZING GRAPHS OF FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which
PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
Outcome R3 Quadratic Functions McGraw-Hill 3.1, 3.2 Key Terms: Quadratic function Parabola Vertex (of a parabola) Minimum value Maximum value Axis of symmetry Vertex form (of a quadratic function) Standard
3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,
Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics A Study Guide for Students PREPARING FOR 12 GRADE Nova Scotia Examinations in Mathematics For more information,
Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles
Course Title: Honors Algebra Course Level: Honors Textbook: Algebra Publisher: McDougall Littell The following is a list of key topics studied in Honors Algebra. Identify and use the properties of operations
Definition of the Quadratic Formula The Quadratic Formula uses the a, b and c from numbers; they are the "numerical coefficients"., where a, b and c are just The Quadratic Formula is: For ax 2 + bx + c
Activity: TEKS: Overview: Materials: Grouping: Time: Square Roots & Quadratics: What s the Connection? (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
Lesson 1-7 Graphing Absolute Value Functions Name Objectives: In this activity, students will relate the piecewise function to the graph of the absolute value function and continue their development of
Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number
CCSS Key: The Number System (NS) Expressions & Equations (EE) Functions (F) Geometry (G) Statistics & Probability (SP) Common Core State Standard I Can Statements 8 th Grade Mathematics 8.NS.1. Understand
High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems
Math 1111 Journal Entries Unit I (Sections 1.1-1.2, 1.4-1.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A
1 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a U-shape called a. Standard Form Vertex Form axis
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much
Mathematics Common Core Domain Mathematics Common Core Cluster Mathematics Common Core Standard Number System Know that there are numbers that are not rational, and approximate them by rational numbers.
This is Solving Equations and Inequalities, chapter 6 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/
Chapter 14 Transformations Questions For Thought 1. When you take a picture, how does the real world image become a reduced celluloid or digital image? 2. How are maps of the Earth made to scale? 3. How
Pennsylvania System of School Assessment The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read
Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal o Graph and transform
GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.
2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to firstname.lastname@example.org. Thank you! PLEASE NOTE
Accelerated Mathematics I Frameworks Student Edition Unit 6 Algebraic Investigations: Quadratics and More, Part nd Edition March, 011 Accelerated Mathematics I Unit 6 nd Edition Table of Contents INTRODUCTION:...
Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote
Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close
STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
Math Objectives Students will be able to predict how a specific change in the value of a will affect the shape of the graph of the quadratic ax bxc. Students will be able to predict how a specific change
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
Math Objectives Students will be able to identify a restricted domain interval and use function translations and dilations to choose and position a portion of the graph accurately in the plane to match