Using the same procedure that was used in Project 5, enter matrix A and its label into an Excel spreadsheet:
|
|
- Corey Nash
- 6 years ago
- Views:
Transcription
1 Math 6: Ecel Lab 6 Summer Inverse Matrices, Determinates and Applications: (Stud sections, 6, 7 and in the matri book in order to full understand the topic.) This Lab will illustrate how Ecel can help ou find and use the inverse of a matri. Recall, a square matri A has an inverse matri (denoted A - ) if it is true that the matri multiplication A*A - I (the identit matri). Using Ecel to Find A - : Eample : Given matri A, as shown below, use Ecel to find A -. A Using the same procedure that was used in Project, enter matri A and its label into an Ecel spreadsheet: Since A is a matri, A - will also be a matri. Tpe the label "A^(-)" into the desired cell (in our eample, this is cell A). To the right of the label, highlight the block of cells that will contain the inverse (cells B to D7 in the eample). While the block of cells is still highlighted, tpe the following command: minverse(b:d). Now press the ke combination Shift-Ctrl-Enter, as ou did for matri operations in Project. Check our answer - it should look like the following:
2 When A - eists, the inverse can be used to solve a sstem of equations. Now that ou know how to use Ecel to find the inverse of a matri, eamine the following to see how to solve a sstem of equations using the inverse of the coefficient matri. Solving a Sstem of Equations: Eample : Consider the following sstem of equations: This sstem can be rewritten in matri form AX B where A is the matri obtained from the coefficients of the variables, X is the matri obtained from the variables, and B is the matri obtained from the constants. A, X, B giving the following matri sstem of equations. Given that matri A is invertible (i.e., has an inverse), use Ecel to find A - : The following steps show WHY the process of finding a solution to a sstem of equations AX B works when A - eists.
3 . Original sstem of equations and A - :, A -. Multipl both sides of the sstem of equations b A - :. Recall: A A - I, so simplif the left side of equation and complete the multiplication on the right side: 7 7. Since I is an identit matri, I X X, so simplif the left side of equation So -7, -7, and - is the solution to the sstem of equations since these are variables that make all three equations in the sstem true. Rule: If A is an invertible matri (i.e., A - eists), then the matri equation AX B has the unique solution X A - B. Note: For matri A, ou were told that A - eists. Suppose ou don't know whether the square matri in our problem has an inverse. There are various tests (some of which will be discussed in this course) to determine if a matri has an inverse, but ou can also tell from the output obtained b using Ecel's minverse command. In the following, matri A (which was known NOT to have an inverse) was entered into the Ecel worksheet. The correct steps were followed to generate the inverse of A, but note the contents of the cells that were chosen to contain the inverse. The #NUM! notation that appears in those cells indicates that no inverse eists.
4 Sstem of Equations (an Eample): (See section in the matri book for additional help, if necessar) The following problem is an eample of an application problem that can be solved using an inverse matri. Eample : John, Mar, and Julie went to the same grocer store to bu sugar, coffee, and butter. John bought pounds of sugar, 6 pounds of coffee, and pounds of butter for $; Mar bought pounds of sugar, pounds of coffee, and pounds of butter for $; while Julie paid $ for pounds of sugar, pounds of coffee, and 6 pounds of butter. Find the selling price per pound of each item. Let X where the price (in dollars) per pound of the price (in dollars) per pound of the price (in dollars) per pound of sugar coffee. butter John, therefore, paid $ for sugar, $6 for coffee, and $ for butter. So he spent a total of $( 6 ). Since it is given that he spent $, the first equation becomes 6. Following the same technique for Mar's and Julie's purchases to obtain the other two equations, we get the following sstem of equations: 6 6 This can easil be converted into a sstem of equations in matri form as follows: 6 6 Setting up and solving the sstem in Ecel gives the following: A 6 B 6 A^(-) X A^(-)*B.6.. Using our definition of X and the variables,, and, ou determine that the selling price for sugar is $.6 per pound; coffee sells for $. per pound; and butter is $. per pound.
5 NOTE:When finding inverses, ou ma find that ou want to change the appearance of some of the number values in our spreadsheet. There are two tpes of numbers for which this ma be especiall desirable.. You often will have unwield decimal answers and ma wish to reduce the number of decimal places that are displaed. This does not change the actual value that is retained in memor, onl the decimal representation that is displaed on the screen and printed in our report. To make this change, highlight the cells containing the displa that ou want to change and click on the decimal icons in the toolbar. The decimal icon shows decimals, eros, and arrows. There is one for increasing the number of decimals displaed and another for decreasing the number of decimals. To reduce the number of decimals displaed, simpl click on the appropriate icon. Continue clicking on the icon until ou have the decimal representation ou desire. An alternative method of adjusting the number of decimal places is to highlight the desired cells and select Format-Cells-Number from the toolbar. Then choose the desired number of decimals.. The second tpe of number that ou ma wish to change is a value that is displaed in a format similar to.7e-6. This particular value is scientific notation for.7*^(-6). The corresponding decimal representation for this value is.7. For all practical purposes, because this number is so etremel small, it ma be interpreted as ero. (In fact, values such as this often show up in cells where ou would obtain a ero value if ou were finding the inverse b hand but are displaed in Ecel as etremel small non-ero values due to rounding.) You ma wish to replace values such as this with ero. You must be careful, however! Tring to edit an output matri in Ecel will cause our Ecel program to "jam". In order to edit the output values, ou must convert the commands in the output cells to actual numerical values. To do this, first highlight the entire matri, then select Edit-Cop. Leave the entire matri highlighted (remember, ou are pasting over the same cells that ou copied from) and select Edit-Paste Special. A dialogue bo will pop up on the screen. You should select Values and then click on OK. This converts all of the cells in the matri to regular numerical values. Now ou ma delete a value and replace it with another value such as ero. Problems to turn in: (Part ) Work through the eample problems in the lab introduction before attempting these problems. Do not turn in the worked eamples from the introduction. Also, do not hand in this document with our lab. Work all problems in order. Do not sa see attached or other notation referring the grader to another location in the lab for part of a problem. Label problems and parts of problems with appropriate numbers and/or letters. Everthing on this lab should be tped ecept where specificall stated that hand-written work is allowed. In those cases, leave space in our document to NEATLY write the appropriate solution b hand. Use Ecel and Ecel's matri operations to complete all of the following problems.. Find the inverse for each of the following matrices, if that inverse eists. If the inverse does not eist, clearl sa so. (a) A (b) B 6
6 . A television manufacturing compan makes three models of television sets. The processes used in building the TVs are wiring, assembl, and testing. The time (in hours) that each model requires is provided in the following table. Wiring Time Assembl Time Testing Time Super Model 6 Delue Model Regular Model Hours available var from month to month, so the compan needs to determine the number of each tpe of TV that will be produced during a given month. The number of hours available during Januar are 6 hours of wiring time, hours of assembl time, and 8 hours of testing time. (a) Set up the sstem of linear equations to represent this problem (see following note). Use the variables of our choice, but don't forget to define our variables. You should leave space in our printed work and neatl write the sstem b hand on our printout. Note: Unlike the eamples that we have done in class the information ou need is found in the columns rather than the rows. When information is in a table, it ma or ma not be used in eactl that position. Because the total amount of wiring time available for assembl is 6 hours, the wiring time used for all models must add up to that amount. This requires looking at the wiring time column. To solve the sstem and find the number of TVs that will be manufactured during Januar, the method of solution will use the inverse of the coefficient matri. This will also allow future production levels to be quickl determined. (b) Solve the sstem to determine the number of TVs that can be made during Januar. State our answer in a complete sentence using terminolog appropriate to the contet of the problem. (c) The number of hours available during Februar are 67 hours of wiring time, hours of assembl time, and 7 hours of testing time. Solve this new sstem using the same inverse that ou found before (do not find the inverse again). State our answer in a complete sentence using terminolog appropriate to the contet of the problem. Determinants and Cramer's Rule In previous projects ou have eplored the use of Ecel to perform a variet of matri operations and using these operations to solve sstems of equations. In this project ou will be studing a function associated with matrices called the determinant. The input into the determinant function is a square matri and the output is a real number. The notation det(a) is used to indicate the determinant of matri A. 6
7 Determinant of a Matri: There is a simple rule for finding the determinant of a matri b hand: Rule: If M is the matri a M c b d, then det(m) ad - bc. Eample : Given Solution: M, find det(m). det(m) ()(-) - (-)() - 6. Note: Another standard notation that is used to indicate the determinant of a matri is to replace the square brackets surrounding the matri with a pair of parallel lines. For instance, the determinant of the matri in Eample can be written as det(m) (as shown in the eample) or as. Using Ecel to Find the Determinant of a Matri: As demonstrated in the above eample, finding the determinant of a matri is a simple process, but finding the determinant of a larger matri can be ver tedious. Ecel, however, has a command that makes it eas to find the determinant of an square matri. Eample : Given C, use Ecel to find det(c). Begin b entering our matri into an Ecel spreadsheet: 7
8 In order to use Ecel to find the determinant, tpe in the label "det(c) ". To the right of the cell containing the label, highlight one empt cell, tpe in the command mdeterm(b:d), then hold down Shift-Ctrl kes and hit Enter. This leads to the following results: C det(c) - Therefore, the determinant for C is given b det(c) -. The determinant of a square matri is a tool that has man useful applications. This project will be focusing on two of those applications. The determinant of a square matri, A, can be used to discover if A - eists, and determinants can be used to solve certain tpes of sstems of equations. 8
9 Solving a Sstem of Equations Using Determinants: Determinants can be used to solve a sstem of equations if the coefficient matri of the sstem has an inverse. This method of using determinants to solve a sstem of equations is called Cramer s Rule. Eample : Consider the following sstem of equations: When using Cramer's Rule to solve a sstem of equations such as this, first write the matri equation for the sstem in the form AX B: Net find det(a) to see if A - eists. If it does not, this sstem cannot be solved using Cramer's Rule. Note that A is the same matri as matri C in Eample. Since det(a) det(c) -, A is invertible. Now create new square matrices. You ma name the matrices with an variable names that ou wish (as long as the names ou choose are not A, X, or B, since those are alread used in this problem). For purposes of eplanation, in this eample the matrices are named A, A, and A. Net create matri A b replacing column in matri A with the entries in matri B; create matri A b replacing column in matri A with the entries in matri B; and create A b replacing column in matri A with the entries in matri B. A, A, and A Now find the determinant for each of the three new matrices:
10 A B det(a) - A det(a) - A det(a) - Using Ecel (as shown above), ou should find that: det(a ) -, det(a ) -, and det(a ) - (Notice the determinant arra is alwas written with straight sides). Now ou should have determinants: det(a) -, det(a ) -, det(a ) -, and det(a ) -. Using these determinants, and the following equations, ou are able to find the solution to the sstem of equations: In general, det(a), det(a) det(a ), and det(a) det(a ), so for this problem, det(a),, and Cramer s Rule ma be used to solve the same tpes of sstems of equations that can be solved using the method of inverse matrices. That is, both methods are onl useful in solving sstems that have a unique solution.
11 Problems to turn in: (Part ). Find the determinant of each of the following matrices using Ecel. (a) A. Each week at a furniture factor there are work hours available in the construction department, work hours available in the painting department, and work hours available in the packing department. Producing a chair requires hours of construction, hour of painting, and hours for packing. Producing a table requires hours of construction, hours of painting, and hours for packing. Producing a chest requires 8 hours of construction, 6 hours of painting, and hours for packing. You want to use Ecel to determine if all available time is used in ever department, how man of each item are to be produced each week. Using,, and to represent the number of chairs, tables, and chests, respectivel results in the following sstem of equations: 8 6 Solve the sstem using Cramer's Rule. Clearl state our final answer in a complete sentence using appropriate units.
Matrix Representations
CONDENSED LESSON 6. Matri Representations In this lesson, ou Represent closed sstems with transition diagrams and transition matrices Use matrices to organize information Sandra works at a da-care center.
More information2.4 Polynomial and Rational Functions
Polnomial Functions Given a linear function f() = m + b, we can add a square term, and get a quadratic function g() = a 2 + f() = a 2 + m + b. We can continue adding terms of higher degrees, e.g. we can
More informationImage Metamorphosis By Affine Transformations
Image Metamorphosis B Affine Transformations Tim Mers and Peter Spiegel December 16, 2005 Abstract Among the man was to manipulate an image is a technique known as morphing. Image morphing is a special
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More information12.4 The Ellipse. Standard Form of an Ellipse Centered at (0, 0) (0, b) (0, -b) center
. The Ellipse The net one of our conic sections we would like to discuss is the ellipse. We will start b looking at the ellipse centered at the origin and then move it awa from the origin. Standard Form
More informationy = f(x) x (x, f(x)) f(x) g(x) = f(x) + 2 (x, g(x)) 0 (0, 1) 1 3 (0, 3) 2 (2, 3) 3 5 (2, 5) 4 (4, 3) 3 5 (4, 5) 5 (5, 5) 5 7 (5, 7)
0 Relations and Functions.7 Transformations In this section, we stud how the graphs of functions change, or transform, when certain specialized modifications are made to their formulas. The transformations
More informationGraphs and Functions
CHAPTER Graphs and Functions. Graphing Equations. Introduction to Functions. Graphing Linear Functions. The Slope of a Line. Equations of Lines Integrated Review Linear Equations in Two Variables.6 Graphing
More informationMath 1526: Excel Lab 5 Summer 2004
Math 5: Excel Lab 5 Summer Matrix Algebra In order to complete matrix operations using Excel, you must learn how to use arrays. First you must learn how to enter a matrix into a spreadsheet. For practice,
More informationLINEAR PROGRAMMING. Straight line graphs LESSON
LINEAR PROGRAMMING Traditionall we appl our knowledge of Linear Programming to help us solve real world problems (which is referred to as modelling). Linear Programming is often linked to the field of
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More informationPre-Algebra Notes Unit 8: Graphs and Functions
Pre-Algebra Notes Unit 8: Graphs and Functions The Coordinate Plane A coordinate plane is formed b the intersection of a horizontal number line called the -ais and a vertical number line called the -ais.
More informationWeek 10. Topic 1 Polynomial Functions
Week 10 Topic 1 Polnomial Functions 1 Week 10 Topic 1 Polnomial Functions Reading Polnomial functions result from adding power functions 1 together. Their graphs can be ver complicated, so the come up
More informationVocabulary. Term Page Definition Clarifying Example. dependent variable. domain. function. independent variable. parent function.
CHAPTER 1 Vocabular The table contains important vocabular terms from Chapter 1. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. dependent variable Term Page
More informationExam January? 9:30 11:30
UNIT STATISTICS Date Lesson TOPIC Homework Dec. 9 Dec. Dec. Jan. 9 Jan. 0 Jan....... Representing Data WS. Histograms WS. Measures of Central Tendenc Find the mean, median, and mode of the data sets on
More information3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately.
Math 65 Weekl Activit 1 (50 points) Name: Simplif the following epressions. Make sure to use the = smbol appropriatel. Due (1) (a) - 4 (b) ( - ) 4 () 8 + 5 6 () 1 5 5 Evaluate the epressions when = - and
More informationscience. In this course we investigate problems both algebraically and graphically.
Section. Graphs. Graphs Much of algebra is concerned with solving equations. Man algebraic techniques have been developed to provide insights into various sorts of equations and those techniques are essential
More informationA Rational Existence Introduction to Rational Functions
Lesson. Skills Practice Name Date A Rational Eistence Introduction to Rational Functions Vocabular Write the term that best completes each sentence.. A is an function that can be written as the ratio of
More information3.1 Functions. The relation {(2, 7), (3, 8), (3, 9), (4, 10)} is not a function because, when x is 3, y can equal 8 or 9.
3. Functions Cubic packages with edge lengths of cm, 7 cm, and 8 cm have volumes of 3 or cm 3, 7 3 or 33 cm 3, and 8 3 or 5 cm 3. These values can be written as a relation, which is a set of ordered pairs,
More informationMATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED
FOM 11 T9 GRAPHING LINEAR EQUATIONS REVIEW - 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) -INTERCEPT = the point where the graph touches or crosses the -ais. It occurs when = 0. ) -INTERCEPT = the
More informationLesson 12: Sine 5 = 15 3
Lesson 12: Sine How did ou do on that last worksheet? Is finding the opposite side and adjacent side of an angle super-duper eas for ou now? Good, now I can show ou wh I wanted ou to learn that first.
More informationPrecalculus Unit 6 Practice
Lesson 34-. What is the sum of 0 6 6 0 9 6 3 7 0? 0 9 8 A. 0 6 6 4 3 0 9 8 Precalculus Unit 6 Practice Model with mathematics. A cellphone compan offers three different models with two different plans.
More informationSystems of Linear Equations
Sstems of Linear Equations Gaussian Elimination Tpes of Solutions A linear equation is an equation that can be written in the form: a a a n n b The coefficients a i and the constant b can be real or comple
More informationIt s Not Complex Just Its Solutions Are Complex!
It s Not Comple Just Its Solutions Are Comple! Solving Quadratics with Comple Solutions 15.5 Learning Goals In this lesson, ou will: Calculate comple roots of quadratic equations and comple zeros of quadratic
More informationMath 1525 Excel Lab 9 Fall 2000 This lab is designed to help you discover how to use Excel to identify relative extrema for a given function.
Math 1525 Excel Lab 9 Fall 2 This lab is designed to help ou discover how to use Excel to identif relative extrema for a given function. Example #1. Stud the data table and graph below for the function
More informationand 16. Use formulas to solve for a specific variable. 2.2 Ex: use the formula A h( ), to solve for b 1.
Math A Intermediate Algebra- First Half Fall 0 Final Eam Stud Guide The eam is on Monda, December 0 th from 6:00pm 8:00pm. You are allowed a scientific calculator and a 5" b " inde card for notes. On our
More informationCHECK Your Understanding
CHECK Your Understanding. State the domain and range of each relation. Then determine whether the relation is a function, and justif our answer.. a) e) 5(, ), (, 9), (, 7), (, 5), (, ) 5 5 f) 55. State
More informationDetermine Whether Two Functions Are Equivalent. Determine whether the functions in each pair are equivalent by. and g (x) 5 x 2
.1 Functions and Equivalent Algebraic Epressions On September, 1999, the Mars Climate Orbiter crashed on its first da of orbit. Two scientific groups used different measurement sstems (Imperial and metric)
More informationPROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS
Topic 21: Problem solving with eponential functions 323 PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS Lesson 21.1 Finding function rules from graphs 21.1 OPENER 1. Plot the points from the table onto the
More informationLESSON 3.1 INTRODUCTION TO GRAPHING
LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered
More informationUnit 2: Function Transformation Chapter 1
Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph.
More informationWhy? Identify Functions A function is a relationship between input and output. In a 1 function, there is exactly one output for each input.
Functions Stopping Distance of a Passenger Car Then You solved equations with elements from a replacement set. (Lesson -5) Now Determine whether a relation is a function. Find function values. Wh? The
More informationTopic 2 Transformations of Functions
Week Topic Transformations of Functions Week Topic Transformations of Functions This topic can be a little trick, especiall when one problem has several transformations. We re going to work through each
More informationLaurie s Notes. Overview of Section 6.3
Overview of Section.3 Introduction In this lesson, eponential equations are defined. Students distinguish between linear and eponential equations, helping to focus on the definition of each. A linear function
More informationIntroduction to Homogeneous Transformations & Robot Kinematics
Introduction to Homogeneous Transformations & Robot Kinematics Jennifer Ka, Rowan Universit Computer Science Department Januar 25. Drawing Dimensional Frames in 2 Dimensions We will be working in -D coordinates,
More informationSECTION 3-4 Rational Functions
20 3 Polnomial and Rational Functions 0. Shipping. A shipping bo is reinforced with steel bands in all three directions (see the figure). A total of 20. feet of steel tape is to be used, with 6 inches
More informationGraphing Calculator Graphing with the TI-86
Graphing Calculator Graphing with the TI-86 I. Introduction The TI-86 has fift kes, man of which perform multiple functions when used in combination. Each ke has a smbol printed on its face. When a ke
More informationWhat s the Point? # 2 - Geo Fashion
What s the Point? # 2 - Geo Fashion Graph the points and connect them with line segments. Do not connect points with DNC between them. Start (-4,1) (-5,5) (-2,2) (-4,1) DNC (2,-4) (3,-3) (4,-3) (5,-4)
More informationUsing Characteristics of a Quadratic Function to Describe Its Graph. The graphs of quadratic functions can be described using key characteristics:
Chapter Summar Ke Terms standard form of a quadratic function (.1) factored form of a quadratic function (.1) verte form of a quadratic function (.1) concavit of a parabola (.1) reference points (.) transformation
More informationLearning Objectives for Section Graphs and Lines. Cartesian coordinate system. Graphs
Learning Objectives for Section 3.1-2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.
More informationGraphs, Linear Equations, and Functions
Graphs, Linear Equations, and Functions. The Rectangular R. Coordinate Fractions Sstem bjectives. Interpret a line graph.. Plot ordered pairs.. Find ordered pairs that satisf a given equation. 4. Graph
More informationIntroduction to Homogeneous Transformations & Robot Kinematics
Introduction to Homogeneous Transformations & Robot Kinematics Jennifer Ka Rowan Universit Computer Science Department. Drawing Dimensional Frames in 2 Dimensions We will be working in -D coordinates,
More informationREMARKS. 8.2 Graphs of Quadratic Functions. A Graph of y = ax 2 + bx + c, where a > 0
8. Graphs of Quadratic Functions In an earlier section, we have learned that the graph of the linear function = m + b, where the highest power of is 1, is a straight line. What would the shape of the graph
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More informationInvestigation Free Fall
Investigation Free Fall Name Period Date You will need: a motion sensor, a small pillow or other soft object What function models the height of an object falling due to the force of gravit? Use a motion
More information20 Calculus and Structures
0 Calculus and Structures CHAPTER FUNCTIONS Calculus and Structures Copright LESSON FUNCTIONS. FUNCTIONS A function f is a relationship between an input and an output and a set of instructions as to how
More informationGraphing Cubic Functions
Locker 8 - - - - - -8 LESSON. Graphing Cubic Functions Name Class Date. Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) + k and f () = ( related to the graph of f ()
More informationConnecticut Common Core Algebra 1 Curriculum. Professional Development Materials. Unit 4 Linear Functions
Connecticut Common Core Algebra Curriculum Professional Development Materials Unit 4 Linear Functions Contents Activit 4.. What Makes a Function Linear? Activit 4.3. What is Slope? Activit 4.3. Horizontal
More informationName Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =
Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations
More informationThe Graph of an Equation
60_0P0.qd //0 :6 PM Page CHAPTER P Preparation for Calculus Archive Photos Section P. RENÉ DESCARTES (96 60) Descartes made man contributions to philosoph, science, and mathematics. The idea of representing
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More informationM O T I O N A N D D R A W I N G
2 M O T I O N A N D D R A W I N G Now that ou know our wa around the interface, ou re read to use more of Scratch s programming tools. In this chapter, ou ll do the following: Eplore Scratch s motion and
More informationSection 1.6: Graphs of Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative
Section.6: Graphs of Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
More information2-1. The Language of Functions. Vocabulary
Chapter Lesson -1 BIG IDEA A function is a special tpe of relation that can be described b ordered pairs, graphs, written rules or algebraic rules such as equations. On pages 78 and 79, nine ordered pairs
More informationHere are some guidelines for solving a linear programming problem in two variables in which an objective function is to be maximized or minimized.
Appendi F. Linear Programming F F. Linear Programming Linear Programming: A Graphical Approach Man applications in business and economics involve a process called optimization, in which ou are asked to
More informationACTIVITY: Graphing a Linear Equation. 2 x x + 1?
. Graphing Linear Equations How can ou draw its graph? How can ou recognize a linear equation? ACTIVITY: Graphing a Linear Equation Work with a partner. a. Use the equation = + to complete the table. (Choose
More informationSection 4.2 Graphing Lines
Section. Graphing Lines Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif collinear points. The order of operations (1.) Graph the line
More informationImplicit differentiation
Roberto s Notes on Differential Calculus Chapter 4: Basic differentiation rules Section 5 Implicit differentiation What ou need to know alread: Basic rules of differentiation, including the chain rule.
More informationA Rational Shift in Behavior. Translating Rational Functions. LEARnIng goals
. A Rational Shift in Behavior LEARnIng goals In this lesson, ou will: Analze rational functions with a constant added to the denominator. Compare rational functions in different forms. Identif vertical
More informationAppendix F: Systems of Inequalities
A0 Appendi F Sstems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit The statements < and are inequalities in two variables. An ordered pair
More information2.3. Horizontal and Vertical Translations of Functions. Investigate
.3 Horizontal and Vertical Translations of Functions When a video game developer is designing a game, she might have several objects displaed on the computer screen that move from one place to another
More informationLESSON 5.3 SYSTEMS OF INEQUALITIES
LESSON 5. SYSTEMS OF INEQUALITIES LESSON 5. SYSTEMS OF INEQUALITIES OVERVIEW Here s what ou ll learn in this lesson: Solving Linear Sstems a. Solving sstems of linear inequalities b graphing As a conscientious
More informationPage 1 of Translate to an algebraic expression. The translation is. 2. Use the intercepts to graph the equation.
1. Translate to an algebraic epression. The product of % and some number The translation is. (Tpe the percentage as a decimal. Use to represent some number.) 2. Use the intercepts to graph the equation.
More informationEXAMPLE A {(1, 2), (2, 4), (3, 6), (4, 8)}
Name class date Understanding Relations and Functions A relation shows how one set of things is related to, or corresponds to, another set. For instance, the equation A 5 s shows how the area of a square
More information6-1: Solving Systems by Graphing
6-1: Solving Sstems b Graphing Objective: To solve sstems of linear equations b graphing Warm Up: Graph each equation using - and -intercepts. 1. 1. 4 8. 6 9 18 4. 5 10 5 sstem of linear equations: two
More informationReview 2. Determine the coordinates of the indicated point on the graph. 1) G A) (-3, 0) B) (0, 3) C) (0, -3) D) (3, 0)
Review Determine the coordinates of the indicated point on the graph. D A B E C M G F - L J H K I - 1) G A) (-3, 0) B) (0, 3) C) (0, -3) D) (3, 0) 1) Name the quadrant or ais in which the point lies. )
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 5. Graph sketching
Roberto s Notes on Differential Calculus Chapter 8: Graphical analsis Section 5 Graph sketching What ou need to know alread: How to compute and interpret limits How to perform first and second derivative
More informationAlgebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES
UNIT LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY:
More informationSolving Systems of Equations Using Matrices With the TI-83 or TI-84
Solving Systems of Equations Using Matrices With the TI-83 or TI-84 Dimensions of a matrix: The dimensions of a matrix are the number of rows by the number of columns in the matrix. rows x columns *rows
More informationFind the Relationship: An Exercise in Graphical Analysis
Find the Relationship: An Eercise in Graphical Analsis In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables. For eample,
More information1-1. Functions. Lesson 1-1. What You ll Learn. Active Vocabulary. Scan Lesson 1-1. Write two things that you already know about functions.
1-1 Functions What You ll Learn Scan Lesson 1- Write two things that ou alread know about functions. Lesson 1-1 Active Vocabular New Vocabular Write the definition net to each term. domain dependent variable
More informationCore Connections, Course 3 Checkpoint Materials
Core Connections, Course 3 Checkpoint Materials Notes to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactl the same wa at the same time. At
More informationEssential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1.
Locker LESSON 4.4 Graphing Eponential Functions Common Core Math Standards The student is epected to: F-IF.7e Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
More informationECE241 - Digital Systems. University of Toronto. Lab #2 - Fall Introduction Computer-Aided Design Software, the DE2 Board and Simple Logic
ECE24 - Digital Sstems Universit of Toronto Lab #2 - Fall 28 Introduction Computer-Aided Design Software, the DE2 Board and Simple Logic. Introduction The purpose of this eercise is to introduce ou to
More information1.3 Introduction to Functions
. Introduction to Functions. Introduction to Functions One of the core concepts in College Algebra is the function. There are man was to describe a function and we begin b defining a function as a special
More informationPATTERNS AND ALGEBRA. He opened mathematics to many discoveries and exciting applications.
PATTERNS AND ALGEBRA The famous French philosopher and mathematician René Descartes (596 65) made a great contribution to mathematics in 67 when he published a book linking algebra and geometr for the
More informationDetermining the 2d transformation that brings one image into alignment (registers it) with another. And
Last two lectures: Representing an image as a weighted combination of other images. Toda: A different kind of coordinate sstem change. Solving the biggest problem in using eigenfaces? Toda Recognition
More informationTransforming Linear Functions
COMMON CORE Locker LESSON 6. Transforming Linear Functions Name Class Date 6. Transforming Linear Functions Essential Question: What are the was in which ou can transform the graph of a linear function?
More informationCMSC 425: Lecture 10 Basics of Skeletal Animation and Kinematics
: Lecture Basics of Skeletal Animation and Kinematics Reading: Chapt of Gregor, Game Engine Architecture. The material on kinematics is a simplification of similar concepts developed in the field of robotics,
More informationDEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO ECE-238L: Computer Logic Design Fall 2013.
ECE-8L: Computer Logic Design Fall Notes - Chapter BINARY NUMBER CONVERSIONS DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-b-digit representation of a positive integer number (powers
More informationSection 4.3 Features of a Line
Section.3 Features of a Line Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif the - and -intercepts of a line. Plotting points in the --plane
More informationLinear Programming. Revised Simplex Method, Duality of LP problems and Sensitivity analysis
Linear Programming Revised Simple Method, Dualit of LP problems and Sensitivit analsis Introduction Revised simple method is an improvement over simple method. It is computationall more efficient and accurate.
More informationTHE INVERSE GRAPH. Finding the equation of the inverse. What is a function? LESSON
LESSON THE INVERSE GRAPH The reflection of a graph in the line = will be the graph of its inverse. f() f () The line = is drawn as the dotted line. Imagine folding the page along the dotted line, the two
More informationDiscussion: Clustering Random Curves Under Spatial Dependence
Discussion: Clustering Random Curves Under Spatial Dependence Gareth M. James, Wenguang Sun and Xinghao Qiao Abstract We discuss the advantages and disadvantages of a functional approach to clustering
More informationPre-Lab Excel Problem
Pre-Lab Excel Problem Read and follow the instructions carefully! Below you are given a problem which you are to solve using Excel. If you have not used the Excel spreadsheet a limited tutorial is given
More informationIntegrating ICT into mathematics at KS4&5
Integrating ICT into mathematics at KS4&5 Tom Button tom.button@mei.org.uk www.mei.org.uk/ict/ This session will detail the was in which ICT can currentl be used in the teaching and learning of Mathematics
More information7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it
0_0705.qd /5/05 9:5 AM Page 5 Section 7.5 7.5 Sstems of Inequalities 5 Sstems of Inequalities What ou should learn Sketch the graphs of inequalities in two variables. Solve sstems of inequalities. Use
More informationReady To Go On? Skills Intervention 4-1 Graphing Relationships
Read To Go On? Skills Intervention -1 Graphing Relationships Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular continuous graph discrete graph Relating Graphs to Situations
More informationLearning Worksheet Fundamentals
1.1 LESSON 1 Learning Worksheet Fundamentals After completing this lesson, you will be able to: Create a workbook. Create a workbook from a template. Understand Microsoft Excel window elements. Select
More informationExponential Functions
6. Eponential Functions Essential Question What are some of the characteristics of the graph of an eponential function? Eploring an Eponential Function Work with a partner. Cop and complete each table
More informationChapter12. Coordinate geometry
Chapter1 Coordinate geometr Contents: A The Cartesian plane B Plotting points from a table of values C Linear relationships D Plotting graphs of linear equations E Horizontal and vertical lines F Points
More informationNote that ALL of these points are Intercepts(along an axis), something you should see often in later work.
SECTION 1.1: Plotting Coordinate Points on the X-Y Graph This should be a review subject, as it was covered in the prerequisite coursework. But as a reminder, and for practice, plot each of the following
More information[ ] [ ] Orthogonal Transformation of Cartesian Coordinates in 2D & 3D. φ = cos 1 1/ φ = tan 1 [ 2 /1]
Orthogonal Transformation of Cartesian Coordinates in 2D & 3D A vector is specified b its coordinates, so it is defined relative to a reference frame. The same vector will have different coordinates in
More informationMath 1050 Lab Activity: Graphing Transformations
Math 00 Lab Activit: Graphing Transformations Name: We'll focus on quadratic functions to eplore graphing transformations. A quadratic function is a second degree polnomial function. There are two common
More informationContents. How You May Use This Resource Guide
Contents How You Ma Use This Resource Guide ii 16 Higher Degree Equations 1 Worksheet 16.1: A Graphical Eploration of Polnomials............ 4 Worksheet 16.2: Thinking about Cubic Functions................
More informationChapter 3. Interpolation. 3.1 Introduction
Chapter 3 Interpolation 3 Introduction One of the fundamental problems in Numerical Methods is the problem of interpolation, that is given a set of data points ( k, k ) for k =,, n, how do we find a function
More informationTransformations using matrices
Transformations using matrices 6 sllabusref eferenceence Core topic: Matrices and applications In this cha 6A 6B 6C 6D 6E 6F 6G chapter Geometric transformations and matri algebra Linear transformations
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) Chapter Outline. Eponential Functions. Logarithmic Properties. Graphs of Eponential
More information11.4. You may have heard about the Richter scale rating. The Richter scale was. I Feel the Earth Move Logarithmic Functions KEY TERMS LEARNING GOALS
I Feel the Earth Move Logarithmic Functions. LEARNING GOALS KEY TERMS In this lesson, ou will: Graph the inverses of eponential functions with bases of, 1, and e. Recognize the inverse of an eponential
More informationSCIE 4101, Spring Math Review Packet #4 Algebra II (Part 1) Notes
SCIE 4101, Spring 011 Miller Math Review Packet #4 Algebra II (Part 1) Notes Matrices A matrix is a rectangular arra of numbers. The order of a matrix refers to the number of rows and columns the matrix
More informationFour Ways to Represent a Function: We can describe a specific function in the following four ways: * verbally (by a description in words);
MA19, Activit 23: What is a Function? (Section 3.1, pp. 214-22) Date: Toda s Goal: Assignments: Perhaps the most useful mathematical idea for modeling the real world is the concept of a function. We eplore
More information