LCD: 2 ( ) ( ) Interval Notation: (
|
|
- Scott Simpson
- 6 years ago
- Views:
Transcription
1 MTH 065 Class notes Lecture 3 (1.4b 1.6) Section 1.4: Linear Inequalities Graphing Linear Inequalities and Interval Notation We will look at two ways to write the solution set for a linear inequality: graphing the solution on a number line and writing it in interval notation. When you graph a linear inequality, you draw a number line and mark all the solutions. One end of the solution, the boundary, is marked with a parenthesis or bracket (depending on the inequality). Refer to the examples on top of page 34 for further illustration. Interval notation is similar to graphing; however you identify both ends of the solution set. One end is the number you already have, called the boundary. The other end is headed off to infinity in either the positive or negative direction. Then, just like with graphing, you use a parenthesis or bracket at the boundary. The end that is infinity always gets a parenthesis. Let s looking at some examples of interval notation: Note: The end with the infinity symbol always gets a parenthesis. The other end depends on the inequality symbol. Practice 4 Solve the inequality given. Then graph the solution on a number line, and write the solution set in interval notation. ( LCD: 2 OR Interval Notation: (
2 Checking Solutions to Linear Inequalities There are two parts to our answer that can be wrong: the boundary number and the inequality symbol. Practice 5 Solve the inequality Check your solution set. One possible solution: 0. We will plug this into our inequality and it should result in a true statement. True Section 1.5: Relations and Functions A relation as a rule that relates one set of numbers to another set of numbers. We call the first set of numbers the set of inputs and the second set of numbers, the set of outputs. A function is a relation where each input is assigned exactly one output. We call this relation 1 to 1. Your graphing (or scientific) calculator is another way we can generate a relation that is a function. You type in an input, you choose the rule and the calculator gives the output. Suppose you type in the number 5 for an input value. Now press the key x 2 (Pressing the key tells the calculator what rule to use to get the output.) The result, or output value, is 25.
3 Practice 1 Use the following table for A. x y=x A. How many values did you record in each cell in the table? One value B. Name at least two other function keys on your calculator. There are several other function keys on your calculator. Here are some examples: When working with relations, it is helpful to identify which variable is the input and which is the output. We do this by assigning the names independent and dependent to the variables. The variable, x, that we used in the table above to represent the input values is called the independent variable. The variable, y, that represents the output values is called the dependent variable. A way to remember this is the dependent variable depends upon both the input value and the rule to get its output value. Practice 2 For the relation given below, identify the independent and dependent variables. Jamie buys apples at a price of $1.29 a pound. The total price he pays is related to the number of pounds of apples he buys. Independent Variable: Number of pounds of apples Dependent Variable: Total cost of apples Domain and Range When working with relations we have two sets of numbers. The set of all input values is called the domain. The set of all output values is called the range. When looking at a graph, the domain is always associated with the horizontal axis and the range is associated with the vertical axis. It is common to use the graph of a function to determine what the domain and range are. When working with a table of values, unless otherwise indicated, the first row (or column) in a table contains the independent variable s values: the domain. The second row (or column) in the table contains the dependent variable s values: the range. Input Independent Variable Domain Horizontal Axis Output Dependent Variable Range Vertical Axis
4 Practice 3 The table defines a function between two variables. s t (a) Identify the independent and dependent variables. Independent Variable: s Dependent Variable: t (b) Determine the domain and range. Domain: {-4, -3, 0, 3, 4} Range: {-5, -4, -1, 2, 3} Practice 4 Given the relations in the tables below, answer the following questions. A. x y B. t p (a) Does the table represent a function? Explain. A. Yes because for each input there is exactly one output. B. No because the input -1 does not have one output, -1 and -2. (b) What is the domain? A. {0, 1, -1, 2, -2} B. {0, -1, -2, 1} (c) What is the range? A. {1, 2, 5} B. {1, -1, 2, -2} Identifying Functions from a Graph We want to be able to tell from the graph whether a relation is a function or not. The definition of a function tells us that for each input value, we have exactly one output value. Graphically this means that two different points on the graph of a function can t have the same x-coordinate. A simple test to determine whether a relation is a function is called the vertical line test. The Vertical Line Test A relation is not a function if a vertical line could be drawn that intersects the graph in more than one point. Otherwise the relation is a function.
5 A. B. C. If it is possible to draw even one vertical line that intersects in more than one place, then the relation is not a function. So note that both Figure A and Figure B are not functions. However, figure 3 is a function because a vertical line placed anywhere on the graph will intersect the graph in exactly one point. Practice 5 For each of the following graphs, (1) Determine whether the graph represents a function. Explain. (2) Give the domain and range. A. B. (1) No this graph is not a function, does not pass vertical line test at. Note that this can be generalized that no vertical line can be a function. (2) Domain: {2} Range: (1) Yes, This graph does represent a function because a vertical line anywhere on the grid will intersect the graph in exactly one point. (2) Domain: Range: Practice 6 A. Heather is driving is driving south on Interstate 5. In 30 minutes she drove 35 miles. After 48 minutes she had driven 56 miles and in one hour she had driven 70 miles. Is the distance Heather drove a function of the time she had been driving? Explain. Time (min.) Distance (mil.) Yes, every input has one output
6 B. Jeremy, Chris and Tom all went to the post office today to buy some stamps. Jeremy bought 20 stamps and paid $7.80. Chris bought 20 stamps and paid $9.00. Tom bought eight stamps and paid $1.92. Is the amount they paid for stamps a function of the number of stamps? Explain. $6.00 C o s t ( D o l l a r s ) $5.00 $4.00 $3.00 $2.00 $1.00 $ Number of stamps No because our graph does not pass the vertical line test. Increasing, Decreasing, and Constant Functions As the value of the independent variable increases, if the value of the dependent variable: always increases, the function is increasing. always decreases, the function is decreasing. remains the same, the function is constant. If we were to look at the graph, then Look at the graph from left to right. Is it going up? (Increasing) Is it going down? (Decreasing) Is it a horizontal line? (Constant) Practice 7 Determine whether the relation is increasing, decreasing, constant, or a combination. Increasing; as x increases, y also increases.
7 Section 1.6: Function Notation and Linear Functions Function Notation Not every equation is a function (where each input has exactly one output) so it is helpful to have a specific notation for equations that are functions. The letter f (for function), is most commonly used when writing functions. Other commonly used letters are g, h, r and s but you can use any letter you like. If the independent variable is n, then the dependent variable would be written as. Function Notation means f is a function of n is read f of n or "f acting on n". One of the benefits of function notation is that you can tell which of the variables is the independent variable. The variable that appears inside the parentheses on the left side is the independent variable of the equation (input value). With an equation you don t know, unless it is specified. Practice 1 Write each equation in function notation. (a), where a is the independent variable. (b), where t is the independent variable. So, why use this notation? Although it appears easier to use the original equation, there are several reasons that function notation is useful. One simple reason is that when we see function notation being used, we know that the equation represents a function. We do not have to use the definition of a function to test it. Another convenience of function notation is that it provides a shorter way of saying that we want to evaluate a function for specific values. Practice 2 Given the function (a). Find each of the following: (b)
8 (c) In order for a graph to show every possible solution, we do not have to solve several different points. Once we see a trend in the data, we can generalize by connecting solutions with a solid line of points. Eventually this leads to an example like the one below. Practice 3 Jamie buys apples at a price of $1.29 a pound. The total price he pays is related to the number of pounds of apples he buys. An equation for the total price he pays is: where n is the number of pounds that Jamie buys and is the total price he pays. (a) Identify the independent and dependent variables. Independent Variable: n Dependent Variable: T(n) (b) Create a table of values for the function. n 0 1 T(n) (c) Graph the function. $10.00 $8.00 $6.00 Cost T(n) $4.00 $2.00 $ Number of pounds of apples (n) 8
9 Linear Functions For a function to be linear: Equal steps in the independent variable will always produce equal steps for the dependent variable. Equal steps means that the difference between any two pairs of consecutive entries in the table are equal. Let's look at two tables that represent relations and determine whether they are linear functions. Look for equal steps in the input and equal steps in the output. That makes it a linear function. Practice 4 Do the tables below represent linear functions? (a) x y Yes because equal steps of 1 in x produced equal steps of 2 in y. (b) c d No, because equal steps of 3 in x did not produce equal steps in y.
Unit 1 Algebraic Functions and Graphs
Algebra 2 Unit 1 Algebraic Functions and Graphs Name: Unit 1 Day 1: Function Notation Today we are: Using Function Notation We are successful when: We can Use function notation to evaluate a function This
More informationMath 7 Notes - Unit 4 Pattern & Functions
Math 7 Notes - Unit 4 Pattern & Functions Syllabus Objective: (.) The student will create tables, charts, and graphs to etend a pattern in order to describe a linear rule, including integer values. Syllabus
More informationMath 7 Notes - Unit 4 Pattern & Functions
Math 7 Notes - Unit 4 Pattern & Functions Syllabus Objective: (3.2) The student will create tables, charts, and graphs to extend a pattern in order to describe a linear rule, including integer values.
More informationMATH 021 UNIT 2 HOMEWORK ASSIGNMENTS
MATH 021 UNIT 2 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usually, the part (A) questions ask for explanations,
More informationGraphing Linear Equations
Graphing Linear Equations Question 1: What is a rectangular coordinate system? Answer 1: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system,
More informationMath 20 Practice Exam #2 Problems and Their Solutions!
Math 20 Practice Exam #2 Problems and Their Solutions! #1) Solve the linear system by graphing: Isolate for in both equations. Graph the two lines using the slope-intercept method. The two lines intersect
More informationFOA/Algebra 1. Unit 2B Review - Linear Functions
FOA/Algebra Unit B Review Name: Date: Block: Unit B Review - Linear Functions What you need to know & be able to do. Determine if a relation is a Things to remember Every input only has one output (each
More informationSection 3.1 Objective 1: Plot Points in the Rectangular Coordinate System Video Length 12:35
Section 3.1 Video Guide The Rectangular Coordinate System and Equations in Two Variables Objectives: 1. Plot Points in the Rectangular Coordinate System 2. Determine If an Ordered Pair Satisfies an Equation
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationSummer Review for Students Entering Algebra 1
1. Order of Operations 2. Evaluating Expressions 3. Unit Rates and Proportions 4. Solving Equations 5. Multiple Representations of Linear Equations 6. Scatterplots 7. Box Plots A TI 84-Plus Graphing Calculator
More information5.7 Solving Linear Inequalities
5.7 Solving Linear Inequalities Objectives Inequality Symbols Graphing Inequalities both simple & compound Understand a solution set for an inequality Solving & Graphing a Simple Linear Inequality Solving
More information1.1 Defining Functions
1.1 Defining Functions Functions govern many interactions in our society today. Whether buying a cup of coffee at the local coffee shop or playing a video game, we are using a function in some fashion.
More informationUnit 3, Activity 1, Vocabulary Self-Awareness Chart
Unit 3, Activity, Vocabulary Self-Awareness Chart Vocabulary Self-Awareness Chart Word + - Example Definition Relation Function Domain Range Graph Vertical line test F(x) input output independent dependent
More informationUnit 0: Extending Algebra 1 Concepts
1 What is a Function? Unit 0: Extending Algebra 1 Concepts Definition: ---Function Notation--- Example: f(x) = x 2 1 Mapping Diagram Use the Vertical Line Test Interval Notation A convenient and compact
More informationNOTES: ALGEBRA FUNCTION NOTATION
STARTER: 1. Graph f by completing the table. f, y -1 0 1 4 5 NOTES: ALGEBRA 4.1 FUNCTION NOTATION y. Graph f 4 4 f 4 4, y --5-4 - - -1 0 1 y A Brief Review of Function Notation We will be using function
More informationEssential Questions. Key Terms. Algebra. Arithmetic Sequence
Linear Equations and Inequalities Introduction Average Rate of Change Coefficient Constant Rate of Change Continuous Discrete Domain End Behaviors Equation Explicit Formula Expression Factor Inequality
More informationQUADRATIC AND CUBIC GRAPHS
NAME SCHOOL INDEX NUMBER DATE QUADRATIC AND CUBIC GRAPHS KCSE 1989 2012 Form 3 Mathematics Working Space 1. 1989 Q22 P1 (a) Using the grid provided below draw the graph of y = -2x 2 + x + 8 for values
More informationRecognizing a Function
Recognizing a Function LAUNCH (7 MIN) Before Why would someone hire a dog walking service? During Do you know exactly what it would cost to hire Friendly Dog Walking? After How does each service encourage
More informationPut the Graphs for Each Health Plan on the Same Graph
At the conclusion of the technology assignment on graphing the total annual cost, you had a graph of each of health insurance plans you are examining. In this technology assignment, you ll combine those
More informationMath 111: Midterm 1 Review
Math 111: Midterm 1 Review Prerequisite material (see review section for additional problems) 1. Simplify the following: 20a 2 b 4a 2 b 1 ( 2x 3 y 2 ) 2 8 2 3 + ( 1 4 ) 1 2 2. Factor the following: a)
More informationSlide 1 / 220. Linear Relations and Functions
Slide 1 / 220 Linear Relations and Functions Slide 2 / 220 Table of Contents Domain and Range Discrete v Continuous Relations and Functions Function Notation Linear Equations Graphing a Linear Equation
More informationMath 1201 Unit 5: Relations & Functions. Ch. 5 Notes
Math 1201 Unit 5: Relations & Functions Read Building On, Big Ideas, and New Vocabulary, p. 254 text. 5.1 Representing Relations (0.5 class) Read Lesson Focus p. 256 text. Outcomes Ch. 5 Notes 1. Define
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationState the domain and range of the relation. EX: {(-1,1), (1,5), (0,3)} 1 P a g e Province Mathematics Southwest TN Community College
A relation is a set of ordered pairs of real numbers. The domain, D, of a relation is the set of all first coordinates of the ordered pairs in the relation (the xs). The range, R, of a relation is the
More information1.1 - Functions, Domain, and Range
1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain
More information3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System
3. The Rectangular Coordinate System Interpret a line graph. Objectives Interpret a line graph. Plot ordered pairs. 3 Find ordered pairs that satisfy a given equation. 4 Graph lines. 5 Find x- and y-intercepts.
More informationMini-Project 1: The Library of Functions and Piecewise-Defined Functions
Name Course Days/Start Time Mini-Project 1: The Library of Functions and Piecewise-Defined Functions Part A: The Library of Functions In your previous math class, you learned to graph equations containing
More informationPractice Test - Chapter 6
1. Write each system of equations in triangular form using Gaussian elimination. Then solve the system. Align the variables on the left side of the equal sign. Eliminate the x-term from the 2nd equation.
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationSection 7D Systems of Linear Equations
Section 7D Systems of Linear Equations Companies often look at more than one equation of a line when analyzing how their business is doing. For example a company might look at a cost equation and a profit
More informationSlide 1 / 96. Linear Relations and Functions
Slide 1 / 96 Linear Relations and Functions Slide 2 / 96 Scatter Plots Table of Contents Step, Absolute Value, Piecewise, Identity, and Constant Functions Graphing Inequalities Slide 3 / 96 Scatter Plots
More informationSec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.
Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical
More informationIntroduction : Identifying Key Features of Linear and Exponential Graphs
Introduction Real-world contexts that have two variables can be represented in a table or graphed on a coordinate plane. There are many characteristics of functions and their graphs that can provide a
More information9.1 Linear Inequalities in Two Variables Date: 2. Decide whether to use a solid line or dotted line:
9.1 Linear Inequalities in Two Variables Date: Key Ideas: Example Solve the inequality by graphing 3y 2x 6. steps 1. Rearrange the inequality so it s in mx ± b form. Don t forget to flip the inequality
More informationQuadratic Equations Group Acitivity 3 Business Project Week #5
MLC at Boise State 013 Quadratic Equations Group Acitivity 3 Business Project Week #5 In this activity we are going to further explore quadratic equations. We are going to analyze different parts of the
More informationInequalities and you 3
Inequalities and you 3 NAME: This worksheet will provide practice for solving absolute value, polynomial, and rational inequalities. We will also work on understanding why the procedures work. We will
More informationMINI LESSON. Lesson 1a Introduction to Functions
MINI LESSON Lesson 1a Introduction to Functions Lesson Objectives: 1. Define FUNCTION 2. Determine if data sets, graphs, statements, or sets of ordered pairs define functions 3. Use proper function notation
More informationA theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
Write an Equation An equation is a term used to describe a collection of numbers and variables related through mathematical operators. An algebraic equation will contain letters that relate to real quantities
More information3.2 Graphs of Linear Equations
3.2 Graphs of Linear Equations Learning Objectives Graph a linear function using an equation. Write equations and graph horizontal and vertical lines. Analyze graphs of linear functions and read conversion
More informationGEOMETRY HONORS COORDINATE GEOMETRY PACKET
GEOMETRY HONORS COORDINATE GEOMETRY PACKET Name Period Homework Lesson Assignment Day 1 - Slopes of Perpendicular WKSHT and Parallel Lines Day 2 - Writing an Equation of a Line HW- Honors TXTBK pages 615-617
More informationMA30SA Applied Math Unit D - Linear Programming Revd:
1 Introduction to Linear Programming MA30SA Applied Math Unit D - Linear Programming Revd: 120051212 1. Linear programming is a very important skill. It is a brilliant method for establishing optimum solutions
More information1.1 THIS IS LINES 1.2 FUNCTIONS
GOOGLE SHEETS 1.1 THIS IS LINES 1.2 FUNCTIONS I CAN LEARN HOW TO EVALUATE FUNCTIONS AND FIND THEIR DOMAINS. I HAVE A VIDEO POSTED ONLINE THAT HELPS YOU THROUGH THE MIRE OF GOOGLE SHEETS. ON THE VIDEO I
More informationMATH 115: Review for Chapter 1
MATH 115: Review for Chapter 1 Can you use the Distance Formula to find the distance between two points? (1) Find the distance d P, P between the points P and 1 1, 6 P 10,9. () Find the length of the line
More information6.5. SYSTEMS OF INEQUALITIES
6.5. SYSTEMS OF INEQUALITIES What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities in two variables to model and solve real-life
More informationSkill 3 Relations and Functions
Skill 3 Relations and Functions 3a: Use Interval and Set Notation 3b: Determine the domain and range of a relation given a set of ordered pairs, a graph, or an equation 3c: Determine whether a relation
More informationMATH ALGEBRA AND FUNCTIONS 5 Performance Objective Task Analysis Benchmarks/Assessment Students:
Students: 1. Use information taken from a graph or Which table, a or b, matches the linear equation to answer questions about a graph? problem situation. y 1. Students use variables in simple expressions,
More informationActivity: page 1/10 Introduction to Excel. Getting Started
Activity: page 1/10 Introduction to Excel Excel is a computer spreadsheet program. Spreadsheets are convenient to use for entering and analyzing data. Although Excel has many capabilities for analyzing
More informationMATH 081 FINAL EXAM REVIEW
MATH 081 FINAL EXAM REVIEW 1. Evaluate: 10 15 f. 4 ( ) d. 7 g. 6 56 8 4 100 5 7 h. 6 ( ) ( 5) i. e. 5( 7 16 ) j.. Perform the indicated operation: 8 5 15 1 16 d. e. 8 5 45 h. 4 8 1 i. f. g. 5(8 10) [1
More informationSpreadsheet Case 2. Clarkson Cosmetics
27 Spreadsheet Case 2 Clarkson Cosmetics Problem: Management skills: PC skills: File: Evaluate the effectiveness of an e-commerce company s Web site and advertising sites Analyzing Organizing Formulas
More informationPart I. Fill in the blank. 2 points each. No calculators. No partial credit
Math 108 (105) Final Exam Page 1 Spring 2015 Part I. Fill in the blank. 2 points each. No calculators. No partial credit 1) Fill in the blank a) 2 8 h) 5 0 21 4 b) 5 7 i) 8 3 c) 2 3 = j) 2 7 d) The additive
More informationAn Example of a Class Frequency Histogram. An Example of a Class Frequency Table. Freq
Section A: uency Histograms for Discrete Quantitative Data The data in a uency Table can be made more visual by creating a graph of the classes and their frequencies. One type of graph used for this purpose
More informationSpreadsheet Management Software Cases. Evaluate the effectiveness of an e-commerce company s Web site and advertising sites
31 Spreadsheet Case 2 Athena Beauty Products Problem: Management skills: Excel skills: File: Evaluate the effectiveness of an e-commerce company s Web site and advertising sites Analyzing Organizing Formulas
More informationCCNY Math Review Chapter 2: Functions
CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and
More informationProperties of Operations
" Properties of Operations When you learn new types of numbers, you want to know what properties apply to them. You know that rational numbers are commutative for addition and multiplication. 1 1 1 1 +
More informationI(g) = income from selling gearboxes C(g) = cost of purchasing gearboxes The BREAK-EVEN PT is where COST = INCOME or C(g) = I(g).
Page 367 I(g) = income from selling gearboxes C(g) = cost of purchasing gearboxes The BREAK-EVEN PT is where COST = INCOME or C(g) = I(g). PROFIT is when INCOME > COST or I(g) > C(g). I(g) = 8.5g g = the
More informationVocabulary Unit 2-3: Linear Functions & Healthy Lifestyles. Scale model a three dimensional model that is similar to a three dimensional object.
Scale a scale is the ratio of any length in a scale drawing to the corresponding actual length. The lengths may be in different units. Scale drawing a drawing that is similar to an actual object or place.
More informationThis is a function because no vertical line can be drawn so that it intersects the graph more than once.
Determine whether each relation is a function. Explain. 1. A function is a relation in which each element of the domain is paired with exactly one element of the range. So, this relation is a function.
More informationLesson 4 Exponential Functions I
Lesson 4 Exponential Functions I Lesson 4 Exponential Functions I Exponential functions play a major role in our lives. Population growth and disease processes are real-world problems that involve exponential
More informationSeptember 18, B Math Test Chapter 1 Name: x can be expressed as: {y y 0, y R}.
September 8, 208 62B Math Test Chapter Name: Part : Objective Questions [ mark each, total 2 marks]. State whether each of the following statements is TRUE or FALSE a) The mapping rule (x, y) (-x, y) represents
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationWarm-up for Integrated Geometry and Algebra I
Summer Assignment Warm-up for Integrated Geometry and Algebra I Who should complete this packet? Students who will be taking Integrated Geometry and Algebra I in the fall of 018. Due Date: The first day
More informationTable of Contents. Student Practice Pages. Number Lines and Operations Numbers. Inverse Operations and Checking Answers... 40
Table of Contents Introduction... Division by Tens... 38 Common Core State Standards Correlation... Division of -Digit Numbers... 39 Student Practice Pages Number Lines and Operations Numbers Inverse Operations
More informationUnit 6 Quadratic Functions
Unit 6 Quadratic Functions 12.1 & 12.2 Introduction to Quadratic Functions What is A Quadratic Function? How do I tell if a Function is Quadratic? From a Graph The shape of a quadratic function is called
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More informationEquations and Functions, Variables and Expressions
Equations and Functions, Variables and Expressions Equations and functions are ubiquitous components of mathematical language. Success in mathematics beyond basic arithmetic depends on having a solid working
More information3.1 Graphing Linear Inequalities
3.1 Graphing Linear Inequalities I. Inequalities A. Introduction Many mathematical descriptions of real situations are best expressed as inequalities rather than equations. For example, a firm might be
More informationStudy Guide for Exam 2
Math 233A Intermediate Algebra Fall 202 Study Guide for Exam 2 Exam 2 is scheduled for Wednesday, October 3 rd. You may use a 3" 5" note card (both sides) and a scientific calculator. You are expected
More informationTangent line problems
You will find lots of practice problems and homework problems that simply ask you to differentiate. The following examples are to illustrate some of the types of tangent line problems that you may come
More informationName: Unit 3 Beaumont Middle School 8th Grade, Introduction to Algebra
Unit 3 Beaumont Middle School 8th Grade, 2016-2017 Introduction to Algebra Name: I can identify a function, the domain and range. I can identify a linear relationship from a situation, table, graph and
More informationLesson 6.1 Matrix Representations
Lesson. Matrix Representations. Supply the missing entries in each transition matrix..7 m r.9..7 a. [M] b. [R] c. [T] t. m. A survey of registered voters showed that of those people who voted in the presidential
More informationLesson 18: There is Only One Line Passing Through a Given Point with a Given
Lesson 18: There is Only One Line Passing Through a Given Point with a Given Student Outcomes Students graph equations in the form of using information about slope and intercept. Students know that if
More informationName Class Date. Using Graphs to Relate Two Quantities
4-1 Reteaching Using Graphs to Relate Two Quantities An important life skill is to be able to a read graph. When looking at a graph, you should check the title, the labels on the axes, and the general
More informationBell Ringer Write each phrase as a mathematical expression. Thinking with Mathematical Models
Bell Ringer Write each phrase as a mathematical expression. 1. the sum of nine and eight 2. the sum of nine and a number 3. nine increased by a number x 4. fourteen decreased by a number p 5. the product
More information2.6: Solving Systems of Linear Inequalities
Quick Review 2.6: Solving Systems of Linear Inequalities = - What is the difference between an equation and an inequality? Which one is shaded? Inequality - When is the line solid?, - When is the line
More informationStudy Guide and Review - Chapter 1
State whether each sentence is true or false If false, replace the underlined term to make a true sentence 1 A function assigns every element of its domain to exactly one element of its range A function
More informationCollege Algebra Exam File - Fall Test #1
College Algebra Exam File - Fall 010 Test #1 1.) For each of the following graphs, indicate (/) whether it is the graph of a function and if so, whether it the graph of one-to one function. Circle your
More information1.6 Modeling with Equations
1.6 Modeling with Equations Steps to Modeling Problems with Equations 1. Identify the variable you want to solve for. 2. Express all unknown quantities in terms of this variable. 3. Set up the model by
More informationDerivatives and Graphs of Functions
Derivatives and Graphs of Functions September 8, 2014 2.2 Second Derivatives, Concavity, and Graphs In the previous section, we discussed how our derivatives can be used to obtain useful information about
More informationChapter 4 Linear Programming
Chapter Objectives Check off these skills when you feel that you have mastered them. From its associated chart, write the constraints of a linear programming problem as linear inequalities. List two implied
More informationName: Dr. Fritz Wilhelm Lab 1, Presentation of lab reports Page # 1 of 7 5/17/2012 Physics 120 Section: ####
Name: Dr. Fritz Wilhelm Lab 1, Presentation of lab reports Page # 1 of 7 Lab partners: Lab#1 Presentation of lab reports The first thing we do is to create page headers. In Word 2007 do the following:
More informationCoached Instruction Supplement
Practice Coach PLUS Coached Instruction Supplement Mathematics 5 Practice Coach PLUS, Coached Instruction Supplement, Mathematics, Grade 5 676NASP Triumph Learning Triumph Learning, LLC. All rights reserved.
More informationhp calculators HP 35s Using Algebraic Mode Calculation modes Functions of a single number in algebraic A simple example in algebraic
Calculation modes Functions of a single number in algebraic A simple example in algebraic Arithmetic calculations with two numbers Another example - the area of a piece of carpet Algebraic mode in detail
More informationSpecific Objectives Students will understand that that the family of equation corresponds with the shape of the graph. Students will be able to create a graph of an equation by plotting points. In lesson
More informationGraphical Analysis. Figure 1. Copyright c 1997 by Awi Federgruen. All rights reserved.
Graphical Analysis For problems with 2 variables, we can represent each solution as a point in the plane. The Shelby Shelving model (see the readings book or pp.68-69 of the text) is repeated below for
More informationMATH NATION SECTION 4 H.M.H. RESOURCES
MATH NATION SECTION 4 H.M.H. RESOURCES SPECIAL NOTE: These resources were assembled to assist in student readiness for their upcoming Algebra 1 EOC. Although these resources have been compiled for your
More informationIntermediate Mathematics League of Eastern Massachusetts
Meet # January 010 Intermediate Mathematics League of Eastern Massachusetts Meet # January 010 Category 1 - Mystery Meet #, January 010 1. Of all the number pairs whose sum equals their product, what is
More information4.2 Linear Equations in Point-Slope Form
4.2 Linear Equations in Point-Slope Form Learning Objectives Write an equation in point-slope form. Graph an equation in point-slope form. Write a linear function in point-slope form. Solve real-world
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationLearning Log Title: CHAPTER 2: FRACTIONS AND INTEGER ADDITION. Date: Lesson: Chapter 2: Fractions and Integer Addition
Chapter : Fractions and Integer Addition CHAPTER : FRACTIONS AND INTEGER ADDITION Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter : Fractions and Integer Addition Date: Lesson:
More informationChapter 7: Linear Functions and Inequalities
Chapter 7: Linear Functions and Inequalities Index: A: Absolute Value U4L9 B: Step Functions U4L9 C: The Truth About Graphs U4L10 D: Graphs of Linear Inequalities U4L11 E: More Graphs of Linear Inequalities
More information9. MATHEMATICIANS ARE FOND OF COLLECTIONS
get the complete book: http://wwwonemathematicalcatorg/getfulltextfullbookhtm 9 MATHEMATICIANS ARE FOND OF COLLECTIONS collections Collections are extremely important in life: when we group together objects
More informationMCS 118 Quiz 1. Fall (5pts) Solve the following equations for x. 7x 2 = 4x x 2 5x = 2
MCS 8 Quiz Fall 6. (5pts) Solve the following equations for. 7 = 4 + 3. (5pts) Solve the following equations for. 3 5 = 3. (5pts) Factor 3 + 35 as much as possible. 4. (5pts) Simplify +. 5. (5pts) Solve
More informationMATH 1075 Final Exam
Autumn 2018 Form C Name: Signature: OSU name.#: Lecturer: Recitation Instructor: Recitation Time: MATH 1075 Final Exam Instructions: You will have 1 hour and 45 minutes to take the exam. Show ALL work
More informationNOTES Linear Equations
NOTES Linear Equations Linear Parent Function Linear Parent Function the equation that all other linear equations are based upon (y = x) Horizontal and Vertical Lines (HOYY VUXX) V vertical line H horizontal
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part I. 4 th Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part I 4 th Nine Weeks, 2016-2017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationSection 2.0: Getting Started
Solving Linear Equations: Graphically Tabular/Numerical Solution Algebraically Section 2.0: Getting Started Example #1 on page 128. Solve the equation 3x 9 = 3 graphically. Intersection X=4 Y=3 We are
More informationFinite Math - J-term Homework. Section Inverse of a Square Matrix
Section.5-77, 78, 79, 80 Finite Math - J-term 017 Lecture Notes - 1/19/017 Homework Section.6-9, 1, 1, 15, 17, 18, 1, 6, 9, 3, 37, 39, 1,, 5, 6, 55 Section 5.1-9, 11, 1, 13, 1, 17, 9, 30 Section.5 - Inverse
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationGraphing Linear Equations
Graphing Linear Equations A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. What am I learning today? How to graph a linear
More informationGRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.
GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),
More information