ACTIVITY 9 Continued Lesson 9-2

Size: px
Start display at page:

Download "ACTIVITY 9 Continued Lesson 9-2"

Transcription

1 Continued Lesson 9- Lesson 9- PLAN Pacing: 1 class period Chunking the Lesson Eample A Eample B #1 #3 Lesson Practice M Notes Learning Targets: Graph on a coordinate plane the solutions of a linear inequalit in two variables. Interpret the graph of the solutions of a linear inequalit in two variables. SUGGESTED LEARNING STRATEGIES: Vocabular Organizer, Interactive Word Wall, Create Representations, Identif a Subtask, Construct an Argument TEACH Bell-Ringer Activit Ask students to think of all the places where the know about or have seen boundar lines. These might include football fields, baseball diamonds, tennis courts, propert lines and state lines. Ask students to write a working definition of a boundar line based on their eamples. Eample A Interactive Word Wall, Vocabular Organizer, Activating Prior Knowledge, Create Representations, Identif a Subtask Guide students through this first eample of how to graph a linear inequalit in two variables. Ask students to discuss how graphing a linear inequalit is similar to and how it is different from graphing a linear equation. Allow students to make the connection between using solid and open circles when graphing inequalities on the number line and using solid and dashed lines in the coordinate plane. MATH TERMS A line that separates the coordinate plane into two regions is a boundar line. The two regions are half-planes. The origin (0, 0) is usuall an eas point to test if it is not on the boundar line. The solutions of a linear inequalit in two variables can be represented in the coordinate plane. Eample A Graph the linear inequalit + 3. Step 1: Graph the corresponding linear equation = + 3. The line ou graphed is the boundar line. Step : Test a point in one of the half-planes to see if it is a solution of the inequalit. Using (0, 0), 0 (0) + 3 is a true statement. So (0, 0) is a solution. Step 3: If the point ou tested is a solution, shade the half-plane in which it lies. If it is not, shade the other half-plane. (0, 0) is a solution, so shade the half-plane containing the solution point (0, 0). 017 College Board. All rights reserved. MATH TERMS A closed half-plane includes the points on the boundar line. An open half-plane does not include the points on the boundar line. 15 SpringBoard Integrated Mathematics I, Unit Linear Functions In Eample A, the solution set includes the points on the boundar line. The solution set is a closed half-plane. In an inequalit containing < or > the solution set does not include the points on the boundar line, so the boundar line is dashed, and the solution set is an open half-plane. 017 College Board. All rights reserved. 15 SpringBoard Integrated Mathematics I, Unit Linear Functions

2 017 College Board. All rights reserved. 017 College Board. All rights reserved. Lesson 9- Eample B Graph + >. Step 1: Solve the inequalit for. + > Subtract from each side. > + > + Divide each side b. > 1 + Simplif. Step : Graph the boundar line, = 1 +. Since the inequalit uses the > smbol, the solution set is an open half-plane. Draw a dashed line. Step 3: Check the test point (0, 0): 0 > 1 ( 0) + is false. Shade the half-plane that does not contain the point (0, 0). Tr These A B Graph each inequalit. a. 3 1 b. < 3 c. > + d When graphing a linear inequalit, is it possible for a test point located on the boundar line to determine which half-plane should be shaded? Eplain. No; this will determine onl if the line itself is part of the solution set, which it is for and inequalities. To determine which half-plane to shade, the test point must be above or below the boundar line. M Notes When multipling or dividing each side of an inequalit b a negative number, ou must reverse the inequalit smbol. Activit 9 Writing and Graphing Inequalities 155 Continued Eample B Activating Prior Knowledge, Create Representations, Identif a Subtask Eample B requires the additional step of solving the inequalit for. Ask students if and wh this step is or is not necessar. Tr These A B a. b. c. d. Activit 9 Writing and Graphing Inequalities 155

3 Continued 1 Activating Prior Knowledge, Create Representations, Think-Pair- Share, Construct an Argument These items lead students to understand that a boundar line and the two half-planes it divides are separate entities. The union of the two half-planes and the boundar line makes up the entire coordinate plane. It is important for students to understand that while the points on the boundar line are sometimes solutions to the inequalit, these points cannot be used to determine in which half-plane the other solutions lie. 3 Discussion Groups, Identif a Subtask, Create Representations, Debriefing These items bring students back to the original problem contet and mirror earlier questions, providing reinforcement. Students ma determine equations and inequalities using different methods. Some ma use the slope and -intercept as determined from the graph, while some ma choose points on the line to determine the slope. Monitor group discussions carefull to be sure students understand the reasons wh the graphs are restricted to the first quadrant. To make connections to future learning, a discussion of how the contet limits the graph of a linear inequalit ma be revisited. M Notes Lesson 9-. Critique the reasoning of others. Al tried to graph the inequalit + <. He first graphed the linear equation + =. He then chose the test point (, 1) and used his result to shade above the line. Eplain Al s mistake and how he should correct it. 3. Al and Aneeza tr a new plan in which the can upload no more than 350 terabtes per month. On the grids below, the -ais represents the number of terabtes that Al can upload and the -ais represents the number of terabtes that Aneeza can upload. a. Suppose Aneeza does not upload anthing for a month. Write 3 an inequalit that represents the amount 70 of data that Al can 0 upload during that month. Graph the 10 inequalit on the grid. 350 b. Suppose Al does not upload anthing for a month. Write an inequalit that represents the amount of data that Aneeza can upload during that month. Graph the inequalit on the grid SpringBoard Integrated Mathematics I, Unit Linear Functions Al s test point (, 1) lies on the boundar line: () + (1) =. Al must choose a test point that is not on the boundar line to determine the half-plane to shade. Al could use (0, 0); then the inequalit would give (0) + (0) <, which is true. Since (0, 0) lies below the graph of + =, Al should shade below the line MINI-LESSON: Equations and Inequalities in One Variable Follow these steps to eplore equations and inequalities in one variable. 1. On grid paper, graph = 1, which means for an value of, = 1.. Graph = 3, which means for an value of, = When the -coordinate of each point on a line is the same, the line is vertical.. When the -coordinate of each point on a line is the same, the line is horizontal. 5. Which parts of the coordinate plane can be described with inequalities in one variable? Eplain our answer. (When the one variable is, a region above or below a horizontal line is shaded. When the one variable is, a region to the left or right of a vertical line is shaded.) College Board. All rights reserved. 017 College Board. All rights reserved. 15 SpringBoard Integrated Mathematics I, Unit Linear Functions

4 Lesson 9-. The graph below represents another plan that Al and Aneeza considered. The -ais represents the number of terabtes of data from photos that can be uploaded each month. The -ais represents the number of terabtes of data from tet files. M Notes Continued 3 () As this portion of the activit is debriefed, discuss how the equation of the boundar line is related to the inequalit. Encourage students to describe the meaning of the -and -intercepts of the graph within the provided contet. 0 Differentiating Instruction 017 College Board. All rights reserved. 017 College Board. All rights reserved. Tet Files (TB) Photos (TB) a. Identif the -intercept and -intercept. What do the represent in this contet? The -intercept is (10, 0); this represents the maimum number of terabtes of photos that can be uploaded during a month in which no tet files are uploaded. The -intercept is (0, ); this represents the maimum number of terabtes of tet files that can be uploaded during a month in which no photos are uploaded. b. Determine the equation for the boundar line of the graph. Justif our response. = + ; The slope of the line is =, and the 10 -intercept is (0, ). c. Model with mathematics. Write a linear inequalit to represent this situation. Then describe the plan in our own words. + ; The plan allows up to terabtes of uploaded information, but no more than 10 terabtes can be from photos. The -intercept of a graph is the point where the graph crosses the -ais. The -intercept is the point where the graph crosses the -ais. Activit 9 Writing and Graphing Inequalities Support students who forget to change the direction of the inequalit smbol when multipling or dividing both sides b a negative number. For the following inequalities, have students predict the direction of the inequalit smbol when solved for with appearing on the left side of the inequalit. Then have them confirm their predictions b solving the inequalities for [ 3 ]. 9 3 < [ > 3 ] [ 1 + 3]. 5 < [ < 5+ ] Etend students thinking b asking them to evaluate the following statement as alwas, sometimes or never true and give an eplanation for their response. The solution of a linear inequalit includes ordered pairs in each of the four quadrants of a coordinate plane < 0 Activit 9 Writing and Graphing Inequalities 157

5 Continued Debrief students answers b asking them how the know which half-plane to shade when graphing an inequalit. Have students identif the boundar lines for Items 5 7. Have them share their boundar line graphs with others before shading. Pa close attention to the shading of the half-planes. See previous page. ASSESS Students answers to the Lesson Practice items will provide a formative assessment of their understanding of graphing and interpreting a linear inequalit in two variables, and of students abilit to appl their learning. Short-ccle formative assessment items for Lesson 9- are also available in the Assessment section on SpringBoard Digital. Refer back to the graphic organizer the class created when unpacking Embedded Assessment 3. Ask students to use the graphic organizer to identif the concepts or skills the learned in this lesson. LESSON 9- PRACTICE 9. ADAPT Check students answers to the Lesson Practice to ensure that students can correctl graph an inequalit in two variables on the coordinate plane. If students are struggling to shade the correct half-plane, ask them to plot their test point before the test it. If the test ields a true statement, the should shade toward the point the have graphed. If the test results ield a false statement, the should shade on the side of the boundar line that does not contain the test point. Encourage students to shade lightl so that corrections can easil be made if necessar. See the Activit Practice on page 13 and the Additional Unit Practice in the Teacher Resources on SpringBoard Digital for additional problems for this lesson. You ma wish to use the Teacher Assessment Builder on SpringBoard Digital to create custom assessments or additional practice. M Notes LESSON 9- PRACTICE Graph each inequalit on the coordinate plane. 9.. > > 13. Make sense of problems. Write the inequalit whose solutions are shown in the graph Lesson 9-5. Graph the linear inequalit < 3 on the coordinate plane.. Graph the linear inequalit > on the coordinate plane. 7. Graph the linear inequalit 0 on the coordinate plane.. Write an inequalit whose solutions are all points in the second and third quadrants. 15 SpringBoard Integrated Mathematics I, Unit Linear Functions College Board. All rights reserved. 017 College Board. All rights reserved. 15 SpringBoard Integrated Mathematics I, Unit Linear Functions

Graphing Linear Inequalities

Graphing Linear Inequalities Graphing Linear Inequalities Basic Mathematics Review 837 Linear inequalities pla an important role in applied mathematics. The are used in an area of mathematics called linear programming which was developed

More information

Name: Thus, y-intercept is (0,40) (d) y-intercept: Set x = 0: Cover the x term with your finger: 2x + 6y = 240 Solve that equation: 6y = 24 y = 4

Name: Thus, y-intercept is (0,40) (d) y-intercept: Set x = 0: Cover the x term with your finger: 2x + 6y = 240 Solve that equation: 6y = 24 y = 4 Name: GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES SHOW ALL WORK AND JUSTIFY ALL ANSWERS. 1. We will graph linear inequalities first. Let us first consider 2 + 6 240 (a) First, we will graph the boundar

More information

LINEAR PROGRAMMING. Straight line graphs LESSON

LINEAR 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 information

Graphing Equations Case 1: The graph of x = a, where a is a constant, is a vertical line. Examples a) Graph: x = x

Graphing Equations Case 1: The graph of x = a, where a is a constant, is a vertical line. Examples a) Graph: x = x 06 CHAPTER Algebra. GRAPHING EQUATIONS AND INEQUALITIES Tetbook Reference Section 6. &6. CLAST OBJECTIVE Identif regions of the coordinate plane that correspond to specific conditions and vice-versa Graphing

More information

Name Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =

Name 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 information

Lesson 5.2 Exercises, pages

Lesson 5.2 Exercises, pages Lesson 5. Eercises, pages 6 68 A. Determine whether each point is a solution of the given inequalit. a) - -16 A(-, ) In the inequalit, substitute:, L.S.: ( ) () 17 R.S. 16 Since the L.S.

More information

Section 4.2 Graphing Lines

Section 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 information

Content Standards Two-Variable Inequalities

Content Standards Two-Variable Inequalities -8 Content Standards Two-Variable Inequalities A.CED. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales.

More information

The Marching Cougars Lesson 9-1 Transformations

The Marching Cougars Lesson 9-1 Transformations The Marching Cougars Lesson 9-1 Learning Targets: Perform transformations on and off the coordinate plane. Identif characteristics of transformations that are rigid motions and characteristics of transformations

More information

Graphing Method. Graph of x + y < > y 10. x

Graphing Method. Graph of x + y < > y 10. x Graphing Method Eample: Graph the inequalities on the same plane: + < 6 and 2 - > 4. Before we graph them simultaneousl, let s look at them separatel. 10-10 10 Graph of + < 6. ---> -10 Graphing Method

More information

This lesson gives students practice in graphing

This lesson gives students practice in graphing NATIONAL MATH + SCIENCE INITIATIVE Mathematics 9 7 5 1 1 5 7 LEVEL Grade, Algebra 1, or Math 1 in a unit on solving sstems of equations MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and

More information

Graph each pair of functions on the same coordinate plane See margin. Technology Activity: A Family of Functions

Graph each pair of functions on the same coordinate plane See margin. Technology Activity: A Family of Functions - What You ll Learn To analze translations To analze stretches, shrinks, and reflections...and Wh To analze a fabric design, as in Eample Families of Functions Check Skills You ll Need G for Help Lessons

More information

Derivatives 3: The Derivative as a Function

Derivatives 3: The Derivative as a Function Derivatives : The Derivative as a Function 77 Derivatives : The Derivative as a Function Model : Graph of a Function 9 8 7 6 5 g() - - - 5 6 7 8 9 0 5 6 7 8 9 0 5 - - -5-6 -7 Construct Your Understanding

More information

Graphs, Linear Equations, and Functions

Graphs, 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 information

7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it

7.5. 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 information

Laurie s Notes. Overview of Section 6.3

Laurie 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 information

Essential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1.

Essential 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 information

CHECK Your Understanding

CHECK 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 information

6.1. Graphing Linear Inequalities in Two Variables. INVESTIGATE the Math. Reflecting

6.1. Graphing Linear Inequalities in Two Variables. INVESTIGATE the Math. Reflecting 6.1 Graphing Linear Inequalities in Two Variables YOU WILL NEED graphing technolog OR graph paper, ruler, and coloured pencils EXPLORE For which inequalities is (3, 1) a possible solution? How do ou know?

More information

The Graph Scale-Change Theorem

The Graph Scale-Change Theorem Lesson 3-5 Lesson 3-5 The Graph Scale-Change Theorem Vocabular horizontal and vertical scale change, scale factor size change BIG IDEA The graph of a function can be scaled horizontall, verticall, or in

More information

Math 1050 Lab Activity: Graphing Transformations

Math 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 information

1 Teaching Objective(s) *Lesson Plan designed for 3-5 days The student will: II (c): complete a function based on a given rule.

1 Teaching Objective(s) *Lesson Plan designed for 3-5 days The student will: II (c): complete a function based on a given rule. ! "# Teaching Objective(s) *Lesson Plan designed for 3-5 das The student will: II (c): complete a function based on a given rule. II (e): appl the principles of graphing in the coordinate sstem. II (f):

More information

Enhanced Instructional Transition Guide

Enhanced Instructional Transition Guide Enhanced Instructional Transition Guide / Unit 04: Suggested Duration: 6 das Unit 04: Geometr: Coordinate Plane, Graphing Transformations, and Perspectives (9 das) Possible Lesson 0 (6 das) Possible Lesson

More information

Sect Linear Inequalities in Two Variables

Sect Linear Inequalities in Two Variables Sect 9. - Linear Inequalities in Two Variables Concept # Graphing a Linear Inequalit in Two Variables Definition Let a, b, and c be real numbers where a and b are not both zero. Then an inequalit that

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate 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 information

Appendix F: Systems of Inequalities

Appendix 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 information

Algebra I Notes Unit Six: Graphing Linear Equations and Inequalities in Two Variables, Absolute Value Functions

Algebra I Notes Unit Six: Graphing Linear Equations and Inequalities in Two Variables, Absolute Value Functions Sllabus Objective.4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Coordinate Plane a plane formed b two real number lines (axes) that

More information

3.7 Graphing Linear Inequalities

3.7 Graphing Linear Inequalities 8 CHAPTER Graphs and Functions.7 Graphing Linear Inequalities S Graph Linear Inequalities. Graph the Intersection or Union of Two Linear Inequalities. Graphing Linear Inequalities Recall that the graph

More information

LESSON 3.1 INTRODUCTION TO GRAPHING

LESSON 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 information

7.6 Solve Linear Systems of

7.6 Solve Linear Systems of 7.6 Solve Linear Sstems of Linear Inequalities Goal p Solve sstems of linear inequalities in two variables. Your Notes VOCABULARY Sstem of linear inequalities Solution of a sstem of linear inequalities

More information

Unit 5 Lesson 2 Investigation 1

Unit 5 Lesson 2 Investigation 1 Name: Investigation 1 Modeling Rigid Transformations CPMP-Tools Computer graphics enable designers to model two- and three-dimensional figures and to also easil manipulate those figures. For eample, interior

More information

4 B. 4 D. 4 F. 3. What are some common characteristics of the graphs of cubic and quartic polynomial functions?

4 B. 4 D. 4 F. 3. What are some common characteristics of the graphs of cubic and quartic polynomial functions? .1 Graphing Polnomial Functions COMMON CORE Learning Standards HSF-IF.B. HSF-IF.C.7c Essential Question What are some common characteristics of the graphs of cubic and quartic polnomial functions? A polnomial

More information

Transformations of y = x 2

Transformations of y = x 2 Transformations of = Parent Parabola Lesson 11-1 Learning Targets: Describe translations of the parent function f() =. Given a translation of the function f() =, write the equation of the function. SUGGESTED

More information

1. A(-2, 2), B(4, -2) 3 EXAMPLE. Graph the line y = Move up 3 units. Quick Check See left.

1. A(-2, 2), B(4, -2) 3 EXAMPLE. Graph the line y = Move up 3 units. Quick Check See left. -. Plan Objectives To graph lines given their equations To write equations of lines Eamples Graphing Lines in Slope- Intercept Form Graphing Lines Using Intercepts Transforming to Slope- Intercept Form

More information

14-1. Translations. Vocabulary. Lesson

14-1. Translations. Vocabulary. Lesson Chapter 1 Lesson 1-1 Translations Vocabular slide, translation preimage translation image congruent figures Adding fied numbers to each of the coordinates of a figure has the effect of sliding or translating

More information

Lesson 11-2 Shrinking, Stretching, and Reflecting Parabolas ACTIVITY 11

Lesson 11-2 Shrinking, Stretching, and Reflecting Parabolas ACTIVITY 11 ACTIVITY 11 Lesson 11- M Notes Unlike a rigid transformation, a vertical stretch or vertical shrink will change the shape of the graph. A vertical stretch stretches a graph awa from the -ais b a factor

More information

Graphing square root functions. What would be the base graph for the square root function? What is the table of values?

Graphing square root functions. What would be the base graph for the square root function? What is the table of values? Unit 3 (Chapter 2) Radical Functions (Square Root Functions Sketch graphs of radical functions b appling translations, stretches and reflections to the graph of Analze transformations to identif the of

More information

Algebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES

Algebra 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 information

3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately.

3x 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 information

5.2 Graphing Polynomial Functions

5.2 Graphing Polynomial Functions Locker LESSON 5. Graphing Polnomial Functions Common Core Math Standards The student is epected to: F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing

More information

Exponential Functions

Exponential 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 information

Lesson 5.3 Exercises, pages

Lesson 5.3 Exercises, pages Lesson 5.3 Eercises, pages 37 3 A. Determine whether each ordered pair is a solution of the quadratic inequalit: 3 - a) (-3, ) b) (, 5) Substitute each ordered pair in» 3. L.S. ; R.S.: 3( 3) 3 L.S. 5;

More information

Lesson 11 Skills Maintenance. Activity 1. Model. The addition problem is = 4. The subtraction problem is 5 9 = 4.

Lesson 11 Skills Maintenance. Activity 1. Model. The addition problem is = 4. The subtraction problem is 5 9 = 4. Lesson Skills Maintenance Lesson Planner Vocabular Development -coordinate -coordinate point of origin Skills Maintenance ddition and Subtraction of Positive and Negative Integers Problem Solving: We look

More information

2-3. Attributes of Absolute Value Functions. Key Concept Absolute Value Parent Function f (x)= x VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

2-3. Attributes of Absolute Value Functions. Key Concept Absolute Value Parent Function f (x)= x VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING - Attributes of Absolute Value Functions TEKS FOCUS TEKS ()(A) Graph the functions f() =, f() =, f() =, f() =,f() = b, f() =, and f() = log b () where b is,, and e, and, when applicable, analze the ke

More information

The Sine and Cosine Functions

The Sine and Cosine Functions Lesson -5 Lesson -5 The Sine and Cosine Functions Vocabular BIG IDEA The values of cos and sin determine functions with equations = sin and = cos whose domain is the set of all real numbers. From the eact

More information

Quadratic Inequalities

Quadratic Inequalities TEKS FCUS - Quadratic Inequalities VCABULARY TEKS ()(H) Solve quadratic inequalities. TEKS ()(E) Create and use representations to organize, record, and communicate mathematical ideas. Representation a

More information

Appendix F: Systems of Inequalities

Appendix F: Systems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit What ou should learn The statements < and ⱖ are inequalities in two variables. An ordered pair 共a, b兲 is a

More information

Parallel and Perpendicular Lines. What are the slope and y-intercept of each equation?

Parallel and Perpendicular Lines. What are the slope and y-intercept of each equation? 6 6-6 What You ll Learn To determine whether lines are parallel To determine whether lines are And Wh To use parallel and lines to plan a bike path, as in Eample Parallel Lines Parallel and Perpendicular

More information

LESSON Constructing and Analyzing Scatter Plots

LESSON Constructing and Analyzing Scatter Plots LESSON Constructing and Analzing Scatter Plots UNDERSTAND When ou stud the relationship between two variables such as the heights and shoe sizes of a group of students ou are working with bivariate data.

More information

SLOPE A MEASURE OF STEEPNESS through 2.1.4

SLOPE A MEASURE OF STEEPNESS through 2.1.4 SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written

More information

Chapter 4 Section 1 Graphing Linear Inequalities in Two Variables

Chapter 4 Section 1 Graphing Linear Inequalities in Two Variables Chapter 4 Section 1 Graphing Linear Inequalities in Two Variables Epressions of the tpe + 2 8 and 3 > 6 are called linear inequalities in two variables. A solution of a linear inequalit in two variables

More information

(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2

(0, 2) y = x 1 2. y = x (2, 2) y = 2x + 2 .5 Equations of Parallel and Perpendicular Lines COMMON CORE Learning Standards HSG-GPE.B.5 HSG-GPE.B. Essential Question How can ou write an equation of a line that is parallel or perpendicular to a given

More information

Vertical and Horizontal Translations. Graph each pair of functions on the same coordinate plane See margin

Vertical and Horizontal Translations. Graph each pair of functions on the same coordinate plane See margin - Lesson Preview What You ll Learn BJECTIVE BJECTIVE To analze vertical translations To analze horizontal translations... And Wh To analze a fabric design, as in Eample BJECTIVE Vertical and Horizontal

More information

Graphing Absolute Value Functions. Objectives To graph an absolute value function To translate the graph of an absolute value function

Graphing Absolute Value Functions. Objectives To graph an absolute value function To translate the graph of an absolute value function 5-8 CC-0 CC-6 Graphing Absolute Value Functions Content Standards F.BF.3 Identif the effect on the graph of replacing f () b f () k, kf (), f (k), and f ( k) for specific values of k (both positive and

More information

Rotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things

Rotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things . Rotations object in a plane? What are the three basic was to move an Rotate A biccle wheel can rotate clockwise or counterclockwise. 0 0 0 9 9 9 8 8 8 7 6 7 6 7 6 ACTIVITY: Three Basic Was to Move Things

More information

Concept: Slope of a Line

Concept: Slope of a Line Concept: Slope of a Line Warm Up Name: The following suggested activities would serve as a review to consolidate previous learning. While promoting rich mathematical dialog, the will also provide students

More information

Name Class Date. Graphing a Linear Inequality

Name Class Date. Graphing a Linear Inequality Name Class Date Solving Linear Inequalities Going Deeper Essential question: How do ou graph a linear inequalit in two variables? A linear inequalit in two variables, such as 2-6, results when ou replace

More information

8.5 Quadratic Functions and Their Graphs

8.5 Quadratic Functions and Their Graphs CHAPTER 8 Quadratic Equations and Functions 8. Quadratic Functions and Their Graphs S Graph Quadratic Functions of the Form f = + k. Graph Quadratic Functions of the Form f = - h. Graph Quadratic Functions

More information

Partial Fraction Decomposition

Partial Fraction Decomposition Section 7. Partial Fractions 53 Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the eamples that follow. Note

More information

The Graph of an Equation

The 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 information

LESSON 5.3 SYSTEMS OF INEQUALITIES

LESSON 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 information

SLOPE A MEASURE OF STEEPNESS through 7.1.5

SLOPE A MEASURE OF STEEPNESS through 7.1.5 SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the

More information

Lesson 2.1 Exercises, pages 90 96

Lesson 2.1 Exercises, pages 90 96 Lesson.1 Eercises, pages 9 96 A. a) Complete the table of values. 1 1 1 1 1. 1 b) For each function in part a, sketch its graph then state its domain and range. For : the domain is ; and the range is.

More information

F8-18 Finding the y-intercept from Ordered Pairs

F8-18 Finding the y-intercept from Ordered Pairs F8-8 Finding the -intercept from Ordered Pairs Pages 5 Standards: 8.F.A., 8.F.B. Goals: Students will find the -intercept of a line from a set of ordered pairs. Prior Knowledge Required: Can add, subtract,

More information

Inclination of a Line

Inclination of a Line 0_00.qd 78 /8/05 Chapter 0 8:5 AM Page 78 Topics in Analtic Geometr 0. Lines What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and

More information

Graphing f ( x) = ax 2 + c

Graphing f ( x) = ax 2 + c . Graphing f ( ) = a + c Essential Question How does the value of c affect the graph of f () = a + c? Graphing = a + c Work with a partner. Sketch the graphs of the functions in the same coordinate plane.

More information

6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz

6.7. Graph Linear Inequalities in Two Variables. Warm Up Lesson Presentation Lesson Quiz 6.7 Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz 6.7 Warm-Up Tell whether the ordered pair is a solution of the equation. 1. x + 2y = 4; (2, 1) no 2. 4x + 3y = 22;

More information

x Check: p. C) 32 8k D) 3t 15

x Check: p. C) 32 8k D) 3t 15 Chapter Notes Alg H -A (Lesson -&) Solving Inequalities p. 0-0 A) n B) Check: n A) B) p When ou multipl or divide b a number, ou must the inequalit sign! A) r B) g 0 C) k D) t Points: Ch Notes Alg H -A

More information

Graph Linear Equations

Graph Linear Equations Lesson 4. Objectives Graph linear equations. Identif the slope and -intercept of linear equations. Graphing Linear Equations Suppose a baker s cookie recipe calls for a miture of nuts, raisins, and dried

More information

This lesson gives students practice in graphing

This lesson gives students practice in graphing NATIONAL MATH + SCIENCE INITIATIVE 9 Mathematics Solving Systems of Linear Equations 7 5 3 1 1 3 5 7 LEVEL Grade, Algebra 1, or Math 1 in a unit on solving systems of equations MODULE/CONNECTION TO AP*

More information

Section 4.3 Features of a Line

Section 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 information

Rationale. Is it feasible?

Rationale. Is it feasible? Learning Targets: Represent constraints by equations or inequalities. Use a graph to determine solutions of a system of inequalities. SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Interactive Word Wall,

More information

ACTIVITY: Graphing a Linear Equation. 2 x x + 1?

ACTIVITY: 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 information

Slope Fields Introduction / G. TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System

Slope Fields Introduction / G. TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System Math Objectives Students will describe the idea behind slope fields in terms of visualization of the famil of solutions to a differential equation. Students will describe the slope of a tangent line at

More information

Enhanced Instructional Transition Guide

Enhanced Instructional Transition Guide Enhanced Instructional Transition Guide Grade / Unit : Suggested Duration: das Unit : Statistics ( das) Possible Lesson 0 ( das) Possible Lesson 0 ( das) POSSIBLE LESSON 0 ( das) This lesson is one approach

More information

Matrix Representations

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 information

Time To Hit The Slopes. Exploring Slopes with Similar Triangles

Time To Hit The Slopes. Exploring Slopes with Similar Triangles Time To Hit The Slopes Eploring Slopes with Similar Triangles Learning Goals In this lesson, ou will: Use an equation to complete a table of values. Graph an equation using a table of values. Use transformations

More information

Section 2.2: Absolute Value Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a

Section 2.2: Absolute Value Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Section.: Absolute Value 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 information

Pre-Algebra Notes Unit 8: Graphs and Functions

Pre-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 information

Functions. Name. Use an XY Coordinate Pegboard to graph each line. Make a table of ordered pairs for each line. y = x + 5 x y.

Functions. Name. Use an XY Coordinate Pegboard to graph each line. Make a table of ordered pairs for each line. y = x + 5 x y. Lesson 1 Functions Name Use an XY Coordinate Pegboard to graph each line. Make a table of ordered pairs for each line. 1. = + = + = 2 3 = 2 3 Using an XY Coordinate Pegboard, graph the line on a coordinate

More information

L3 Rigid Motion Transformations 3.1 Sequences of Transformations Per Date

L3 Rigid Motion Transformations 3.1 Sequences of Transformations Per Date 3.1 Sequences of Transformations Per Date Pre-Assessment Which of the following could represent a translation using the rule T (, ) = (, + 4), followed b a reflection over the given line? (The pre-image

More information

Unit 2: Function Transformation Chapter 1

Unit 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 information

Graphing Cubic Functions

Graphing 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 information

INEQUALITIES Graphing Linear Inequalities Common Core Standard

INEQUALITIES Graphing Linear Inequalities Common Core Standard F Inequalities, Lesson 4, Graphing Linear Inequalities (r. 2018) INEQUALITIES Graphing Linear Inequalities Common Core Standard A-REI.12 Graph the solutions to a linear inequality in two variables as a

More information

Essential Question How many turning points can the graph of a polynomial function have?

Essential Question How many turning points can the graph of a polynomial function have? .8 Analzing Graphs of Polnomial Functions Essential Question How man turning points can the graph of a polnomial function have? A turning point of the graph of a polnomial function is a point on the graph

More information

Mathematics Stage 5 PAS5.1.2 Coordinate geometry

Mathematics Stage 5 PAS5.1.2 Coordinate geometry Mathematics Stage PAS.. Coordinate geometr Part Graphing lines Acknowledgments This publication is copright New South Wales Department of Education and Training (DET), however it ma contain material from

More information

4.1 The Coordinate Plane

4.1 The Coordinate Plane 4. The Coordinate Plane Goal Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin -ais -ais Ordered pair -coordinate -coordinate Quadrant Scatter plot Copright McDougal Littell, Chapter

More information

Graphing Equations. The Rectangular Coordinate System

Graphing Equations. The Rectangular Coordinate System 3.1 Graphing Equations The Rectangular Coordinate Sstem Ordered pair two numbers associated with a point on a graph. The first number gives the horizontal location of the point. The second gives the vertical

More information

Rational Numbers and the Coordinate Plane

Rational Numbers and the Coordinate Plane Rational Numbers and the Coordinate Plane LAUNCH (8 MIN) Before How can you use the numbers placed on the grid to figure out the scale that is used? Can you tell what the signs of the x- and y-coordinates

More information

Lesson 8.1 Exercises, pages

Lesson 8.1 Exercises, pages Lesson 8.1 Eercises, pages 1 9 A. Complete each table of values. a) -3 - -1 1 3 3 11 8 5-1 - -7 3 11 8 5 1 7 To complete the table for 3, take the absolute value of each value of 3. b) - -3 - -1 1 3 3

More information

Graphing Radical Functions

Graphing Radical Functions 17 LESSON Graphing Radical Functions Basic Graphs of Radical Functions UNDERSTAND The parent radical function, 5, is shown. 5 0 0 1 1 9 0 10 The function takes the principal, or positive, square root of.

More information

Using a Table of Values to Sketch the Graph of a Polynomial Function

Using a Table of Values to Sketch the Graph of a Polynomial Function A point where the graph changes from decreasing to increasing is called a local minimum point. The -value of this point is less than those of neighbouring points. An inspection of the graphs of polnomial

More information

3.2 Polynomial Functions of Higher Degree

3.2 Polynomial Functions of Higher Degree 71_00.qp 1/7/06 1: PM Page 6 Section. Polnomial Functions of Higher Degree 6. Polnomial Functions of Higher Degree What ou should learn Graphs of Polnomial Functions You should be able to sketch accurate

More information

What s the Point? # 2 - Geo Fashion

What 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 information

PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS

PROBLEM 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 information

Translations, Reflections, and Rotations

Translations, Reflections, and Rotations Translations, Reflections, and Rotations The Marching Cougars Lesson 9-1 Transformations Learning Targets: Perform transformations on and off the coordinate plane. Identif characteristics of transformations

More information

Section Graphs and Lines

Section 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 information

PATTERNS AND ALGEBRA. He opened mathematics to many discoveries and exciting applications.

PATTERNS 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 information

8.6 Three-Dimensional Cartesian Coordinate System

8.6 Three-Dimensional Cartesian Coordinate System SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 69 What ou ll learn about Three-Dimensional Cartesian Coordinates Distance and Midpoint Formulas Equation of a Sphere Planes and Other Surfaces

More information

Graphs and Functions

Graphs 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 information