Like with systems of linear equations, we can solve linear inequalities by graphing.
|
|
- Gertrude Green
- 7 years ago
- Views:
Transcription
1 Like with systems of linear equations, we can solve linear inequalities by graphing. Section 1: Reminder on Graphing Word Bank smaller equation slope slope-intercept y-intercept bigger The of a line looks like this: y = mx + b. The above form of a line s equation is called m represents the slope b represents the y-intercept form, where: is the steepness of a line. If a line is very steep, the slope is. However, if a line is not very steep (more flat), the slope is. is where a line crosses the y-axis. - Think of it as where the line intercepts (or meets with) the y-axis. Example: The line y = 1 x 1 has a slope of and a y-intercept of. 2 To graph this line, we start by placing a point on the y-intercept. This point should be at (0,-1). Next, we will use the slope of the line to find additional points on the line. If a line has a slope of 1, then we know that from the location of our first plotted 2 point, we must: rise: space(s) and run: space(s) Note: when we say run on a graph, we mean: move to the (right / left).
2 Example 2: For a line with a slope of 3 we would rise: space(s) and run: space(s). 4 That means, move space(s) (up / down). And, move space(s) (right / left). Example 3: For a line with a slope of 1 2 That means, move space(s) (up / down). we would rise: space(s) and run: space(s). And, move space(s) (right / left). Note: we rise a (positive / negative) amount. Example 4: For a line with a slope of 3 we would rise: space(s) and run: space(s). 4 That means, move space(s) (up / down). And, move space(s) (right / left). Note: we run a (positive / negative) amount. Careful on this one. Since we are running a negative amount, we will actually be moving to the left, not the right (Note: yes, I realize I just answered the previous blank for you thanks for noticing!) Example 5: For a line with a slope of - 4 the negative sign can go in either the numerator (top) or the 5 denominator (bottom). If we put it in both parts, however, we would end up with 4 and when you divide a negative by a negative like this, you will end up with a positive. That is to say, 4 5 is the exact same thing as 4 5. And 4 5 is equal to 4 5. So, we will rise: space(s) and run: space(s). 5 Example 6: For a line with a slope of 2 we would rise: space(s) and run: space(s). Note: 2 is equal to 2 1
3 Practice: On the coordinate plane to the left, graph the line y = 4x +2. Your y-intercept will be at. So, your first point should be at: (, ) To find your next point, you will rise: space(s) and run: space(s). Repeat this procedure to find additional points. In the practice problem above, complete the line on your graph so that it crosses the entire coordinate plane. For the next section of this activity, it will be important that you do this when graphing all lines. For instance, the graph of y = 1 3 x 2 should look like this: Now, consider all the points that are less than the line y = 1 3 x 2 Do you think you will find these points above or below the line? (above / below) With a colored pencil or highlighter, color all of the points on the above graph that are less than the line y = 1 x 2. To represent all 3 of these points, along with the line, we will write y 1 3 x 2. Practice: On the axis to the right, graph y 2x 4 Did you color your graph above or below the line?
4 What is the difference between y 2x 4 and y < 2x 4? Section 2: Graphing Inequalities If we wanted to make a graph of all of the points that are less than y = 2x -4, without including the line, we would write y < 2x 4. In this case, we would draw a dotted line instead of a regular line, and then color beneath the dotted line. It should look like this: Practice: Now, you try. On the coordinate plane below, graph y < 1 x What is the difference between y < 1 x + 2 and y > 1 x + 2? 3 3 Where do you think you should color when y is greater than 1 x + 2? Above or Below the line: 3 Now, on the coordinate plane below, graph y > 1 3 x + 2 What is the difference between y > 1 3 x + 2 and y 1 3 x + 2? Which one has a dotted line? Which one has a complete line?
5 Section 3: Graphing Systems of Inequalities Word Box Intersection Ordered Pair Divide Standard Slope-Intercept Common Two or More Equations A system of equations is a set of two or more. When we were solving systems of linear equations, we graphed both lines and found the. That is to say, the place where the two lines cross (a.k.a. the point that the two lines have in ). A system of inequalities is a set of inequalities. Just like with systems of equations, we can use graphing to solve a system of inequalities. Practice: On the coordinate plane to the right, graph both of these inequalities: y 3x 1 y -2x + 2 (Use different colors to fill in the colored areas of your graph). Next, in dark pencil or pen, outline the areas that the two graphs have in common. That is the solution to your system of inequalities the part your two graphs have in common. Now, on the following grid, graph your two inequalities, and only color the parts that they have in common.
6 Practice For the following systems of inequalities, graph to find the solution. y > x + 2 y < 2x + 3 x + y -2 y < 1 4 x - 1 y x -3 y < -4x + 5 y 1 2 x + 4 y > 2x 3 Note: If you are given an inequality in the form 3x 4y > 1 which is in form, you must first put the inequality into form by solving for y. Example: 3x 4y > 1 To solve for y, I want y by itself, so move everything else to the other side of the inequality. 3x 4y > 1-3x -3x (Subtract 3x from both sides of the inequality) - 4y > -3x + 1-4y > -3x + 1 (Divide both sides by -4) -4-4 y < 3 x + 1 (Note: the > becomes < when you by a negative number. 4 4 y < 3 4 x 1 4
x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =
Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationDetermine If An Equation Represents a Function
Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The
More informationPLOTTING DATA AND INTERPRETING GRAPHS
PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they
More informationMake sure you look at the reminders or examples before each set of problems to jog your memory! Solve
Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More informationSolving Systems of Linear Equations Graphing
Solving Systems of Linear Equations Graphing Outcome (learning objective) Students will accurately solve a system of equations by graphing. Student/Class Goal Students thinking about continuing their academic
More informationGraphing Linear Equations
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationWrite the Equation of the Line Review
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections
More informationHigh School Algebra Reasoning with Equations and Inequalities Solve systems of equations.
Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student
More information7. Solving Linear Inequalities and Compound Inequalities
7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing
More informationLinear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber
Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83
More informationThe Point-Slope Form
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationWriting the Equation of a Line in Slope-Intercept Form
Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing
More informationAnswer Key Building Polynomial Functions
Answer Key Building Polynomial Functions 1. What is the equation of the linear function shown to the right? 2. How did you find it? y = ( 2/3)x + 2 or an equivalent form. Answers will vary. For example,
More informationSlope-Intercept Equation. Example
1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine
More informationTennessee Department of Education. Task: Sally s Car Loan
Tennessee Department of Education Task: Sally s Car Loan Sally bought a new car. Her total cost including all fees and taxes was $15,. She made a down payment of $43. She financed the remaining amount
More informationLesson 4: Solving and Graphing Linear Equations
Lesson 4: Solving and Graphing Linear Equations Selected Content Standards Benchmarks Addressed: A-2-M Modeling and developing methods for solving equations and inequalities (e.g., using charts, graphs,
More information1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
More informationLet s explore the content and skills assessed by Heart of Algebra questions.
Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,
More informationTemperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.
Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationA synonym is a word that has the same or almost the same definition of
Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given
More informationCoordinate Plane, Slope, and Lines Long-Term Memory Review Review 1
Review. What does slope of a line mean?. How do you find the slope of a line? 4. Plot and label the points A (3, ) and B (, ). a. From point B to point A, by how much does the y-value change? b. From point
More informationGraphing - Slope-Intercept Form
2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationWhy should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate
More informationEffects of changing slope or y-intercept
Teacher Notes Parts 1 and 2 of this lesson are to be done on the calculator. Part 3 uses the TI-Navigator System. Part 1: Calculator Investigation of changing the y-intercept of an equation In your calculators
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationLines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan
Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.
More informationUsing Mathematics to Solve Real World Problems
Using Mathematics to Solve Real World Problems Creating a mathematical model: Creating a mathematical model: We are given a word problem Creating a mathematical model: We are given a word problem Determine
More informationGraphs of Proportional Relationships
Graphs of Proportional Relationships Student Probe Susan runs three laps at the track in 12 minutes. A graph of this proportional relationship is shown below. Explain the meaning of points A (0,0), B (1,4),
More information1.2 Linear Equations and Rational Equations
Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationPart 1: Background - Graphing
Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationWarm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.
Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is
More informationSolving Equations Involving Parallel and Perpendicular Lines Examples
Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationCurrent California Math Standards Balanced Equations
Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationActivity 6 Graphing Linear Equations
Activity 6 Graphing Linear Equations TEACHER NOTES Topic Area: Algebra NCTM Standard: Represent and analyze mathematical situations and structures using algebraic symbols Objective: The student will be
More informationGraphs of Proportional Relationships
Graphs of Proportional Relationships Student Probe Susan runs three laps at the track in 12 minutes. A graph of this proportional relationship is shown below. Explain the meaning of points A (0,0), B (1,),
More informationAim: How do we find the slope of a line? Warm Up: Go over test. A. Slope -
Aim: How do we find the slope of a line? Warm Up: Go over test A. Slope - Plot the points and draw a line through the given points. Find the slope of the line.. A(-5,4) and B(4,-3) 2. A(4,3) and B(4,-6)
More informationPolynomial and Rational Functions
Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving
More informationFive Ways to Solve Proportion Problems
Five Ways to Solve Proportion Problems Understanding ratios and using proportional thinking is the most important set of math concepts we teach in middle school. Ratios grow out of fractions and lead into
More informationIV. ALGEBRAIC CONCEPTS
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationAcademic Support Center. Using the TI-83/84+ Graphing Calculator PART II
Academic Support Center Using the TI-83/84+ Graphing Calculator PART II Designed and Prepared by The Academic Support Center Revised June 2012 1 Using the Graphing Calculator (TI-83+ or TI-84+) Table of
More informationGraphing Equations. with Color Activity
Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each
More informationUnit 1: Integers and Fractions
Unit 1: Integers and Fractions No Calculators!!! Order Pages (All in CC7 Vol. 1) 3-1 Integers & Absolute Value 191-194, 203-206, 195-198, 207-210 3-2 Add Integers 3-3 Subtract Integers 215-222 3-4 Multiply
More informationFractions and Linear Equations
Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps
More information10.1 Systems of Linear Equations: Substitution and Elimination
726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations
More informationTI-83/84 Plus Graphing Calculator Worksheet #2
TI-83/8 Plus Graphing Calculator Worksheet #2 The graphing calculator is set in the following, MODE, and Y, settings. Resetting your calculator brings it back to these original settings. MODE Y Note that
More informationTessellations. Practice 1 Identifying Tessellations. In each tessellation, color the repeated shape. Example
Name: Chapter Date: Practice 1 Identifying In each tessellation, color the repeated shape. Example 1. 2. 3. Lesson 14.1 Identifying 133 Is each pattern a tessellation of a single repeated shape? Write
More informationUnit 1 Equations, Inequalities, Functions
Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious
More informationSlope-Intercept Form of a Linear Equation Examples
Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation
More informationSolving Systems of Two Equations Algebraically
8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x
More informationExample 1. Rise 4. Run 6. 2 3 Our Solution
. Graphing - Slope Objective: Find the slope of a line given a graph or two points. As we graph lines, we will want to be able to identify different properties of the lines we graph. One of the most important
More informationCHAPTER 1 Linear Equations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,
More informationGraphing - Parallel and Perpendicular Lines
. Graphing - Parallel and Perpendicular Lines Objective: Identify the equation of a line given a parallel or perpendicular line. There is an interesting connection between the slope of lines that are parallel
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information(Least Squares Investigation)
(Least Squares Investigation) o Open a new sketch. Select Preferences under the Edit menu. Select the Text Tab at the top. Uncheck both boxes under the title Show Labels Automatically o Create two points
More informationMathematical goals. Starting points. Materials required. Time needed
Level C1 of challenge: D C1 Linking the properties and forms of quadratic of quadratic functions functions Mathematical goals Starting points Materials required Time needed To enable learners to: identif
More informationEL-9650/9600c/9450/9400 Handbook Vol. 1
Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations
More informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationLinear Approximations ACADEMIC RESOURCE CENTER
Linear Approximations ACADEMIC RESOURCE CENTER Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations
More informationF.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions
F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions Analyze functions using different representations. 7. Graph functions expressed
More informationSection 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5
Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.
More informationBecause the slope is, a slope of 5 would mean that for every 1cm increase in diameter, the circumference would increase by 5cm.
Measurement Lab You will be graphing circumference (cm) vs. diameter (cm) for several different circular objects, and finding the slope of the line of best fit using the CapStone program. Write out or
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationSection 1.5 Linear Models
Section 1.5 Linear Models Some real-life problems can be modeled using linear equations. Now that we know how to find the slope of a line, the equation of a line, and the point of intersection of two lines,
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationElements of a graph. Click on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on
More informationMATHEMATICS Y6 Geometry 6750 Use co-ordinates and extend to 4 quadrants Equipment MathSphere www.mathsphere.co.uk
MATHEMATICS Y6 Geometry 675 Use co-ordinates and etend to quadrants Paper, pencil, ruler Equipment MathSphere 675 Use co-ordinates and etend to quadrants. Page Concepts Children should be familiar with
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationMSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions
MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSPIRIT 2.0 Lesson: A Point Of Intersection
SPIRIT 2.0 Lesson: A Point Of Intersection ================================Lesson Header============================= Lesson Title: A Point of Intersection Draft Date: 6/17/08 1st Author (Writer): Jenn
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationIntroduction to the TI-Nspire CX
Introduction to the TI-Nspire CX Activity Overview: In this activity, you will become familiar with the layout of the TI-Nspire CX. Step 1: Locate the Touchpad. The Touchpad is used to navigate the cursor
More informationDrawing Lines with Pixels. Joshua Scott March 2012
Drawing Lines with Pixels Joshua Scott March 2012 1 Summary Computers draw lines and circles during many common tasks, such as using an image editor. But how does a computer know which pixels to darken
More informationGrade 6 Mathematics Performance Level Descriptors
Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this
More information