# 2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7

Save this PDF as:

Size: px
Start display at page:

Download "2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7"

## Transcription

1 DETAILED SOLUTIONS AND CONCEPTS GRAPHICAL REPRESENTATION OF LINEAR INEQUALITIES IN TWO VARIABLES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to Thank you! PLEASE NOTE THAT YOU CANNOT USE A CALCULATOR ON THE ACCUPLACER - ELEMENTARY ALGEBRA TEST! YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR! Linear Inequalities in Two Variables An inequality in which two letters are used for variables and the exponent of the variables is 1. Following are a few examples of linear inequalities in two variables: 2x - y 4 y -3x - 6 y < 2x 5x - 3y > 7 y 0 x 0 These inequalities look different from the previous four examples. There is only one variable in each one! See below! Strategy for Graphing Linear Inequalities in Two Variables Replace the inequality sign with an equal sign and graph this linear equation. This is our "boundary" line. Make a solid line if the boundary line is included ( or ) or a dashed line if the boundary line is not included (< or >). Pick a test point that is not on the boundary line and replace the x- and y-variable with it. If the test point makes a true statement with the original inequality, shade the side containing the test point. If the test point makes a false statement with the original inequality, shade the side opposite the side containing the test point.

2 Problem 1: Graph the linear inequality 2x - y First we will graph the equality 2x - y = 4. Coordinates of the x-intercept (when y = 0): 2x - 0 = 4 x = 2 The coordinates of the x-intercept are (2, 0). Coordinates of the y-intercept (when x = 0): 2(0) - y = 4 y = -4 The coordinates of the y-intercept are (0, -4). Coordinates of one additional point using the Point-by-Point Plotting Method: Let's choose x = 4. Please note that we could have chosen any other value, but looking at the existing points, it seemed that 4 was a reasonable choice. Next, we will replace the x in 2x - y = 4 with 4. That is, 2(4) - y = y = 4 and y = 4 The coordinates of the additional point are (4, 4). Since we have a "greater than or equal" which is our boundary line. inequality, we draw a solid line,

3 2. Next we are going to pick a test point. The easiest one is (0, 0) since it is not on the boundary line. Problem 2: Graph the linear inequality y -3x First we will graph the equality y = -3x - 6. Coordinates of the x-intercept (when y = 0): 0 = -3x - 6 3x = -6 x = -2 The coordinates of the x-intercept are (-2, 0). Coordinates of the y-intercept (when x = 0): y = -3(0) - 6 = -6 The coordinates of the y-intercept are (0, -6). Coordinates of one additional point using the Point-by-Point Plotting Method: Let's choose x = -4. Please note that we could have chosen any other value, but looking at the existing points, it seemed that -4 was a reasonable choice. Next, we will replace the x in y = -3x - 6 with -4. Using the original inequality 2x - y 4 and (0, 0), we will write the following statement: 2(0) and 0 4. This is a false statement since 0 is not greater than or equal to 4. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables!

4 That is, y = -3(-4) - 6 y = 12-6 and y = 6 The coordinates of the additional point are (-4, 6). Since we have a "less than or equal" is our boundary line. inequality, we draw a solid line, which 2. Next we are going to pick a test point. The easiest one is (0, 0) since it is not on the boundary line. Problem 3: Graph the linear inequality y > 2x. 1. First we will graph the equality y = 2x. Coordinates of the x-intercept (when y = 0): 0 = 2x x = 0 The coordinates of the x-intercept are (0, 0). Using the original inequality y -3x - 6 and (0, 0), we will write the following statement: 0-3(0) - 6 and 0-6. This is a false statement since 0 is not less than or equal to -6. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables!

5 Coordinates of the y-intercept (when x = 0): y = 2(0) y = 0 The coordinates of the y-intercept are (0, 0). We find that the coordinates of the x- and y-intercept are the same. That is, we didn't get two data points... and without two data points we cannot graph our linear equation. Here we definitely have to use the the Point-by-Point Plotting Method to find the coordinates of some other points. Let's choose a positive and a negative value for x. Not too close to the point (0,0) so that we can draw an accurate line. Let's use x = 4, then y = 2(4) and y = 8 The coordinates of the additional point are (4, 8). Next, let's use x = -4, then y = 2(-4) and y = -8 The coordinates of the second point are (-4, -8). Since we have strictly a "less than" < inequality, we draw a dashed line, which is our boundary line.

6 2. Next we are going to pick a test point. Please note that the test point (0, 0) is on the boundary line, therefore, we must use a different test point. Any point will do, as long as it is not on the boundary line. Let's use (4, 4). Problem 4: Graph the linear equation 3y > -9x. 1. First we will graph the equality 3y = -9x. Coordinates of the x-intercept (when y = 0): 3(0) = -9x x = 0 The coordinates of the x-intercept are (0, 0). Coordinates of the y-intercept (when x = 0): 3y = -9(0) 0 = 0 The coordinates of the y-intercept are (0, 0). Using the original inequality y > 2x and (4, 4), we will write the following statement: 4 > 2(4) and 4 > 8. This is a false statement since 4 is not greater than 8. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables! We find that the coordinates of the x- and y-intercept are the same. That is, we didn't get two data points... and without two data points we cannot graph our linear equation. Here we definitely have to use the the Point-by-Point Plotting Method to find the coordinates of some other points.

7 Let's choose a positive and a negative value for x. Not too close to the point (0,0) so that we can draw an accurate line. Let's use x = 2, then 3y = -9(2) and y = -6 The coordinates of the additional point are (2, -6). Next, let's use x = -2, then 3y = -9(-2) and y = 6 The coordinates of the second point are (-2, 6). Since we have strictly a "greater than" > inequality, we draw a dashed line, which is our boundary line. 2. Next we are going to pick a test point. Please note that the test point (0, 0) is on the boundary line, therefore, we must use a different test point. Any point will do, as long as it is not on the boundary line. Let's use (-4, 4). Using the original inequality 3y > -9x and (-4, 4), we will write the following statement: 3(4) > -9(-4) and 12 > 36. This is a false statement since 12 is not greater than 36. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables!

8 Problem 5: Graph the linear inequality y 0. This inequality looks different from the previous four examples. There is only one variable in it! So how can we call it a "linear inequality in two variables"?????? Oddly enough, it still fall into this category with the "missing" variable being allowed to take on any value. Recall the following: Equation of Horizontal Lines The equation of a horizontal line is y = b, where b is a constant and the y-coordinate of any point on the line. Horizontal lines do not have an x-intercept, but they do have a y- intercept at. We know that the graph of y = 0 is a horizontal line, specifically, it is the equation of the x-axis. Since we have a "greater than or equal" inequality the x-axis is included! Next we are going to pick a test point. Since the point (0, 0) is on the boundary line, we must use a different test point. Any point will do, as long as it is not on the boundary line. Let's use (4, -4). Using the original inequality y 0 and (4, -4), we will write the following statement: This is a false statement since -4 is not greater than 0. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables!

9 Problem 6: Graph the linear inequality x 0. This inequality also looks different from Examples 1-4. Recall the following: Equation of Vertical Lines The equation of a vertical line is x = a, where a is a constant and the x-coordinate of any point on the line. Vertical lines do not have a y-intercept, but they do have an x- intercept at. We know that the graph of x = 0 is a vertical line, specifically, it is the equation of the y-axis. Since we have a "less than or equal" inequality the y-axis is included! Next we are going to pick a test point. Since the point (0, 0) is on the boundary line, we must use a different test point. Any point will do, as long as it is not on the boundary line. Let's use (2, 2). Using the original inequality x (2, 2), we will write the following statement: 0 and 2 0. This is a false statement since 2 is not less than 0. Therefore, we shaded the side opposite the side containing the test point. The boundary line together with the shaded area represents the graph of the inequality. Sometimes, this is also referred to as the "solution" of the inequality in two variables!

10 Problem 7: Solve the following system of linear inequalities: 3y > -9x y > 2x A system of linear inequalities can only be solved graphically. That is, we graph each inequality on the same pair of axes. The solution set of the system will be the overlapping region of the solution sets of the two inequalities. We have already graphed the two inequalities separately in Problems 3 and 4 above. Below you can find their graphs one more time.

11 Now we can determine the solution of the system of these two inequalities by observing where the shaded regions overlap. Problem 8: Solve the following system of linear inequalities: y 0 x 0 We have already graphed the two inequalities separately in Problems 5 and 6 above. Below you can find their graphs one more time.

12 Now we can determine the solution of the system of these two inequalities by observing where the shaded regions overlap.

13 Problem 9: Find the inequality that represents the shaded region. First, let's find the equation of the boundary line. Since it is solid, we know that we have either a "greater than or equal" or a "less than or equal" situation. We note that the boundary line has one intercept at the origin (0, 0) and a point at (6, 6). This allows us to find the equation as follows. First, we find the slope m given its formula. Next, we'll use the point-slope form of the linear equation and how about the point (0, 0)? y - 0 = 31x - 0) then y = x Now we need to find out if the shaded region is represented by y x or y x. First, let's assume it's y inequality true. x then a point within the shaded region should make this Let's use (1, 0) for simplicity's sake. Is 0 1? No! Therefore, the shaded region is represented by y x.

### GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

### Graphing 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

### EQUATIONS 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

### Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

### 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 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

### The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

### Section 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

### Systems of Linear Equations

DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

### Plot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.

Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line

### Linear 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

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

### Graphing 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

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### Lecture 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

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### 1 You Try. 2 You Try.

1 Simplify. 1 You Try. 1) Simplify. AF 1.2 2 Evaluate if, and. 2 You Try. 2) Evaluate if. 1.0 Page 1 of 19 MDC@ACOE (AUSD) 09/13/10 3 Simplify. 3 You Try. 3) Simplify. NS 1.2 4 Simplify. 4 You Try. 4)

### Section 2.1 Intercepts; Symmetry; Graphing Key Equations

Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point

### 1 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

### Practice Problems for Exam 1 Math 140A, Summer 2014, July 2

Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other

### Writing 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

1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

### Graphing - 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,

### MATH 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

### WARM UP EXERCSE. 1-3 Linear Functions & Straight lines

WARM UP EXERCSE A company makes and sells inline skates. The price-demand function is p (x) = 190 0.013(x 10) 2. Describe how the graph of function p can be obtained from one of the library functions.

### Sect The Slope-Intercept Form

Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

### Algebra 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

### What 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

### The 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

### 1. Graphing Linear Inequalities

Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means

### Algebraic 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

### Slope-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

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### Slope-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

### Write 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

### Answer 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,

### Intro to Linear Equations Algebra 6.0

Intro to Linear Equations Algebra 6.0 Linear Equations: y x 7 y x 5 x y Linear Equations generally contain two variables: x and y. In a linear equation, y is called the dependent variable and x is the

### LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

### MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem. Constant Rate of Change/Slope

MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem Constant Rate of Change/Slope In a Table Relationships that have straight-lined graphs

### EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

### Brunswick 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

### A 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

### Temperature 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

### Rationale/Lesson Abstract: Students will be able to solve a Linear- Quadratic System algebraically and graphically.

Grade Level/Course: Algebra 1 Lesson/Unit Plan Name: Linear- Quadratic Systems Rationale/Lesson Abstract: Students will be able to solve a Linear- Quadratic System algebraically and graphically. Timeframe:

### MSLC 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

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

### Module: Graphing Linear Equations_(10.1 10.5)

Module: Graphing Linear Equations_(10.1 10.5) Graph Linear Equations; Find the equation of a line. Plot ordered pairs on How is the Graph paper Definition of: The ability to the Rectangular Rectangular

### Activity 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

### Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate

### Lines, 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.

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

### Patterns, Equations, and Graphs. Section 1-9

Patterns, Equations, and Graphs Section 1-9 Goals Goal To use tables, equations, and graphs to describe relationships. Vocabulary Solution of an equation Inductive reasoning Review: Graphing in the Coordinate

### Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

9.11 Quadratics - Graphs of Quadratics Objective: Graph quadratic equations using the vertex, x-intercepts, and y-intercept. Just as we drew pictures of the solutions for lines or linear equations, we

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### Warm 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

### Building Polynomial Functions

Building Polynomial Functions NAME 1. What is the equation of the linear function shown to the right? 2. How did you find it? 3. The slope y-intercept form of a linear function is y = mx + b. If you ve

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### Algebra 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

Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

### Vocabulary 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

### BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line

College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University

### Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

### 3.1 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

### Elements 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

### 2.3 Writing Equations of Lines

. Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### 4.4 Transforming Circles

Specific Curriculum Outcomes. Transforming Circles E13 E1 E11 E3 E1 E E15 analyze and translate between symbolic, graphic, and written representation of circles and ellipses translate between different

### Solving 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

### GRAPH OF A RATIONAL FUNCTION

GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find

### Answer 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.

### Make 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

### Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

### CHAPTER 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

### Algebra 1 Chapter 05 Review

Class: Date: Algebra 1 Chapter 05 Review Multiple Choice Identify the choice that best completes the statement or answers the question. Find the slope of the line that passes through the pair of points.

### Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

### ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

### ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Students will understand 1. use numerical bases and the laws of exponents

Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.

COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

### Chapter 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

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### Intersecting Two Lines, Part One

Module 1.4 Page 97 of 938. Module 1.4: Intersecting Two Lines, Part One This module will explain to you several common methods used for intersecting two lines. By this, we mean finding the point x, y)

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### MAT12X Intermediate Algebra

MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

### Multiple Representations of Equations & What We Know. Note that the worked out our turn and your turn charts can also be used as a matching activity.

Multiple Representations of s & What We California State Standards: 7 AF., 7 AF 3.3, 7 AF 3.4, Alg. 6., Alg. 7., Alg. 8. CCSS: 8.EE.6, 8.F., 8.F., 8.F.4 The idea of this lesson is to have students make

### Math 113 Review for Exam I

Math 113 Review for Exam I Section 1.1 Cartesian Coordinate System, Slope, & Equation of a Line (1.) Rectangular or Cartesian Coordinate System You should be able to label the quadrants in the rectangular

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### Algebra 1 Topic 8: Solving linear equations and inequalities Student Activity Sheet 1; use with Overview

Algebra 1 Topic 8: Student Activity Sheet 1; use with Overview 1. A car rental company charges \$29.95 plus 16 cents per mile for each mile driven. The cost in dollars of renting a car, r, is a function

### Part 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