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DrDelMath

Intermediate Algebra 5th Edition
by Elayn Martin-Gay
SUMMARY

Chapter 3: Graphs and Functions

Section 3.1: Graphing Equations in Two Variables

Textbook Objectives
  1. Plot ordered pairs.
  2. Determine whether an ordered pair is a solution to an equation in two variables.
  3. Graph linear equations.
  4. Graph nonlinear equations.
Additional Goals
  1. Construct the Cartesian Coordinate system.
  2. Know the one-to-one correspondence between points in the Cartesian Coordinate system and ordered pairs of real numbers.
  3. Know the definition of graph.
  4. Know the definition of solution.
  5. Recall the two fundamental properties of equations.
  6. Recall the three fundamental properties of inequalities.
  7. Determine the x and y intercepts of a graph when given its equation.

Embedded in the following diagram are some obvious facts about the Cartesian Coordinate system, points in the system, and the Real Number coordinates of those points. Because these facts are obvious, the beginning student has a tendency to overlook them at times when their application might be appropriate. You are advised to study these facts and look for applications.
coordinate System

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Location of Graphs

The preceding six definitions of solution, solution set, and graph make it clear that the graph of an equation in two variables or an inequality in two variables consists of points represented by ordered pairs of real numbers. Because every ordered pair of real numbers is associated with a point in the Cartesian Coordinate System (and visa versa), we deduce that the graph of any equation in two variables or inequality in two variables must appear in the Cartesian Coordinate System.

  • Graphs of Equations and Inequalities in one variable appear on the Real Number Line.
    • One variable -- one dimensional.
    • Solutions are numbers -- numbers are represented on the Real Number Line.
  • Graphs of Equations and Inequalities in two variables appear in the Cartesian Coordinate System.
    • Two variables -- two dimensional.
    • Solutions are ordered pairs of numbers -- ordered pairs of numbers are represented on the Cartesian Coordinate System.

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Properties of Equations:
Properties of Inequalities:

An understanding of the Cartesian Coordinate System permits a development and an understanding of some basic formulas and concepts in an algebraic framework. We begin with an ancient formula (The Pythagorean Theorem) and use it to develop a few useful algebraic formulas.

The first two formulas for a circle should be memorized. They are used frequently in this course and the next as well as applications outside the classroom.

The following are links to other web pages which are related to the topics in this section.
The alternate or supplemental discussion presented on these other pages should be helpful.

About Graphing Equations in Two Variables

Minimal List of Exercises Page 126.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Readiness Check: 1 - 20
Section 3.1 Exercise Set: 1, 5, 6, 11, 13, 15, 17, 19, 21, 25, 27, 33, 39, 45, 49, 57, 59, 61.

Minimal List of Exercises Page 189.

Section 3.7 Exercise Set: 23, 29, 33, 35, 37, 43, 45

Section 3.2: Introduction to Functions
Textbook Objectives
  1. Define relation, domain, and range.
  2. Identify functions.
  3. Use the vertical line test for functions.
  4. Find the domain and range of functions.
  5. Use function notation.
Additional Goals

Minimal List of Exercises Page 141.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 10
Section 3.2 Exercise Set: 3, 5, 6, 9, 12, 17, 20, 22, 23, 25, 29-37, 39, 41, 43, 45, 50, 51, 52, 55, 67-69, 73, 75, 77, 81, 83, 93, 95.

Section 3.3: Graphing Linear Equations in Two Variables
Section 3.7: Graphing Linear Inequalities in Two Variables
Textbook Objectives
  1. Graph linear equations in two variables.
  2. Graph linear equations by finding intercepts.
  3. Graph vertical and horizontal lines.
Textbook Objectives
  1. Graph linear inequalities.
  2. Graph the intersection or union of two linear inequalities
Additional Goals
  1. Know the definitions of solution, solution set, graph, x-intercept, and y-intercept.
  2. Know the definition of linear equation in two variables.
  3. Know how to find x-intercepts and y-intercepts.
  4. Know the standard form for a linear equation in two variables.
  5. Know the graph of a linear equation in two variables is a non-vertical line.
  6. Know that every non-vertical line is the graph of a linear equation in two variables.
Additional Goals
  1. Know the definition of a linear inequality in two variables.
  2. Know the definition of solution of a linear inequality in two variables.
  3. Know the definition of solution set of a linear inequality in two variables.
  4. Know the Law of Trichotomy.
  5. Understand how the Law of Trichotomy relates to solving a linear inequality in two variables.
  6. Know the process for solving and graphing a linear inequality in two variables.
geogebra geogebra

To rewrite an equation or inequality means to change the form of the equation without changing the solution set. Another way to understand the idea of rewriting an equation or inequality is to view it as generating an equivalent equation or inequality.

Therefore the two Properties of Equations and Three Properties of Inequalities which are used to generate equivalent equations and inequalities are again important.

When an equation in two variables is not written in the form y = mx + b, these basic properties are used to determine if it is a linear equation in two variables. Similar statements are valid for inequalities in two variables.

geogebra
The symbols = and < are frequently combined to mean the union of the solution set of the inequality and the solution set of the equality. The result is the solution set of the inequality and its boundary.
Therefore you may see statements of the form y ≤ mx + b or Ax + By ≤ C. Although an equation is involved, these are usually called inequalities.

In the same way the symbols = and > are frequently combined to mean the union of the solution set of the inequality and the solution set of the equality. The result is the solution set of the inequality and its boundary.
Therefore you may see statements of the form y ≥ mx + b or Ax + By ≥ C. Although an equation is involved, these are usually called inequalities.
 
geogebra

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When two linear expressions are compared, the Law of Trichotomy dictates that we consider three cases. We refer to the equation in the triple as the boundary equation because its graph forms a boundary between the graphs of the two inequalities.

When considering linear equations and inequalities in two variables, the graph of the boundary equation divides the coordinate plane into two half-planes, one of which is the graph of the "less-than" inequality and the other is the graph of the "greater-than" inequality.

Sketching the graph of the boundary equation (accomplished by plotting two points) and testing one point determines the solution sets and graphs for each of the three.

Observation: The above procedure considers, and solves, both inequalities as well as the corresponding equation as suggested by The Law of Trichotomy. geogebra

Here are a few sets of examples.
Set 1   Set 2    Set 3    

About Graphing Linear Equations and Inequalities in Two Variables

Minimal List of Exercises Page 150.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 6
Section 3.3 Exercise Set: For the set {1, 7, 9, 11} replace f(x) with y and sketch the resulting linear equation.
For the set {13, 15, 16, 17, 19, 20} sketch the graph only.
35, 39, 43, 51, 57, 75.

Section 3.4: The Slope of a Line
Textbook Objectives
  1. Find the slope of a line given two points on the line.
  2. Find the slope of a line given the equation of the line.
  3. Interpret the slope-intercept form in an application.
  4. Find the slopes of horizontal and vertical lines.
  5. Compare the slopes of parallel and perpendicular lines.
Additional Goals
  1.  

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About Slope of a Line

Minimal List of Exercises Page 162.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 12
Section 3.4 Exercise Set: 9, 10, 11, 13, 16
For the set { 25, 27, 29} replace f(x) with y and then find the slope, the x-intercept, and the y-intercept.
48, 49, 61, 62, 63, 64, 71, 74, 79, 85, 87, 89, 91, 93, 95.

Section 3.5: Equations of Lines
Textbook Objectives
  1. Use slope-intercepts form to write the equation of a line.
  2. Graph a line using its slope and y-intercept.
  3. Use the point-slope form to write the equation of a line.
  4. Write equations of vertical and horizontal lines.
  5. Find equations of parallel and perpendicular lines.
Additional Goals
  1. Know the slope-intercept form for the equation of a line.
  2. Know the point-slope from of the equation of a line.
  3. Find the equation of a line when given a point on the line and its slope.
  4. Find the equation of a line when given two points on the line.

The little apprentice owl tells you how to convert the equation of a line from one form to another. small owl'

In Special Topics there are two short papers about linear equations in two variables.
Each of them contains everything about linear equations in two variables.
Individually or collectively they provide a quick summary of all information about linear equations in two variables. They are Paper 1 and Paper 2.

About Equations of Lines

Minimal List of Exercises Page 173.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 10
Section 3.5 Exercise Set: 3, 5, 9, 13, 15, 17, 19, 21, 25, 27, 29, 35, 37, 39, 41, 43, 65.

Section 3.6: Graphing Piecewise Defined Functions

Textbook Objectives
  1. Graph piecewise-defined functions.
  2. Reflect graphs.
Additional Goals
  1.  
  2.  

 

Minimal List of Exercises Page 183.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check:
Section 3.6 Exercise Set:

Systems of Inequalities GeoGebra