# The Point-Slope Form

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 is m Q(x, y) y y 1 Often in mathematics it is useful to be able to write the equation of a line, given its slope and any point on the line. In this section, we will derive a third special form for a line for this purpose. Suppose that a line has slope m and that it passes through the known point P(x 1, y 1 ). Let Q(x, y) be any other point on the line. Once again we can use the definition of slope and write P(x 1, y 1 ) x x 1 x m y y 1 x x 1 Multiplying both sides of equation (1) by x x 1, we have (1) m(x x 1 ) y y 1 or y y 1 m(x x 1 ) () Equation () is called the point-slope form for the equation of a line, and all points lying on the line [including (x 1, y 1 )] will satisfy this equation. We can state the following general result. Rules and Properties: Point-Slope Form for the Equation of a Line NOTE The equation of a line with undefined slope passing through the point (x 1, y 1 ) is given by x x 1. The equation of a line with slope m that passes through point (x 1, y 1 ) is given by y y 1 m(x x 1 ) Example 1 Write the equation for the line that passes through point (, 1) with a slope of. Letting (x 1, y 1 ) (, 1) and m in point-slope form, we have y ( 1) (x ) or y 1 x 9 We can write the final result in slope-intercept form as y x 10 CHECK YOURSELF 1 Write the equation of the line that passes through point (, 4) with a slope of Write your result in slope-intercept form.. 577

2 578 CHAPTER 7 GRAPHING AND INEQUALITIES Because we know that two points determine a line, it is natural that we should be able to write the equation of a line passing through two given points. Using the point-slope form together with the slope formula will allow us to write such an equation. Example Write the equation of the line passing through (, 4) and (4, 7). First, we find m, the slope of the line. Here m NOTE We could just as well have chosen to let (x 1, y 1 ) (4, 7) The resulting equation will be the same in either case. Take time to verify this for yourself. Now we apply the point-slope form with m and (x 1, y 1 ) (, 4): y 4 (x ) y 4 x y x 1 CHECK YOURSELF Write the equation of the line passing through (, 5) and (1, ). Write your result in slope-intercept form. A line with slope zero is a horizontal line. A line with an undefined slope is vertical. The next example illustrates the equations of such lines. Example (a) Find the equation of a line passing through (7, ) with a slope of zero. We could find the equation by letting m 0. Substituting the ordered pair (7, ) into the slope-intercept form, we can solve for b. y mx b 0(7) b b So, y 0 x or y It is far easier to remember that any line with a zero slope is a horizontal line and has the form y b

3 THE POINT-SLOPE FORM SECTION The value for b will always be the y coordinate for the given point. (b) Find the equation of a line with undefined slope passing through (4, 5). A line with undefined slope is vertical. It will always be of the form x a, in which a is the x coordinate for the given point. The equation is x 4 CHECK YOURSELF (a) Find the equation of a line with zero slope that passes through point (, 5). (b) Find the equation of a line passing through (, 6) with undefined slope. Alternate methods for finding the equation of a line through two points exist and have particular significance in other fields of mathematics, such as statistics. The following example shows such an alternate approach. Example 4 Write the equation of the line through points (, ) and (4, 5). First, we find m, as before: m 5 4 ( ) 6 1 NOTE We substitute these values because the line must pass through (, ). We now make use of the slope-intercept equation, but in a slightly different form. Because y mx b, we can write b y mx Now letting x, y, and m 1 we can calculate b:, b 1 ( ) 11 With m 1 we can apply the slope-intercept form to write the equation of and b 11, the desired line. We have y 1 x 11 CHECK YOURSELF 4 Repeat the Check Yourself exercise, using the technique illustrated in Example 4. We now know that we can write the equation of a line once we have been given appropriate geometric conditions, such as a point on the line and the slope of that line. In some applications the slope may be given not directly but through specified parallel or perpendicular lines.

4 580 CHAPTER 7 GRAPHING AND INEQUALITIES Example 5 Find the equation of the line passing through ( 4, ) and parallel to the line determined by x 4y 1. First, we find the slope of the given parallel line, as before: x 4y 1 4y x 1 NOTE The slope of the given line is 4. y 4 x Now because the slope of the desired line must also be we can use the point-slope form 4, to write the required equation: NOTE The line must pass through ( 4, ), so let (x 1, y 1 ) ( 4, ). y ( ) [x ( 4)] 4 This simplifies to y 4 x 6 and we have our equation in slope-intercept form. CHECK YOURSELF 5 Find the equation of the line passing through (5, 4) and perpendicular to the line with equation x 5y 10. Hint: Recall that the slopes of perpendicular lines are negative reciprocals of each other. The following chart summarizes the various forms of the equation of a line. Equation Form for Line L Conditions Standard ax by c Constants a and b cannot both be zero. Slope-intercept y mx b Line L has y intercept (0, b) with slope m. Point-slope y y 1 m(x x 1 ) Line L passes through point (x 1, y 1 ) with slope m. Horizontal y k Slope is zero. Vertical x h Slope is undefined. CHECK YOURSELF ANSWERS 1. y. y. (a) y 5; (b) x x 11 x 1 4. y 5. y 5 x x 11

5 Name 7. Exercises Section Date Write the equation of the line passing through each of the given points with the indicated slope. Give your results in slope-intercept form, where possible. 1. (0, ), m. (0, 4), m ANSWERS 1... (0, ), m 4. (0, ), m (0, 4), m 0 6. (0, 5), m (0, 5), m 5 8. (0, 4), m (1, ), m 10. ( 1, ), m 11. (, ), m 1. (1, 4), m 4 1. (5, ), m (4, ), m (, ), m is undefined 16. (, 5), m Write the equation of the line passing through each of the given pairs of points. Write your result in slope-intercept form, where possible. 17. (, ) and (5, 6) 18. (, ) and (6, 4) (, ) and (, 0) 0. ( 1, ) and (4, ) 1. (, ) and (4, ). ( 5, ) and (4, 1)

6 ANSWERS.. (, 0) and (0, ) 4. (, ) and (, 4) (0, 4) and (, 1) 6. ( 4, 1) and (, 1) Write the equation of the line L satisfying the given geometric conditions. 7. L has slope 4 and y intercept (0, ). 8. L has slope and y intercept (0, 4). 9. L has x intercept (4, 0) and y intercept (0, ). 0. L has x intercept (, 0) and slope 1. L has y intercept (0, 4) and a 0 slope. 4.. L has x intercept (, 0) and an undefined slope.. L passes through point (, ) with a slope of L passes through point (, 4) with a slope of. 5. L has y intercept (0, ) and is parallel to the line with equation y x L has y intercept (0, ) and is parallel to the line with equation y x L has y intercept (0, 4) and is perpendicular to the line with equation y x L has y intercept (0, ) and is parallel to the line with equation y L has y intercept (0, ) and is parallel to the line with equation y L has y intercept (0, ) and is perpendicular to the line with equation x y L passes through point (, ) and is parallel to the line with equation y x. 4. L passes through point ( 4, ) and is parallel to the line with equation y x L passes through point (, ) and is parallel to the line with equation y 4 x 4. 58

7 ANSWERS 44. L passes through point (, 1) and is perpendicular to the line with equation y x L passes through point (5, ) and is perpendicular to the line with equation y x. 46. L passes through point (, 4) and is perpendicular to the line with equation y 5 x. 47. L passes through (, 1) and is parallel to the line with equation x y L passes through (, 5) and is parallel to the x axis. 49. Describe the process for finding the equation of a line if you are given two points on the line. 50. How would you find the equation of a line if you were given the slope and the x intercept? 51. A temperature of 10 C corresponds to a temperature of 50 F. Also 40 C corresponds to 104 F. Find the linear equation relating F and C. 5. In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a linear equation. Projections are that 100 items will cost \$10,000 to produce and that 00 items will cost \$,000 to produce. Find the equation that relates C and x. 5. A word processing station was purchased by a company for \$10,000. After 4 years it is estimated that the value of the station will be \$4000. If the value in dollars V and the time the station has been in use t are related by a linear equation, find the equation that relates V and t. 54. Two years after an expansion, a company had sales of \$4,000. Four years later the sales were \$10,000. Assuming that the sales in dollars S and the time in years t are related by a linear equation, find the equation relating S and t a. b. c. d. e. f. Getting Ready for Section 7.4 [Section.7] Graph each of the following inequalities. (a) x (b) x (c) x 8 (d) x 9 (e) x 1 (f) x 10 (g) (h) 4 x 6 x 4 g. h. 58

8 Answers 1. y x. y 5. y 4 7. x y 5 4 x 5 9. y x y x 9 1. y 15. x 5 x y x y 1. y. 4 x y x 5. y 5 7. y 4x 9. x 4 y 1 x 1. y 4. y 5x 1 5. y x y 41. y x 8 4. y x y 1 x 4 y 1 x y x F 9 5 C 5. V 1500t 10,000 a. b. 0 0 c. d e. f g. h

### 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

### 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

### 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

### 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.

### Worksheet A5: Slope Intercept Form

Name Date Worksheet A5: Slope Intercept Form Find the Slope of each line below 1 3 Y - - - - - - - - - - Graph the lines containing the point below, then find their slopes from counting on the graph!.

### 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

### 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

### Homework #1 Solutions

Homework #1 Solutions Problems Section 1.1: 8, 10, 12, 14, 16 Section 1.2: 2, 8, 10, 12, 16, 24, 26 Extra Problems #1 and #2 1.1.8. Find f (5) if f (x) = 10x x 2. Solution: Setting x = 5, f (5) = 10(5)

### PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### 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,

### Algebra 2 PreAP. Name Period

Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

### 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

### 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,

### Solving 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

### Unit 5: Coordinate Geometry Practice Test

Unit 5: Coordinate Geometry Practice Test Math 10 Common Name: Block: Please initial this box to indicate you carefully read over your test and checked your work for simple mistakes. What I can do in this

### 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

### 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

### Why 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

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### 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

### 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

### Algebra 2: Q1 & Q2 Review

Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

### 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

### CRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide

Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are

### 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

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### 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

### CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses

### 1.1 Practice Worksheet

Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)

### 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

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 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

### G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam

G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e - C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Answers for the lesson Write Linear Equations in Slope-Intercept Form

LESSON 4.1 Answers for the lesson Write Linear Equations in Slope-Intercept Form Skill Practice 1. slope. You can substitute the slope for m and the y-intercept for b to get the equation of the line..

### List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

### Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form

Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Students will use various media (computer, graphing calculator, paper and pencil) to graph/sketch linear equations.

Title: Lines, Lines, Everywhere!! A discovery/exploration lesson investigating equations of the form y = mx + b to see how the values of b and m affects the graph. Link to Outcomes: Communication/ Cooperation

### Chapter 1 Linear Models page Linear Models Part 1 2 Linear Models Activities 1 17 Linear Models Part 2 21 Linear Models Activities 2 28

Table of Contents Chapter 1 Linear Models page Linear Models Part 1 Linear Models Activities 1 17 Linear Models Part 1 Linear Models Activities 8 Chapter Linear Programming Linear Programming Part 1 34

### Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

### Algebra 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

### Mathematics Curriculum Guide Precalculus 2015-16. Page 1 of 12

Mathematics Curriculum Guide Precalculus 2015-16 Page 1 of 12 Paramount Unified School District High School Math Curriculum Guides 2015 16 In 2015 16, PUSD will continue to implement the Standards by providing

### 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

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

### Graphing Trigonometric Skills

Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

### Algebra 1 End-of-Course Exam Practice Test with Solutions

Algebra 1 End-of-Course Exam Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fill-in Response Items, write your answer in the box provided, placing one digit in

### Tennessee 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

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### How to Graph Trigonometric Functions

How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### REVIEW EXERCISES DAVID J LOWRY

REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and

74 In the Herb Business, Part III Factoring and Quadratic Equations In the herbal medicine business, you and your partner sold 120 bottles of your best herbal medicine each week when you sold at your original

### Chesapeake College. Syllabus. MAT 031 Elementary Algebra (silcox87517) John H. Silcox

Chesapeake College Syllabus MAT 031 Elementary Algebra (silcox87517) John H. Silcox Phone 410-778-4540 (W) Phone 410-348-5370 (H) john.silcox@1972.usna.com Syllabus For MAT 031 I. OFFICE HOURS: One hour

### Section 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,

### Functions Modeling Change: A Precalculus Course. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Functions Modeling Change: A Precalculus Course Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 PREFACE This supplement consists of my lectures of a freshmen-level mathematics class offered

### 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

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Math 1. Month Essential Questions Concepts/Skills/Standards Content Assessment Areas of Interaction

Binghamton High School Rev.9/21/05 Math 1 September What is the unknown? Model relationships by using Fundamental skills of 2005 variables as a shorthand way Algebra Why do we use variables? What is a

### MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

### Blue Pelican Alg II First Semester

Blue Pelican Alg II First Semester Teacher Version 1.01 Copyright 2009 by Charles E. Cook; Refugio, Tx (All rights reserved) Alg II Syllabus (First Semester) Unit 1: Solving linear equations and inequalities

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### Ellington High School Principal

Mr. Neil Rinaldi Ellington High School Principal 7 MAPLE STREET ELLINGTON, CT 0609 Mr. Dan Uriano (860) 896- Fa (860) 896-66 Assistant Principal Mr. Peter Corbett Lead Teacher Mrs. Suzanne Markowski Guidance

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### Prentice Hall MyMathLab Algebra 1, 2011

Prentice Hall MyMathLab Algebra 1, 2011 C O R R E L A T E D T O Tennessee Mathematics Standards, 2009-2010 Implementation, Algebra I Tennessee Mathematics Standards 2009-2010 Implementation Algebra I 3102

### Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

### COLLEGE ALGEBRA LEARNING COMMUNITY

COLLEGE ALGEBRA LEARNING COMMUNITY Tulsa Community College, West Campus Presenter Lori Mayberry, B.S., M.S. Associate Professor of Mathematics and Physics lmayberr@tulsacc.edu NACEP National Conference

### A Year-long Pathway to Complete MATH 1111: College Algebra

A Year-long Pathway to Complete MATH 1111: College Algebra A year-long path to complete MATH 1111 will consist of 1-2 Learning Support (LS) classes and MATH 1111. The first semester will consist of the

### 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

### Linear 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

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### SECTION 2-2 Straight Lines

- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

### Successful completion of Math 7 or Algebra Readiness along with teacher recommendation.

MODESTO CITY SCHOOLS COURSE OUTLINE COURSE TITLE:... Basic Algebra COURSE NUMBER:... RECOMMENDED GRADE LEVEL:... 8-11 ABILITY LEVEL:... Basic DURATION:... 1 year CREDIT:... 5.0 per semester MEETS GRADUATION

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### Answers to Text Questions and Problems. Chapter 22. Answers to Review Questions

Answers to Text Questions and Problems Chapter 22 Answers to Review Questions 3. In general, producers of durable goods are affected most by recessions while producers of nondurables (like food) and services

### 9 Multiplication of Vectors: The Scalar or Dot Product

Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### ( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:

Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (

### Project Scheduling: PERT/CPM

Project Scheduling: PERT/CPM Project Scheduling with Known Activity Times (as in exercises 1, 2, 3 and 5 in the handout) and considering Time-Cost Trade-Offs (as in exercises 4 and 6 in the handout). This

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II

Intermediate Math Circles March 7, 2012 Linear Diophantine Equations II Last week: How to find one solution to a linear Diophantine equation This week: How to find all solutions to a linear Diophantine

### March 2013 Mathcrnatics MATH 92 College Algebra Kerin Keys. Dcnnis. David Yec' Lscture: 5 we ekly (87.5 total)

City College of San Irrancisco Course Outline of Itecord I. GENERAI- DESCRIPI'ION A. Approval Date B. Departrnent C. Course Number D. Course Title E. Course Outline Preparer(s) March 2013 Mathcrnatics

### MAT 116. Algebra 1A. Version 5 12/15/07 MAT 116

- MAT 116 Algebra 1A Version 5 12/15/07 MAT 116 Program Council The Academic Program Councils for each college oversee the design and development of all University of Phoenix curricula. Council members

### CONSUMER PREFERENCES THE THEORY OF THE CONSUMER

CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials