x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =


 Douglas Logan
 4 years ago
 Views:
Transcription
1 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 slope value is away from zero. Another thing to keep in mind is that the slope is the same value from point to point (or between any two points on the line). This is called the average rate of change because it describes how much the line changes from point to point. There are three different ways to find the slope, labeled m. rise 1) m = this is the graphical way run y y1 ) Given two points ( 1, y 1 ) and (, y ): m = 1 3) Calculate how much y changes from point to point and compare that to how much changes from point to point. The third way requires a little eplanation. Let s say we re given the points (3, 6) and (5, 9) and we want to find what the slope of the line is between them. I m going to make a t chart. means change in and y means change in y, or how much and y change from point to point. y y increases by y increases by y 3 Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m = y y = =. I just prefer and y and will use it almost eclusively this quarter because of its adaptation to polynomials in general. There are a couple of other things to notice about slopes. One is that if a slope is positive, it is going uphill from left to right; if a slope is negative, it is going downhill from left to right. In other words, in a positive slope, as gets bigger, y also gets bigger. In a negative slope, as gets bigger, y gets smaller. Since the slope above is positive, it is going uphill from left to right. Let s graph it! y We can also use the slope to find more points on the same line. Keep in mind that the slope is y = =, so that if we + increase y by 3 and increase by, we get another point!
2 rise Notice how, as we go from point to point on the graph, we can identify the. If we go from run (3, 6) to (5, 9), we get y = +3 and = +, so, again, y = though, see how we get the same thing? This time y = 3 and = , so This works because of the nature of ratios! 3. If we go from (5, 9) to (3, 6), y 3 = = Practice  Find the slope of the line between the pairs of points. 1) (, 7) and (1, 1) ) (3, ) and (3, ) 3) (, 5) and (, 3) 3. y y 33 y 5 3 m = y = m = y = m = y = One thing to notice about # is that with a slope of 0, our line is going to be horizontal. See how it s flat and has no incline? In #3, we have an undefined slope, which gives us a vertical line. In this case, our line is so infinitely steep that we don t have a number to represent its incline. We can also eamine the slope of a line based on the given equation of the line. For eample, let f() = 3 +. To find points on this line, we re not going to use the standard ,  1, 0, 1, like we did for graphing basic equations. If we did that, we d end up having fractions as point values, and that s difficult to graph. It would be easier for us to pick values that will cancel with that in the denominator. So, let s pick multiples of, like 0,, 8, and so on. f() = y = 3 + Also, eamine the slope from point to point using and y. What is the slope of this line? y y Do you see that number represented in the equation 0 y = 3 +? 8 Another thing to notice is that when = 0, the yvalue is. Do you see the number represented in the equation y = 3 +?
3 The form y = m + b (when we solve for y ) is called slopeintercept form because if a line is in this form, the coefficient of is the slope and b is the yintercept (the yvalue when = 0 this is where the line crosses the yais). Notice how when we plug in 0 for, the m portion of the equation disappears and we re left with y = b. So, m is our slope and b is our yintercept. Practice Find the slope and the yintercept point on the following lines just by eamination: 1) y = 6 1 ) y = + 3 3) y = + 10 ) y = point point point point The neat thing is that once we get one point, we can use the slope to find other points on the rise line, either by using the on the graph or by using and y on the tchart. This also run means that if we know the yintercept and the slope, we can find the equation of the line easily by simply plugging into our y = m + b format. Be aware, however, that this method only works if we are given a yintercept, or a point where the value = 0. It does NOT work if the point we re given isn t the yintercept! Practice  Find the equation of the line having the given slope and passing through the given yintercept. Notice that the given points all have = 0. 1) m = 3, (0, ) ) m =, (0, 3) 9 7 3) m =, (0, 9) ) m = 9, (0, 1) y = y = y = y = Another linear form that s important is what is commonly called standard form. It looks like this: A + By = C. The way we turn standard form into slopeintercept form is simply by solving for y. Practice Write the following in slopeintercept form ( y = m + b form). 1) 3 + y = 8 ) 5 7y = 1 3) + 3y = 7
4 Practice Graphing Lines that are in SlopeIntercept Form The lines below are given in slopeintercept form. Find the slope, yintercept (b), and then graph the line. 3 1) y = + 3 ) y = 3) y = m = m = m = b = b = b = y y y The advantage to having an equation written in slopeintercept form when graphing is it immediately gives us a point (the yintercept) and a slope, with which we can count out another point either on our y grid or by using and y on a tchart. But what if our line is given to us in standard form (A + By = C)? There are a couple of options. One option is to solve for y and rewrite our equation in slopeintercept form. The other option is to find the  and yintercepts. In other words, plug in = 0 and find y and then plug in y = 0 and find. Eample: Graph 3 y = 1. Way #1: 3 y = 1 y = 3 3 Way #: 3 y = 1 m = b = 0 y 0 0 y Practice Graphing Lines that are in Standard Form Find the  and yintercepts and graph the line. Also, use and y to find the slope of each line. 1) 3 + y = 6 ) y = y y
5 3) 5 + y = 10 ) 6 + 3y = 1 y y 5) y = 6 6) 3 + y = 3 y y In general let s solve the standard form of a line (A + By = C) for y so that we can see what it looks like in slopeintercept form. A + By = C What is the slope of the line in this format? Notice the negative sign! This means that we can look at any line in standard form and, just by eamination, figure out the slope. Practice Find the slope of the lines below. 1) 3 + y = 6 ) y = 3) 5 + y = 10 ) 6 + 3y = 1 5) y = 6 6) 3 + y = 3
6 A This method of finding the slope also works in reverse: if we know what the slope of B our line is, we can split the numerator and denominator into its A and B parts and find the front part of the equation of our line. We re looking at that net. Finding Equations of Lines Without a YIntercept There are several methods we can use to find the equation of a line. If we are given a slope and a yintercept (where = 0), then we can simply insert those values into the y = m + b format and be finished. We have used this method previously. But what do we do if the point we re given is not the yintercept? Well, we re going to use what we learned above, and we re going to avoid fractions! Eamples: Find the beginning of the equation of a line given the following slopes. 1) m = 3 ) m = 3) m = 7 1 ) m = 3 To actually find the entire equation, we need a point to go along with our slope. A slope alone is not enough because there are an infinite number of lines all having the same slope. Eample: The four lines on the grid to the right all have the same slope:. The difference is the value of the actual points the line goes through. We can see that difference most obviously in the values of the y intercepts. Line #1 (top): y = + yintercept = (0, ) Line #: y = + 1 yintercept = (0, 1) Line #3: y = yintercept = (0, 0) Line # (bottom): y = yintercept = (0, ) So, if we are given a point in addition to our slope, then we can pinpoint the equation of the line we re looking for. We use the slope to find the beginning of our equation and the point to find the end of it! The eamples below use the same slopes as the first eamples in this section. Eamples: Find the equation of the line through the given point and having the given slope. 1 1) m =, (, ) ) m =, (5, 6) 3) m =, (1, ) ) m = 3, (3, 5) 3 7
7 Parallel Lines: The advantage of this method of finding equations of lines is evident where parallel lines are concerned. Parallel lines in a plane are lines that do not cross. They also have the same slope, so the lefthand side of the standard equation will, when simplified, be the same for all parallel lines. Eamples: All five of the following lines are parallel. Find the slopes of each line to prove this. 1 3 y = 3 y = 1 6 y = y = 3 + y = 19 Eample: Find the line parallel to5 y = 10 which passes through the point (1, ). Perpendicular Lines: Perpendicular lines are lines that cross in a 90degree angle. In most cases, this means that if one line is going downhill (or has a negative slope), the other will be going uphill (or have a positive slope). The slopes also are reciprocals of each other. If one 3 slope, the slope of a perpendicular line will have a slope of. The only time this doesn t 3 apply is when one of the lines is either horizontal or vertical. In this case, the perpendicular line would be vertical or horizontal, respectively. Once we get the new slope, the rest of the process is the same: Eamples: Find the slopes of the lines perpendicular to the given lines below: 1 3 y = 3 y = 1 6 y = y = 3 + y = 19 Perpendicular slopes: Eample: Find the line perpendicular to5 y = 10 which passes through the point (1, ).
8 Given two points: Sometimes, instead of being given a slope and a point, we are given two points and are asked to find the equation of the line that passes through those points. While having two points is preferable in graphing, knowing the slope is preferable in finding the equation of the line. So, in this case, we ll have to find the slope first, and then use that slope and one of the two given points to find the rest of the equation. Eample: Find the equation of the line that passes through the points (, ) and (1, 5). 1) Find the slope: y ) Put the slope into linear form: 3) Use (, ): OR 3) Use (1, 5): (same answer, yes?) As you can see, the first two parts of this process are the same. Also, it doesn t matter which point you pick to plug in to finish finding your equation. I usually pick the numbers that are smaller and/or positive. If the slope is given to us, we can bypass part 1.
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
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 xcomponent of a point in the form (x,y). Range refers to the set of possible values of the ycomponent of a point in
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 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 informationThe PointSlope Form
7. The PointSlope 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 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 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 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 informationA synonym is a word that has the same or almost the same definition of
SlopeIntercept Form Determining the Rate of Change and yintercept Learning Goals In this lesson, you will: Graph lines using the slope and yintercept. Calculate the yintercept of a line when given
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 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 informationWarm Up. Write an equation given the slope and yintercept. Write an equation of the line shown.
Warm Up Write an equation given the slope and yintercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and yintercept From the graph, you can see that the slope is
More informationA 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
More informationSlopeIntercept Form and PointSlope Form
SlopeIntercept Form and PointSlope Form In this section we will be discussing SlopeIntercept Form and the PointSlope Form of a line. We will also discuss how to graph using the SlopeIntercept Form.
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 informationSlopeIntercept 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 yintercept. Determine
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 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 information5. Equations of Lines: slope intercept & point slope
5. Equations of Lines: slope intercept & point slope Slope of the line m rise run SlopeIntercept Form m + b m is slope; b is intercept PointSlope Form m( + or m( Slope of parallel lines m m (slopes
More informationCoordinate Plane, Slope, and Lines LongTerm 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 yvalue change? b. From point
More informationGraphing Rational Functions
Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
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 T37/H37 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 informationWriting the Equation of a Line in SlopeIntercept Form
Writing the Equation of a Line in SlopeIntercept Form SlopeIntercept Form y = mx + b Example 1: Give the equation of the line in slopeintercept form a. With yintercept (0, 2) and slope 9 b. Passing
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationGraphing  SlopeIntercept Form
2.3 Graphing  SlopeIntercept Form Objective: Give the equation of a line with a known slope and yintercept. When graphing a line we found one method we could use is to make a table of values. However,
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 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 informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationWorksheet 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!.
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 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 yintercept in the equation of a line Comparing lines on
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 informationMath 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
More informationFlorida Algebra 1 EndofCourse 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
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationPlot 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
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 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 informationMA107 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
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 information5.1. A Formula for Slope. Investigation: Points and Slope CONDENSED
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationThe fairy tale Hansel and Gretel tells the story of a brother and sister who
Piecewise Functions Developing the Graph of a Piecewise Function Learning Goals In this lesson, you will: Develop the graph of a piecewise function from a contet with or without a table of values. Represent
More informationYears after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540
To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration
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 information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationIndiana 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
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
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 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 xaxis, and the vertical axis or
More informationReview 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
More informationLines, Lines, Lines!!! SlopeIntercept Form ~ Lesson Plan
Lines, Lines, Lines!!! SlopeIntercept Form ~ Lesson Plan I. Topic: SlopeIntercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.
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 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 informationWhy should we learn this? One realworld 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 realworld connection is to find the rate
More informationSlopeIntercept Form of a Linear Equation Examples
SlopeIntercept 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 yintercept of AB. Suppose you want to find an equation
More informationMidterm 2 Review Problems (the first 7 pages) Math 1235116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 15116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationHomework #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)
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
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 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 informationAlgebra 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
More informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More informationCRLS 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
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 information1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient
Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear
More information25 Rational Functions
5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More information3. Solve the equation containing only one variable for that variable.
Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated
More information7.3 Solving Systems by Elimination
7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need
More informationUnit 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
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
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 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 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 informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line () 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationHIBBING 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,
More informationG 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
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 informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationOverview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: SlopeIntercept Form
Name Date Linear Functions: SlopeIntercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,
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 information2 Reason: This is harder. I'll give an argument in an Addendum to this handout.
LINES Slope The slope of a nonvertical line in a coordinate plane is defined as follows: Let P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) be any two points on the line. Then slope of the line = y 2 y 1 change in
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
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 informationIntro 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
More information6.2 Solving Nonlinear Equations
6.2. SOLVING NONLINEAR EQUATIONS 399 6.2 Solving Nonlinear Equations We begin by introducing a property that will be used extensively in this and future sections. The zero product property. If the product
More informationLecture 9: Lines. m = y 2 y 1 x 2 x 1
Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending
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 informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationMA.8.A.1.2 Interpret the slope and the x and yintercepts when graphing a linear equation for a realworld problem. Constant Rate of Change/Slope
MA.8.A.1.2 Interpret the slope and the x and yintercepts when graphing a linear equation for a realworld problem Constant Rate of Change/Slope In a Table Relationships that have straightlined graphs
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
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 information