coordinate system: (0, 2), (0, 0), (0, 3).
|
|
- Bridget Dean
- 6 years ago
- Views:
Transcription
1 Lesson. Objectives Find the slope of a line from the graph of the line. Find the slope of a line given two points on the line. Activity The Slope of a Line A surveyor places two stakes, A and B, on the side of a hill. Stake A is 0 feet lower than Stake B. If the horizontal distance between the stakes is 00 feet, what is the slope of the hill? The y-axis Graph these three points on the same Cartesian coordinate system: (0, ), (0, 0), (0, ). Describe the location of the three points. All three points lie on the y-axis (the vertical axis). The x-value of each point is zero. However, the y-values are different. Any ordered pair that has an x-value equal to zero must identify a point somewhere along the y-axis. Thus, the equation x 0 describes the y-axis. Activity The x-axis Graph these three points on the same Cartesian coordinate system: (, 0), (0, 0), (, 0). Describe the location of the three points. What is true about the y-value for each of these points? What is the equation that describes the x-axis? Activity Equal Coordinates Graph these three points on the same Cartesian coordinate system: (, ), (0, 0), (, ). Describe the location of the three points. Do all of these points lie on one straight line? What is true of the x-values and y-values of all these points? Write an equation that describes this line.. The Slope of a Line 07
2 In the preceeding Activities, you have used equations to describe three lines. You can also graph the lines on the same coordinate system. y-axis y = x 6 (x = 0) (y = 0) x-axis Slope Imagine that each of the three equations (x 0, y 0, and y x) represents a hill. You have to climb each hill. Which hill is easiest to walk up? The y 0 hill (or x-axis) is the easiest it has no rise at all. You can say that the line y 0 has a steepness of zero. The steepness of a line is called the slope. Thus, the line y 0 has a slope of zero. Which equation represents a hill so steep that it is impossible to climb? You cannot walk up the x 0 hill (or y-axis) at all. The slope of this hill is so great that you cannot assign a number to it. The slope of the line x 0 is undefined. Which equation represents a hill that is fairly steep, but one you could still climb? The y x hill has a slope somewhere between the x-axis (with a slope of zero) and the y-axis (with a slope that is undefined). How can you find the slope of the line y x? Rise Run The slope of a line is a measure of its steepness or tilt. The steepness of a line (or a hill) is found by comparing its vertical rise to its horizontal run. A very steep hill has a large amount of vertical rise for the given amount of horizontal run shown. 08 Chapter Linear Equations
3 A road with a gentle slope has a small amount of vertical rise for the same amount of horizontal run. Run Rise Slope The slope of a line is the ratio of the distance of the rise to the distance of the run, where the distances are measured with the same units. y Run is to the right, so the slope is positive. Run = B Rise = 6 D Run Rise A C x Lines have slopes that are positive, negative, zero, or undefined. To find the slope of a line, you need two points on the line. Imagine yourself walking from point A to point B. However, in this imaginary walk, you must first move up (or down), and then go right or left; you cannot go diagonally. Count the steps (or units) going up to find the rise. Then count the steps (or units) either left or right (the run) to reach B. If you move to the right, the number for the run is positive; if you move to the left, the number for the run is negative. Thus, slope is a rate of change. Once you know the value of the rise and run, write a fraction with the rise as the numerator and the run as the denominator. This fraction representing the ratio rise run is the slope of the line. The slope of line AB is 6 or.. The Slope of a Line 09
4 Critical Thinking Why is the slope of the line passing through C and D negative? Example Critical Thinking Finding Slope Critical Why is Thinking the slope of Why the line is the passing slope through of the line C passing through Refer and to the Dopening negative? paragraph From and CDto in negative? D, this the rise lesson is From positive, about C to the and D, the two the rise run stakes is positive, negative. the surveyor and the run is negative. placed. What is the slope of the hill? EXAMPLE Finding EXAMPLE Slope Finding Slope Solution A surveyor places two A surveyor stakes, Aplaces and B, two on the stakes, side Aof and a hill. B, on Stake the Draw Aside a of d a hill. Stake A 0 Draw is a diagram. 0 feet lower The rise than is Stake 0 feet. B. lower If The the run horizontal than is Stake 00 feet. distance B. If The the slope between horizontal is the distance 00 or 0 between th or 0.. stakes is 00 feet, what stakes is the is 00 slope feet, of what the hill? is the slope of the hill? SOLUTION SOLUTION Draw a diagram. The Draw rise is a diagram. 0 feet. The The run rise is is 00 feet. The run slope is is 00 feet. The slope 0 0 Stake B 00 or 0 or or 0 or 0.. Stake A 0 ft Activity 00 ft Stake B Discovering the Slope Formula Stake A Stake A 0 ft 0 Find the slope of AC. How many units did you rise? How many 00 ft 00 ft units did you run? ACTIVITY Discovering ACTIVITY the Discovering Slope Formula the Slope Formula Use the y-elements of the Find ordered the pairs slope for of points AC. Find How the A and slope many C. of units AC. did How you many rise? units Howdid you rise? How many Look for units a relationship did you many run? with units ;rise the 8 did units; you run run? 6 units ;rise 8 units; run 6 units rise units named in Step. Use the y-elements Use of the y-elements ordered pairs of for the points ordered A and pairs C, for points A and C, Do look the for x-elements a relationship look of the with for a the relationship rise units with named the in rise Step units. named in Step. the difference between ordered pairs for points the and difference is 8. A and between C and is 8. Does the x-elements have the same relationship Does of the ordered x-elements pairs with of for the points ordered A and pairs C for points A and C have the same relationship the run units named have in the with Step same the? relationship run units named with the in run Step units? named in Step yes; the difference between yes; the and difference is 6. between and is 6. Explain why a rise Explain why rise Explain of 8 units of units why and and a rise run run of of 86 units equals and equals run a of slope slope 6 units equals a slope of of. 8 6 reduces to. of. 8 6 reduces to. 8 6 reduces to Repeat Steps Repeat Steps Repeat for CB. for CB. ; Steps ; rise rise 6 units; 6 units; for CB. run run ; units; units; rise yes; yes; 6 the the units; difference run units; yes; the difference is 6;Yes; the difference difference is is 6;Yes; is ; yes; 6 the difference is ; 6 reduces to. reduces to. 66 Repeat Steps 6 Repeat for AB. AB. Steps ;rise 9; units; for run AB. 9 ;rise units; 9 yes; units; the difference run 9 units; is ; yes; the difference yes; the difference is ; yes; is 9; the because difference difference the is ; is rise 9; and yes; since run the the are difference rise both and negative is run 9; are since both the slope is the rise and run are both 7 Generalize negative the slope how is to positive. negative find the the slope is of positive. a line using the ordered pairs. 7 Generalize the 7how Generalize to find the the slope how of to a find line the using slope theof a line using the ordered pairs. Subtract ordered the y elements pairs. Subtract for the numerator the y elements and subtract for the numerator the and subtract the x elements for the denominator. 0 Chapter Linear Equations x elements for the denominator. Stake
5 The Slope Formula The coordinates of any two points on a line determine its slope. The difference between the y-coordinates is the rise. The difference between the x-coordinates is the run. This gives a formula for finding slope. Because the slope is the ratio rise run, the slope can also be written in the following way slope difference of y-coordinates difference of x-coordinates. To find the slope of a line between two points, use the slope formula. Slope Formula If A(x, y ) and B(x, y ) are two points on line AB, then the slope of AB. x x y y When you use the slope formula to find the slope of a line between two points, be sure to subtract the coordinates in the same order. Example The Slope Between Two Points Find the slope of the line that contains A(, ) and B(, ). Solution Method Make a sketch. Start at the lower point, A. Move up to find a rise of 7 units. To reach point B, move to the right units. This is a run is i of. Thus, the slope is 7. y 6 run is B (, ) rise is (, ) A 6 x Method Use the slope formula. y y Slope of AB x ( ) 7 x ( ) The slope is the ratio 7.. The Slope of a Line
6 Chapter Think and Discuss Think see and margin Discuss see margin Describe the points Describe that are the located points that on the are y-axis located and on on the y-axis and on the x-axis. the x-axis. LESSON ASSESSMENT Lesson Assessment Think Think Explain and Discuss and what Discuss the slope of a line means. see Explain see margin margin what the slope of a line means. Explain. how Describe the points that are located on the y-axis and on the x-axis. to find Explain the the points slope how that to of find a are are line the located if slope you know on on of the a line its y-axis rise if you and know on on its rise and its run. the x-axis.. Explain what and the its slope run. of a line means. Explain. Explain how Explain how to use what Explain to find the how the slope slope to formula of of use a of line the a to line slope means. find if you the formula know slope to of its find rise the slope of a line. and its run. a line. Explain how to to find the slope of of a line if if if if you know its its rise Explain. Explain why and its how a its slope run. Explain to use can why the be positive slope a slope formula can negative, be to positive find and the or describe slope negative, of a line. and describe the line that models each slope.. Explain why the a slope line to that can models be positive each slope. Explain how to use the slope formula negative, to to find the and slope describe of of a Practice and Problem line that Solving models each slope. Practice a line. and Problem Solving Practice Find the Explain slope and Problem why of Find a line a the slope for slope Solving can the of be given be a positive line rise for and or the or negative, run. given rise and and describe run. 6. rise, the run line that rise, run 8. rise 8, run 0 Find the slope of a line for the given rise and run. 6. rise, run rise rise models, run each slope. 7. rise, run undefined 9. rise 0, run 0. rise 0, run 7.. rise,, run run rise 8. rise 8, run 0 undefined 9. rise 0, run 0. rise 0, run. 0 Practice and Problem Solving rise, run 0 rise 0 0 rise Find 8. rise the Find 8, slope run the 0 of Find undefined slope a each the of of slope line. a line of for for a each the 9. given line. rise 0, rise and run run.. 6., ,. ris 0. rise , rise run,, run rise 0., rise run, run rise 0 8, 8, run rise, run undefined rise 0, 0,. run 0. rise. 0, 0, run rise,. run rise, 0 run 0 00 Find the Find slope the of slope each of line. of a each line A fencing contractor. A fencing uses contractor a scale drawing uses a on scale a drawing on a coordinate plane coordinate to calculate plane a bid to on calculate a job. Part a bid of on thisa job. Part of this calculation includes calculation finding includes the slopes finding of the the lines slopes in a of the lines in a drawing. Find the slopes of the line segments with the drawing. Find the slopes of the line segments given with the given endpoints below... A fencing endpoints contractor below. uses a scale drawing on on a a. A(, ), B(, ) b. M(, ), N(, ) a. A(, ), B(, ) M(, 6. A fencing contractor uses a scale drawing on a coordinate ), N(, plane ) coordinate plane to to calculate a bid bid on on a job. Part of of this c. C(, to calculate ), calculation D(, ) a c. 0 bid on a job. Part C(, includes ), D(, finding ) d. of 0 the X(0, this slopes 0), calculation Y(, of of ) the d. includes finding X(0, lines 0), Y(, a ) the drawing. slopes of Find the the lines slopes in a drawing. of of the line Find segments the slopes with of the line given e. S( 6, ), T( 6, ) undefined f. R( 8, ), S(, ) e. S( 6, ), T( 6, ) undefined f. R( 8, segments endpoints with below. the given endpoints below. ), S(, ) g. L(, a. a. a. a. ), A(, A(, M(, ), ) ), ) g. ), L(, B(, ) 7 ), M(, ) h. P(7, b. 0), b. M(, Q( 7, ), 0) N(, 0 ) b. M(, 7 h. ), P(7, N(, 0), ) Q( 7, 0) 0 i. G(, c. c. c. c. ), c. C(, H(, ), ), ), ) i. G(, D(, ) ) ) ) 0 ), 00 H(, ) j. E(, d. ), F(, 7) d. d. X(0, X(0, 0), 0), j. 0), Y(, E(, Y(, ) ) ) ), F(, 7) e. e. S(6, ), T(6, ) f. f. f. R(8, ), ) e. e. S( 6, ), T( 6, ) ) undefined f. f. R( 8, ), S(, ). The Slope of a. Line The Slope of a Line g. ), g. g. L(, L(, ), M(, ) ) 7 h. 7 h. P(7, 0), Q(7, 0) 7 h. P(7, 0), Q( 7, 0) Chapter Linear Equations i. i. ), j. i. i. i. G(, G(, ), H(, ) ) j. 7) j. 7) j. E(, ), j. E(, ), F(, 7).. The The Slope Slope of of of of a a Line Line Cha
7 7. It is 0 miles from Johnson City to Putnam. The elevation of Johnson City is,000 feet. The elevation of Putnam is,00 feet. What is the average rate of increase in elevation per mile from Johnson City to Putnam? 8. What is the positive slope of the roof at the right? 8 ft 6 ft 9. In a landing approach, an airplane maintains a constant rate of descent of 0 feet for every 00 feet traveled horizontally. What is the positive slope of the line that represents the landing approach of the plane? 0 Without graphing, determine if each set of points lie on the same line. 0. (, ), (, ), (, ). (7, 7), (, ), (, ). (8, ), (6, ), (, 6), (, 7). (0, 0), (, ), (, ), (0, ) Mixed Review For each situation, write and solve an equation.. The amount of water flowing over a dam at noon is. million gallons per hour more than its rate at mid-morning. When the water flow was tested at noon, it had reached 8 million gallons per hour. What was the rate of the water flow at mid-morning?. Keshia sells her inventory for twice what she pays. After expenses of $0 are deducted, Keshia finds she has $680 left. What did Keshia pay for her initial inventory? 6. Ramon is carpeting a rectangular room with a perimeter of 0 feet. One side of the room is feet longer than the other. Find the length of the longer side. Solve each equation. Check your answer. 7. d () 8. x% of r. The Slope of a Line
LINEAR EQUATIONS IN TWO VARIABLES
LINEAR EQUATIONS IN TWO VARIABLES What You Should Learn Use slope to graph linear equations in two " variables. Find the slope of a line given two points on the line. Write linear equations in two variables.
More informationSolving Equations and Graphing
Solving Equations and Graphing Question 1: How do you solve a linear equation? Answer 1: 1. Remove any parentheses or other grouping symbols (if necessary). 2. If the equation contains a fraction, multiply
More informationChapter 9 Linear equations/graphing. 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane
Chapter 9 Linear equations/graphing 1) Be able to graph points on coordinate plane 2) Determine the quadrant for a point on coordinate plane Rectangular Coordinate System Quadrant II (-,+) y-axis Quadrant
More informationIn this section, we find equations for straight lines lying in a coordinate plane.
2.4 Lines Lines In this section, we find equations for straight lines lying in a coordinate plane. The equations will depend on how the line is inclined. So, we begin by discussing the concept of slope.
More informationLesson 15: The Slope of a Non Vertical Line
Classwork Opening Exercise Example Graph A Graph B a. Which graph is steeper? b. Write directions that explain how to move from one point on the graph to the other for each of Graph A and Graph B. c. Write
More informationSection 1.3. Slope formula: If the coordinates of two points on the line are known then we can use the slope formula to find the slope of the line.
MATH 11009: Linear Functions Section 1.3 Linear Function: A linear function is a function that can be written in the form f(x) = ax + b or y = ax + b where a and b are constants. The graph of a linear
More informationGraphs, Linear Equations and Functions
Graphs, Linear Equations and Functions There are several ways to graph a linear equation: Make a table of values Use slope and y-intercept Use x and y intercepts Oct 5 9:37 PM Oct 5 9:38 PM Example: Make
More information4.4 Slope and Graphs of Linear Equations. Copyright Cengage Learning. All rights reserved.
4.4 Slope and Graphs of Linear Equations Copyright Cengage Learning. All rights reserved. 1 What You Will Learn Determine the slope of a line through two points Write linear equations in slope-intercept
More information3-5 Slopes of Lines. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Geometry
3-5 Slopes of Lines Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Find the value of m. 1. 2. 3. 4. undefined 0 Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular
More information4 The Cartesian Coordinate System- Pictures of Equations
The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the Pythagorean
More informationE. Slope-Intercept Form and Direct Variation (pp )
and Direct Variation (pp. 32 35) For any two points, there is one and only one line that contains both points. This fact can help you graph a linear equation. Many times, it will be convenient to use the
More informationStudy Guide: Slope and Linear Equations
Rates and Unit Rates A rate is a proportional relationship between two quantities. Unit rate is a rate where the second quantity is 1. Example: Pauline can mow 35 square feet of lawn is 2.5 minutes. (this
More informationPlotting Points in 2-dimensions. Graphing 2 variable equations. Stuff About Lines
Plotting Points in 2-dimensions Graphing 2 variable equations Stuff About Lines Plotting Points in 2-dimensions Plotting Points: 2-dimension Setup of the Cartesian Coordinate System: Draw 2 number lines:
More informationSlope-Intercept Form. Find the x- and y-intercepts. 1. y 3x 6 2. y 2x 8. Graph each equation. 3. y 1 x 3 4. y 5x 5 5. y x 4
Practice A Slope-Intercept Form Find the x- and y-intercepts. 1. y 3x 6. y x 8 _ Graph each equation. 3. y 1 x 3 4. y 5x 5 5. y x 4 Write the equation of the line in slope-intercept form. 6. 7. _ Practice
More informationMA Lesson 16 Sections 2.3 and 2.4
MA 1500 Lesson 16 Sections.3 and.4 I Piecewise Functions & Evaluating such Functions A cab driver charges $4 a ride for a ride less than 5 miles. He charges $4 plus $0.50 a mile for a ride greater than
More informationName: Date: Block: Mid-Unit 4 Test Review All work must be shown for full credit.
Name: Date: Block: Mid-Unit 4 Test Review All work must be shown for full credit. 1) How do you have to walk so the motion detector graphs a straight line? Explain as clearly as you can. 2) What determines
More informationSlope. Domain 2 Lesson 11. Getting the Idea
Domain Lesson Slope Common Core Standard: 8.EE. Getting the Idea The graph of a linear equation is a straight line. The steepness of the line is called its slope. The slope shows the rate at which two
More informationGraphs of linear equations will be perfectly straight lines. Why would we say that A and B are not both zero?
College algebra Linear Functions : Definition, Horizontal and Vertical Lines, Slope, Rate of Change, Slopeintercept Form, Point-slope Form, Parallel and Perpendicular Lines, Linear Regression (sections.3
More informationPage 1 of 1-7 Equations Teks Focus TEKS (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity
More informationMath 154 :: Elementary Algebra
Math :: Elementary Algebra Section. Section. Section. Section. Section. Math :: Elementary Algebra Section. The Rectangular (Cartesian) Coordinate System. The variable x usually represents the independent
More informationMath 10 Lesson 4-1 Slope of a Line
I. Lesson Objectives: Math 10 Lesson 4-1 Slope of a Line 1) Determine the slope of a line segment and a line. II. Rate of change slope In Lesson 3-6 we learned about the rate of change for a linear function.
More informationSlope The slope m of a line is a ratio of the change in y (the rise) to the change in x (the run) between any two points, ), on the line.
. Lesson Lesson Tutorials Ke Vocabular slope, p. 0 rise, p. 0 run, p. 0 Reading In the slope formula, is read as sub one, and is read as sub two. The numbers and in and are called subscripts. Slope The
More informationACTIVITY: Finding the Slope of a Line
. Slope of a Line describe the line? How can ou use the slope of a line to Slope is the rate of change between an two points on a line. It is the measure of the steepness of the line. To find the slope
More informationPearson's Ramp-Up Mathematics
Introducing Slope L E S S O N CONCEPT BOOK See pages 7 8 in the Concept Book. PURPOSE To introduce slope as a graphical form of the constant of proportionality, k. The lesson identifies k as the ratio
More informationAnalytic Geometry ةيليلحتلا ةسدنھلا
Analytic Geometry الھندسة التحليلية نظام اإلحداثيات الديكارتي 1-1 Cartesian Coordinate System The Cartesian coordinate system, or the rectangular coordinate system, is a geometrical system that is used
More informationAnalytic Geometry. The x and y axes divide the Cartesian plane into four regions called quadrants.
Analytic Geometry الھندسة التحليلية نظام اإلحداثيات الديكارتي 1-1 Cartesian Coordinate System The Cartesian coordinate system, or the rectangular coordinate system, is a geometrical system that is used
More informationStudy Guide: Slope and Linear Equations
Rates and Unit Rates A rate is a proportional relationship between two quantities. Unit rate is a rate where the second quantity is 1. Example: Pauline can mow 35 square feet of lawn is 2.5 minutes. (this
More informationSection 3.5. Equations of Lines
Section 3.5 Equations of Lines Learning objectives Use slope-intercept form to write an equation of a line Use slope-intercept form to graph a linear equation Use the point-slope form to find an equation
More informationMATH 021 TEST 2 REVIEW SHEET
TO THE STUDENT: MATH 021 TEST 2 REVIEW SHEET This Review Sheet gives an outline of the topics covered on Test 2 as well as practice problems. Answers for all problems begin on page 8. In several instances,
More informationDetermine the intercepts of the line and ellipse below: Definition: An intercept is a point of a graph on an axis. Line: x intercept(s)
Topic 1 1 Intercepts and Lines Definition: An intercept is a point of a graph on an axis. For an equation Involving ordered pairs (x, y): x intercepts (a, 0) y intercepts (0, b) where a and b are real
More informationSlope. Plug In. Finding the Slope of a Line. m 5 1_ 2. The y-intercept is where a line
LESSON Slope Plug In Finding the Slope of a Line The slope of a line is the ratio of the change in the -values to the change in the corresponding -values. 0 7 8 change in -values Slope change in -values
More informationCumulative Review : MAT-032 (Algebra B) 2013
Perform the indicated operations and simplify: ( 7. 8. 9. Add 10. Subtract from 1 Subtract from the sum of and 1 Subtract the sum of and from 7. 8. 9. 10. 1 1 Factor completely: 7. 8. 7. 8. Factor completely:
More informationSection 1.3. Slope of a Line
Slope of a Line Introduction Comparing the Steepness of Two Objects Two ladders leaning against a building. Which is steeper? We compare the vertical distance from the base of the building to the ladder
More informationChapter 7, Part 1B Equations & Functions
Chapter 7, Part 1B Equations & Functions Fingerstache Fingerstaches cost $7 per box. Copy and complete the table to find the cost of 2, 3, and 4 boxes. Number of Boxes Multiply by 7 Cost 1 1 x 7 $7 2 3
More informationACT Coordinate Geometry Review
ACT Coordinate Geometry Review Here is a brief review of the coordinate geometry concepts tested on the ACT. Note: there is no review of how to graph an equation on this worksheet. Questions testing this
More informationReview Journal 6 Assigned Work: Page 146, All questions
MFM2P Linear Relations Checklist 1 Goals for this unit: I can explain the properties of slope and calculate its value as a rate of change. I can determine y-intercepts and slopes of given relations. I
More informationUnit 5: Graphs. Input. Output. Cartesian Coordinate System. Ordered Pair
Section 5.1: The Cartesian plane Section 5.2: Working with Scale in the Cartesian Plane Section 5.3: Characteristics of Graphs Section 5.4: Interpreting Graphs Section 5.5: Constructing good graphs from
More informationAnswers for the lesson Plot Points in a Coordinate Plane
LESSON 3.1 Answers for the lesson Plot Points in a Coordinate Plane Skill Practice 1. 5; 23 2. No; the point could lie in either Quadrant II or Quadrant IV. 3. (3, 22) 4. (, 21) 5. (4, 4) 6. (24, 3) 7.
More informationDetermine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 A) Even B) Odd C) Neither
Assignment 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the function is even, odd, or neither. 1) f(x) = 8x4 + 7x + 5 1) A)
More informationFair Game Review. Chapter 4. Name Date. Find the area of the square or rectangle Find the area of the patio.
Name Date Chapter Fair Game Review Find the area of the square or rectangle... ft cm 0 ft cm.. in. d in. d. Find the area of the patio. ft 0 ft Copright Big Ideas Learning, LLC Big Ideas Math Green Name
More informationCH 54 SPECIAL LINES. Ch 54 Special Lines. Introduction
479 CH 54 SPECIAL LINES Introduction Y ou may have noticed that all the lines we ve seen so far in this course have had slopes that were either positive or negative. You may also have observed that every
More informationChapter 9. Conic Sections and Analytic Geometry. 9.1 The Ellipse. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Graph ellipses centered at the origin. Write equations of ellipses in standard
More informationChapter 2: Functions and Graphs Lesson Index & Summary
Section 1: Relations and Graphs Cartesian coordinates Screen 2 Coordinate plane Screen 2 Domain of relation Screen 3 Graph of a relation Screen 3 Linear equation Screen 6 Ordered pairs Screen 1 Origin
More informationLesson 16: The Computation of the Slope of a Non Vertical Line
++ Lesson 16: The Computation of the Slope of a Non Vertical Line Student Outcomes Students use similar triangles to explain why the slope is the same between any two distinct points on a non vertical
More informationUnit 5: Moving Straight Ahead
Unit 5: Moving Straight Ahead Investigation 4 Exploring Slope: Connecting Rates and Ratios I can demonstrate understanding that linear relationships are relationships represented by the slope of the line
More informationChapter 3 Graphing Linear Equations
Chapter 3 Graphing Linear Equations Rectangular Coordinate System Cartesian Coordinate System Origin Quadrants y-axis x-axis Scale Coordinates Ex: Plot each point: (0,0), (-1, 3), (1, 3), (1, -3), (-1,
More informationMTH 103 Group Activity Problems (W2B) Name: Equations of Lines Section 2.1 part 1 (Due April 13) platform. height 5 ft
MTH 103 Group Activity Problems (W2B) Name: Equations of Lines Section 2.1 part 1 (Due April 13) Learning Objectives Write the point-slope and slope-intercept forms of linear equations Write equations
More informationAppendix M TERMINOLOGY. Slope of a Line. Slope. Undefined Slope. Slope-Intercept Form
Appendices : Slope of a Line TERMINOLOGY For each of the following terms, provide ) a definition in our own words, 2) the formal definition (as provided b our text or instructor), and ) an example of the
More informationSect Linear Equations in Two Variables
99 Concept # Sect. - Linear Equations in Two Variables Solutions to Linear Equations in Two Variables In this chapter, we will examine linear equations involving two variables. Such equations have an infinite
More information10 GRAPHING LINEAR EQUATIONS
0 GRAPHING LINEAR EQUATIONS We now expand our discussion of the single-variable equation to the linear equation in two variables, x and y. Some examples of linear equations are x+ y = 0, y = 3 x, x= 4,
More informationLesson 6.1 Linear Equation Review
Name: Lesson 6.1 Linear Equation Review Vocabulary Equation: a math sentence that contains Linear: makes a straight line (no Variables: quantities represented by (often x and y) Function: equations can
More informationGrade 8, Unit 3 Practice Problems - Open Up Resources
Grade 8, - Open Up Resources Lesson 1 Priya jogs at a constant speed. The relationship between her distance and time is shown on the graph. Diego bikes at a constant speed twice as fast as Priya. Sketch
More informationAlgebra 1 2 nd Six Weeks
Algebra 1 2 nd Six Weeks Second Six Weeks October 6 November 14, 2014 Monday Tuesday Wednesday Thursday Friday October 6 B Day 7 A Day 8 B Day 9 A Day 10 B Day Elaboration Day Test 1 - Cluster 2 Test Direct
More informationInvestigating the equation of a straight line
Task one What is the general form of a straight line equation? Open the Desmos app on your ipad If you do not have the app, then you can access Desmos by going to www.desmos.com and then click on the red
More informationconstant EXAMPLE #4:
Linear Equations in One Variable (1.1) Adding in an equation (Objective #1) An equation is a statement involving an equal sign or an expression that is equal to another expression. Add a constant value
More informationUnit 10: The Equation of a Linear Function
Section 10.1: The Equation of a Linear Function Section 10.2: Writing Linear Equations in Slope-Intercept Form Section 10.3: Parallel and Perpendicular Lines Section 10.4: Applications Slope-Intercept
More informationGraphing - Slope-Intercept Form
2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,
More informationUnit 3 Algebra What is the y-intercept for the graph of the equation 3x 5y = 15?
Unit 3 lgebra 1 Name: ate: 1. The equation below is used to find (x, y) coordinates. y = 3x + 2 3. ennie is using this pattern to make stars for an laska state flag. Which coordinates could be found using
More informationStudent Exploration: Standard Form of a Line
Name: Date: Student Exploration: Standard Form of a Line Vocabulary: slope, slope-intercept form, standard form, x-intercept, y-intercept Prior Knowledge Questions (Do these BEFORE using the Gizmo.) 1.
More informationAdding & Subtracting Decimals. Multiplying Decimals. Dividing Decimals
1. Write the problem vertically, lining up the decimal points. 2. Add additional zeroes at the end, if necessary, to make the numbers have the same number of decimal places. 3. Add/subtract as if the numbers
More information3.3. You wouldn t think that grasshoppers could be dangerous. But they can damage
Grasshoppers Everywhere! Area and Perimeter of Parallelograms on the Coordinate Plane. LEARNING GOALS In this lesson, you will: Determine the perimeter of parallelograms on a coordinate plane. Determine
More informationThe Picture Tells the Linear Story
The Picture Tells the Linear Story Students investigate the relationship between constants and coefficients in a linear equation and the resulting slopes and y-intercepts on the graphs. This activity also
More informationG.2 Slope of a Line and Its Interpretation
G.2 Slope of a Line and Its Interpretation Slope Slope (steepness) is a very important concept that appears in many branches of mathematics as well as statistics, physics, business, and other areas. In
More informationReview for Mastery. Identifying Linear Functions
Identifying Linear Functions You can determine if a function is linear by its graph, ordered pairs, or equation. Identify whether the graph represents a linear function. Step 1: Determine whether the graph
More informationAnalytical geometry. Multiple choice questions
Analytical geometry Multiple choice questions 1. Temperature readings on any given day in Québec can vary greatly. The temperatures for a fall day in Montreal were recorded over a 10-hour interval. The
More informationUsing Slopes and Intercepts
CODE Name Date Teacher Practice A Using Slopes and Intercepts 1. Name the ordered pair if the x-intercept is 2. 2. Name the ordered pair if the y-intercept is 8. 3. In the ordered pair (9, 0), what is
More informationLine Graphs. Name: The independent variable is plotted on the x-axis. This axis will be labeled Time (days), and
Name: Graphing Review Graphs and charts are great because they communicate information visually. For this reason graphs are often used in newspapers, magazines, and businesses around the world. Sometimes,
More informationHyperbolas Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas p. 583
C H A P T ER Hyperbolas Flashlights concentrate beams of light by bouncing the rays from a light source off a reflector. The cross-section of a reflector can be described as hyperbola with the light source
More informationLesson 1: Understanding Proportional. Relationships
Unit 3, Lesson 1: Understanding Proportional Relationships 1. Priya jogs at a constant speed. The relationship between her distance and time is shown on the graph. Diego bikes at a constant speed twice
More informationSince each element is paired with unique element in the range, it is a function.
1. State the domain and range of the relation {( 3, 2), (4, 1), (0, 3), (5, 2), (2, 7)}. Then determine whether the relation is a function. The domain is the set of x-coordinates. The range is the set
More informationCharacteristics of Linear Relations
HW Mark: 10 9 8 7 6 RE-Submit Characteristics of Linear Relations This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg.
More information6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line.
6.1 Slope of a Line Name: Date: Goal: Determine the slope of a line segment and a line. Toolkit: - Rate of change - Simplifying fractions Main Ideas: Definitions Rise: the vertical distance between two
More information4-7 Point-Slope Form. Warm Up Lesson Presentation Lesson Quiz
Warm Up Lesson Presentation Lesson Quiz Holt Algebra McDougal 1 Algebra 1 Warm Up Find the slope of the line containing each pair of points. 1. (0, 2) and (3, 4) 2. ( 2, 8) and (4, 2) 1 3. (3, 3) and (12,
More informationMotion Graphs. Plotting distance against time can tell you a lot about motion. Let's look at the axes:
Motion Graphs 1 Name Motion Graphs Describing the motion of an object is occasionally hard to do with words. Sometimes graphs help make motion easier to picture, and therefore understand. Remember: Motion
More informationFolding Activity 1. Colored paper Tape or glue stick
Folding Activity 1 We ll do this first activity as a class, and I will model the steps with the document camera. Part 1 You ll need: Patty paper Ruler Sharpie Colored paper Tape or glue stick As you do
More informationGeometry. Practice Pack
Geometry Practice Pack WALCH PUBLISHING Table of Contents Unit 1: Lines and Angles Practice 1.1 What Is Geometry?........................ 1 Practice 1.2 What Is Geometry?........................ 2 Practice
More information8.EE. Development from y = mx to y = mx + b DRAFT EduTron Corporation. Draft for NYSED NTI Use Only
8.EE EduTron Corporation Draft for NYSED NTI Use Only TEACHER S GUIDE 8.EE.6 DERIVING EQUATIONS FOR LINES WITH NON-ZERO Y-INTERCEPTS Development from y = mx to y = mx + b DRAFT 2012.11.29 Teacher s Guide:
More informationMS Algebra A-S-ID-7 Ch. 5.5a Find Slope Given Two Points. Mr. Deyo Find Slope and Rate of Change
MS Algebra A-S-ID-7 Ch. 5.5a Find Slope Given Two Points Mr. Deyo Find Slope and Rate of Change Title: 5.5a Find Slope Given Two Points Date: Learning Target By the end of the period, I will find the slope
More informationAlgebra & Trig. 1. , then the slope of the line is given by
Algebra & Trig. 1 1.4 and 1.5 Linear Functions and Slope Slope is a measure of the steepness of a line and is denoted by the letter m. If a nonvertical line passes through two distinct points x, y 1 1
More information5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs
Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2
More informationBook 10: Slope & Elevation
Math 21 Home Book 10: Slope & Elevation Name: Start Date: Completion Date: Year Overview: Earning and Spending Money Home Travel and Transportation Recreation and Wellness 1. Budget 2. Personal Banking
More informationAlgebra 1 Online:
Dear Algebra 2 Students, Within this packet you will find mathematical concepts and skills learned in Algebra 1 that are the foundation from which Algebra 2 is built. These concepts need to be reviewed
More informationUse the Point-Slope Form to Write the Equation of a Line
Math 90 8.3 "Writing Equations of Lines" Objectives: * Use the point-slope form to write the equation of a line. * Use the slope-intercept form to write the equation of a line. * Use slope as an aid when
More informationAlgebra Success. LESSON 16: Graphing Lines in Standard Form. [OBJECTIVE] The student will graph lines described by equations in standard form.
T328 [OBJECTIVE] The student will graph lines described by equations in standard form. [MATERIALS] Student pages S125 S133 Transparencies T336, T338, T340, T342, T344 Wall-size four-quadrant grid [ESSENTIAL
More informationMath 1023 College Algebra Worksheet 1 Name: Prof. Paul Bailey September 22, 2004
Math 1023 College Algebra Worksheet 1 Name: Prof. Paul Bailey September 22, 2004 Every vertical line can be expressed by a unique equation of the form x = c, where c is a constant. Such lines have undefined
More informationGraphs of sin x and cos x
Graphs of sin x and cos x One cycle of the graph of sin x, for values of x between 0 and 60, is given below. 1 0 90 180 270 60 1 It is this same shape that one gets between 60 and below). 720 and between
More informationGRADE LEVEL: SEVENTH SUBJECT: MATH DATE: CONTENT STANDARD INDICATORS SKILLS ASSESSMENT VOCABULARY ISTEP
GRADE LEVEL: SEVENTH SUBJECT: MATH DATE: 2015 2016 GRADING PERIOD: QUARTER 2 MASTER COPY 10 8 15 CONTENT STANDARD INDICATORS SKILLS ASSESSMENT VOCABULARY ISTEP COMPUTATION Unit Rates Ratios Length Area
More informationLesson 1 Area of Parallelograms
NAME DATE PERIOD Lesson 1 Area of Parallelograms Words Formula The area A of a parallelogram is the product of any b and its h. Model Step 1: Write the Step 2: Replace letters with information from picture
More informationGeometry 2001 part 1
Geometry 2001 part 1 1. Point is the center of a circle with a radius of 20 inches. square is drawn with two vertices on the circle and a side containing. What is the area of the square in square inches?
More informationChapter 3 Parallel and Perpendicular Lines
Chapter 3 Parallel and Perpendicular Lines Parallel Lines Lines Parallel Symbol: Perpendicular Lines Lines that Perpendicular Symbol: Postulate 13: Parallel Postulate For any and a not on the line, there
More informationProblem Solving with the Coordinate Plane
Grade 5 Module 6 Problem Solving with the Coordinate Plane OVERVIEW In this 40-day module, students develop a coordinate system for the first quadrant of the coordinate plane and use it to solve problems.
More informationA slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
More informationUnit 8: Coordinate Plane (including x/y tables), Proportional Reasoning, and Slope
Page 1 CCM6+7+ --Unit 9 Graphing and Slope Unit 8: Coordinate Plane (including x/y tables), Proportional Reasoning, and Slope 2015-16 Name Teacher Projected Test Date Main Topic(s) Page(s) Vocabulary 2-3
More informationThen finding the slope, we can just use the same method that we have done the other ones we get the slope 4 1
169 Graphing Equations with Slope Okay, now that you know how to graph a line by getting some points, and you know how to find the slope between two points, you should be able to find the slope of a line
More information3.2 Exercises. rise y (ft) run x (ft) Section 3.2 Slope Suppose you are riding a bicycle up a hill as shown below.
Section 3.2 Slope 261 3.2 Eercises 1. Suppose ou are riding a biccle up a hill as shown below. Figure 1. Riding a biccle up a hill. a) If the hill is straight as shown, consider the slant, or steepness,
More informationObjective: Investigate patterns in vertical and horizontal lines, and. interpret points on the plane as distances from the axes.
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 5 6 Lesson 6 Objective: Investigate patterns in vertical and horizontal lines, and Suggested Lesson Structure Fluency Practice Application Problem Concept
More informationMath Labs. Activity 1: Rectangles and Rectangular Prisms Using Coordinates. Procedure
Math Labs Activity 1: Rectangles and Rectangular Prisms Using Coordinates Problem Statement Use the Cartesian coordinate system to draw rectangle ABCD. Use an x-y-z coordinate system to draw a rectangular
More informationSurveying & Measurement. Detail Survey Topographic Surveying
Surveying & Measurement Detail Survey Topographic Surveying Introduction Mapping surveys are made to determine the relief of the earth s surface and locate critical points on it. to determine the locations
More information3.4 The Slope of a Line
CHAPTER Graphs and Functions. The Slope of a Line S Find the Slope of a Line Given Two Points on the Line. Find the Slope of a Line Given the Equation of a Line. Interpret the Slope Intercept Form in an
More information1. Write an equation in slope-point for this line.
1. Write an equation in slope-point for this line. 2. Which of the following equations describes the linear relation graphed below? I II! " 2 3 % & 2! ' 4 " 2 )% ' 3* 3 III 3% ' 2! & 2 " 0 A. I, II, and
More information