4.4 Slope and Graphs of Linear Equations. Copyright Cengage Learning. All rights reserved.

Size: px
Start display at page:

Download "4.4 Slope and Graphs of Linear Equations. Copyright Cengage Learning. All rights reserved."

Transcription

1 4.4 Slope and Graphs of Linear Equations Copyright Cengage Learning. All rights reserved. 1

2 What You Will Learn Determine the slope of a line through two points Write linear equations in slope-intercept form and graph the equations Use slopes to determine whether lines are parallel, perpendicular, or neither 2

3 The Slope of a Line 3

4 The Slope of a Line The slope of a nonvertical line is the number of units the line rises or falls vertically for each unit of horizontal change from left to right. For example, the line in Figure 4.21 rises two units for each unit of horizontal change from left to right, and so this line has a slope of m = 2. 4

5 The Slope of a Line The slope m of a nonvertical line that passes through the points (x 1, y 1 ) and (x 2, y 2 ) is where x 1 x 2. 5

6 The Slope of a Line When the formula for slope is used, the order of subtraction is important. Given two points on a line, you are free to label either of them (x 1, y 1 ) and the other (x 2, y 2 ). However, once this has been done, you must form the numerator and denominator using the same order of subtraction. 6

7 Example 1 Sketching the Graph of an Equation Find the slope of the line passing through each pair of points. a. ( 2, 0) and (3, 1) b. (0, 0) and (1, 1) Solution: a. Let (x 1, y 1 ) = ( 2, 0) and (x 2, y 2 ) = (3, 1). The slope of the line through these points is 7

8 Example 1 Sketching the Graph of an Equation cont d Difference in y-values Difference in x-values Simplify. 8

9 Example 1 Sketching the Graph of an Equation b. The slope of the line through (0, 0) and (1, 1) is Difference in y-values Difference in x-values cont d Simplify. Simplify. Here is the line of the graph. 9

10 The Slope of a Line 10

11 The Slope of a Line From the slopes of the lines below, you can make several generalizations about the slope of a line. 11

12 The Slope of a Line 12

13 Example 2 Finding the Slope of a Ladder Find the slope of the ladder leading up to the tree house. 13

14 Example 2 Finding the Slope of a Ladder Solution: Consider the tree trunk as the y-axis and the level ground as the x-axis. The endpoints of the ladder are (0, 12) and (5, 0). So, the slope of the ladder is 14

15 Slope as a Graphing Aid 15

16 Slopes as a Graphing Aid You have seen that before creating a table of values for an equation, it is helpful first to solve the equation for y. When you do this for a linear equation, you obtain some very useful information. Consider: 3x 2y = 4 3x 3x 2y = 3x + 4 2y = 3x + 4 Write original equation. Subtract 3x from each side. Simplify. Divide each side by 2. Simplify. 16

17 Slopes as a Graphing Aid Observe that the coefficient of x is the slope of the graph of this equation. Moreover, the constant term, 2, gives the y-intercept of the graph. This form is called the slope-intercept form of the equation of the line. 17

18 Slopes as a Graphing Aid 18

19 Slopes as a Graphing Aid So far, you have been plotting several points to sketch the equation of a line. However, now that you can recognize equations of lines, you don t have to plot as many points two points are enough. (You might remember from geometry that two points are all that are necessary to determine a line.) The next example shows how to use the slope to help sketch a line. 19

20 Example 3 Using Slope-Intercept Form Use the slope and y-intercept to sketch the graph of x 3y = 6. Solution: First, write the equation in slope-intercept form. x 3y = 6 3y = x 6 Write original equation. Subtract x from each side. Divide each side by 3. Simplify to slope-intercept form. 20

21 Example 3 Using Slope-Intercept Form cont d So, the slope of the line is (0, b) = (0, 2). and the y-intercept is Now you can sketch the graph of the equation. First, plot the y-intercept. 21

22 Example 3 Using Slope-Intercept Form Then, using a slope of cont d locate a second point on the line by moving three units to the right and one unit up (or one unit up and three units to the right). 22

23 Example 3 Using Slope-Intercept Form Finally, obtain the graph by drawing a line through the two points. cont d 23

24 Parallel and Perpendicular Lines 24

25 Parallel and Perpendicular Lines You know from geometry that two lines in a plane are parallel if they do not intersect, and two lines in a plane are perpendicular if they intersect at right angles. 25

26 Example 4 Parallel and Perpendicular Lines (a) a. The lines y = 3x 4 each have a slope of 3. So, the lines are parallel. 26

27 Example 4 Parallel and Perpendicular Lines (b) b. The lines y = 5x + 2 and y = x 4 have slopes that are negative reciprocals of each other. So, the lines are perpendicular. 27

28 Example 5 Parallel or Perpendicular? Determine whether the pairs of lines are parallel, perpendicular, or neither. a. y = 3x 2, y = x + 1 b. y = x + 1, y = x 1 28

29 Example 5 Parallel or Perpendicular? a. The first line has a slope of m 1 = 3 and the second line has a slope of m 2 =. Because these slopes are negative reciprocals of each other, the two lines must be perpendicular. cont d 29

30 Example 5 Parallel or Perpendicular? b. Both lines have a slope of m =. So, the two lines must be parallel. cont d 30

31

LINEAR EQUATIONS IN TWO VARIABLES

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 information

Chapter 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 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 information

Solving Equations and Graphing

Solving 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 information

E. Slope-Intercept Form and Direct Variation (pp )

E. 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 information

Use smooth curves to complete the graph between and beyond the vertical asymptotes.

Use smooth curves to complete the graph between and beyond the vertical asymptotes. 5.3 Graphs of Rational Functions Guidelines for Graphing Rational Functions 1. Find and plot the x-intercepts. (Set numerator = 0 and solve for x) 2. Find and plot the y-intercepts. (Let x = 0 and solve

More information

ACT Coordinate Geometry Review

ACT 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 information

1.7 Parallel and Perpendicular Lines

1.7 Parallel and Perpendicular Lines Section 1.7 Parallel and Perpendicular Lines 11 Eplaining the Concepts 17. Name the five forms of equations of lines given in this section. 18. What tpe of line has one -intercept, but no -intercept? 19.

More information

Find the equation of a line given its slope and y-intercept. (Problem Set exercises 1 6 are similar.)

Find the equation of a line given its slope and y-intercept. (Problem Set exercises 1 6 are similar.) Directions Each problem below is similar to the example with the same number in your textbook. After reading through an example in your textbook, or watching one of the videos of that example on MathTV,

More information

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines

Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Name Period GEOMETRY CHAPTER 3 Perpendicular and Parallel Lines Section 3.1 Lines and Angles GOAL 1: Relationship between lines Two lines are if they are coplanar and do not intersect. Skew lines. Two

More information

Outcome 9 Review Foundations and Pre-Calculus 10

Outcome 9 Review Foundations and Pre-Calculus 10 Outcome 9 Review Foundations and Pre-Calculus 10 Level 2 Example: Writing an equation in slope intercept form Slope-Intercept Form: y = mx + b m = slope b = y-intercept Ex : Write the equation of a line

More information

CH 54 SPECIAL LINES. Ch 54 Special Lines. Introduction

CH 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 information

Block: Date: Name: REVIEW Linear Equations. 7.What is the equation of the line that passes through the point (5, -3) and has a slope of -3?

Block: Date: Name: REVIEW Linear Equations. 7.What is the equation of the line that passes through the point (5, -3) and has a slope of -3? Name: REVIEW Linear Equations 1. What is the slope of the line y = -2x + 3? 2. Write the equation in slope-intercept form. Block: Date: 7.What is the equation of the line that passes through the point

More information

In this section, we find equations for straight lines lying in a coordinate plane.

In 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 information

constant EXAMPLE #4:

constant 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 information

Warm-Up. Complete the second homework worksheet (the one you didn t do yesterday). Please begin working on FBF010 and FBF011.

Warm-Up. Complete the second homework worksheet (the one you didn t do yesterday). Please begin working on FBF010 and FBF011. Warm-Up Complete the second homework worksheet (the one you didn t do yesterday). Please begin working on FBF010 and FBF011. You have 20 minutes at the beginning of class to work on these three tasks.

More information

Section 1.3. Slope of a Line

Section 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 information

Math 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 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 information

Slopes of of Parallel and and Perpendicular Lines Lines Holt Algebra 1

Slopes of of Parallel and and Perpendicular Lines Lines Holt Algebra 1 5-8 Slopes of of Parallel and and Lines Warm Up Lesson Presentation Lesson Quiz Bell Quiz 5-8 Find the reciprocal. 1. 2 2. 1 pt 1 pt 1 pt 3. 2 pts 2 pts 2 pts Find the slope of the line that passes through

More information

Chapter 2: Functions and Graphs Lesson Index & Summary

Chapter 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 information

10 GRAPHING LINEAR EQUATIONS

10 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 information

Algebra & Trig. 1. , then the slope of the line is given by

Algebra & 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 information

A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal

A 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 information

Page 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 information

Name: Date: Period: Activity 4.6.2: Point-Slope Form of an Equation. 0, 4 and moving to another point on the line using the slope.

Name: Date: Period: Activity 4.6.2: Point-Slope Form of an Equation. 0, 4 and moving to another point on the line using the slope. Name: Date: Period: Activity.6.2: Point-Slope Form of an Equation 1.) Graph the equation y x = + starting at ( ) 0, and moving to another point on the line using the slope. 2.) Now, draw another graph

More information

Section 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.

Section 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 information

You MUST know the big 3 formulas!

You MUST know the big 3 formulas! Name 3-13 Review Geometry Period Date Unit 3 Lines and angles Review 3-1 Writing equations of lines. Determining slope and y intercept given an equation Writing the equation of a line given a graph. Graphing

More information

Use the Point-Slope Form to Write the Equation of a Line

Use 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 information

Section 2.3 Task List

Section 2.3 Task List Summer 2017 Math 108 Section 2.3 67 Section 2.3 Task List Work through each of the following tasks, carefully filling in the following pages in your notebook. Section 2.3 Function Notation and Applications

More information

y-intercept remains constant?

y-intercept remains constant? 1. The graph of a line that contains the points ( 1, 5) and (4, 5) is shown below. Which best represents this line if the slope is doubled and the y-intercept remains constant? F) G) H) J) 2. The graph

More information

Chapter 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. 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 information

3-5 Slopes of Lines. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Geometry

3-5 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 information

Hyperbolas Graphs, Equations, and Key Characteristics of Hyperbolas Forms of Hyperbolas p. 583

Hyperbolas 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 information

6.1 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. 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 information

Outcome 7 Review. *Recall that -1 (-5) means

Outcome 7 Review. *Recall that -1 (-5) means Outcome 7 Review Level 2 Determine the slope of a line that passes through A(3, -5) and B(-2, -1). Step 1: Remember that ordered pairs are in the form (x, y). Label the points so you can substitute into

More information

Vocabulary slope, parallel, perpendicular, reciprocal, negative reciprocal, horizontal, vertical, rise, run (earlier grades)

Vocabulary slope, parallel, perpendicular, reciprocal, negative reciprocal, horizontal, vertical, rise, run (earlier grades) Slope Reporting Category Reasoning, Lines, and Transformations Topic Exploring slope, including slopes of parallel and perpendicular lines Primary SOL G.3 The student will use pictorial representations,

More information

Section 3.5. Equations of Lines

Section 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 information

Chapter 6: Linear Relations

Chapter 6: Linear Relations Chapter 6: Linear Relations Section 6. Chapter 6: Linear Relations Section 6.: Slope of a Line Terminolog: Slope: The steepness of a line. Also known as the Rate of Change. Slope = Rise: The change in

More information

2.3 BUILDING THE PERFECT SQUARE

2.3 BUILDING THE PERFECT SQUARE 16 2.3 BUILDING THE PERFECT SQUARE A Develop Understanding Task Quadratic)Quilts Optimahasaquiltshopwhereshesellsmanycolorfulquiltblocksforpeoplewhowant tomaketheirownquilts.shehasquiltdesignsthataremadesothattheycanbesized

More information

Algebra 1B. Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5. COLYER Fall Name: Period:

Algebra 1B. Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5. COLYER Fall Name: Period: Chapter 6: Linear Equations & Their Graphs Sections 6-1 through 6-7 & 7-5 COLYER Fall 2016 Name: Period: What s the Big Idea? Analyzing Linear Equations & Inequalities What can I expect to understand when

More information

Graphing - Slope-Intercept Form

Graphing - Slope-Intercept Form 2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,

More information

Graphs, Linear Equations and Functions

Graphs, 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 information

Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice

Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice Section 2-4: Writing Linear Equations, Including Concepts of Parallel & Perpendicular Lines + Graphing Practice Name Date CP If an equation is linear, then there are three formats typically used to express

More information

4.4 Equations of Parallel and Perpendicular

4.4 Equations of Parallel and Perpendicular www.ck12.org Chapter 4. Determining Linear Equations 4.4 Equations of Parallel and Perpendicular Lines Learning Objectives Determine whether lines are parallel or perpendicular. Write equations of perpendicular

More information

Review for Mastery. Identifying Linear Functions

Review 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 information

Parallel and Perpendicular Lines on the Coordinate Plane

Parallel and Perpendicular Lines on the Coordinate Plane Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane 1.5 Learning Goals Key Term In this lesson, you will: Determine whether lines are parallel. Identify and write the

More information

Use Slope-Intercept Form to Write the Equation of a Line

Use Slope-Intercept Form to Write the Equation of a Line Math 35 2. "Writing Equations of Lines" Objectives: * Use the slope-intercept form to write the equation of a line. * Use the point-slope form to write the equation of a line. * Use slope as an aid when

More information

The Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2

The Ellipse. PF 1 + PF 2 = constant. Minor Axis. Major Axis. Focus 1 Focus 2. Point 3.4.2 Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Focus 1 Focus 2 Major Axis Point PF

More information

Equations of Lines and Linear Models

Equations of Lines and Linear Models 8. Equations of Lines and Linear Models Equations of Lines If the slope of a line and a particular point on the line are known, it is possible to find an equation of the line. Suppose that the slope of

More information

Creating a foldable for Equations of Lines

Creating a foldable for Equations of Lines Creating a foldable for Equations of Lines Equations of Lines Slope Direct Variation Slope-Intercept Form Standard Form Point-Slope Form Equation w/ slope & 1 point Equation w/ 2 points Horizontal & Vertical

More information

Welcome to Math! Put last night s homework on your desk and begin your warm-up (the other worksheet that you chose to save for today)

Welcome to Math! Put last night s homework on your desk and begin your warm-up (the other worksheet that you chose to save for today) Welcome to Math! Put last night s homework on your desk and begin your warm-up (the other worksheet that you chose to save for today) Unit Map - Geometry Thursday - Parallel Lines Cut by a Transversal

More information

Appendix M TERMINOLOGY. Slope of a Line. Slope. Undefined Slope. Slope-Intercept Form

Appendix 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 information

The Picture Tells the Linear Story

The 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 information

Determine the intercepts of the line and ellipse below: Definition: An intercept is a point of a graph on an axis. Line: x intercept(s)

Determine 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 information

MTH 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) 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 information

Analytic Geometry ةيليلحتلا ةسدنھلا

Analytic 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 information

Analytic Geometry. The x and y axes divide the Cartesian plane into four regions called quadrants.

Analytic 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 information

2. To receive credit on any problem, you must show work that explains how you obtained your answer or you must explain how you obtained your answer.

2. To receive credit on any problem, you must show work that explains how you obtained your answer or you must explain how you obtained your answer. Math 50, Spring 2006 Test 2 PRINT your name on the back of the test. Circle your class: MW @ 11 TTh @ 2:30 Directions 1. Time limit: 50 minutes. 2. To receive credit on any problem, you must show work

More information

Math 152 Rodriguez Blitzer 2.5 The Point-Slope Form of the Equation of a Line

Math 152 Rodriguez Blitzer 2.5 The Point-Slope Form of the Equation of a Line Math 152 Rodriguez Blitzer 2.5 The Point-Slope Form of the Equation of a Line I. Point-Slope Form A. Linear equations we have seen so far: 1. standard form: Ax +By=C A, B, and C real numbers 2. slope-intercept

More information

Graphing Lines with a Table

Graphing Lines with a Table Graphing Lines with a Table Select (or use pre-selected) values for x Substitute those x values in the equation and solve for y Graph the x and y values as ordered pairs Connect points with a line Graph

More information

Algebra I Notes Unit Seven: Writing Linear Equations

Algebra I Notes Unit Seven: Writing Linear Equations Sllabus Objective.6 The student will be able to write the equation of a linear function given two points, a point and the slope, table of values, or a graphical representation. Slope-Intercept Form of

More information

4 The Cartesian Coordinate System- Pictures of Equations

4 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 information

Section 7B Slope of a Line and Average Rates of Change

Section 7B Slope of a Line and Average Rates of Change Section 7B Slope of a Line and Average Rates of Change IBM stock had a price of $186.91 at the end of September 2014. Over the next three months the stock price rose and fell and by the end of December

More information

3.1 parallel lines and transversals

3.1 parallel lines and transversals VOCAB Parallel lines- 3.1 parallel lines and transversals Skew lines- Parallel planes- Transversal- Interior < s Transversal Angle Pair Relationships Exterior < s Same side Interior < s (consecutive interiors

More information

Chapter 3 Graphing Linear Equations

Chapter 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 information

MA Lesson 16 Sections 2.3 and 2.4

MA 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 information

Chapter 3 Linear Equations in Two Variables

Chapter 3 Linear Equations in Two Variables Chapter Linear Equations in Two Variables. Check Points. 6. x y x ( x, y) y ( ) 6, 6 y ( ), 0 y (0) 0, y () 0,0 y (),. E(, ) F(,0) G (6,0). a. xy 9 ( ) 9 69 9 9, true (, ) is a solution. b. xy 9 () 9 99

More information

3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes

3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes 3.4 and 4.3 Explain Graphing and Writing Linear Equations in Standard Form - Notes Essential Question: How can you describe the graph of the equation Ax + By = C? How can you write the equation of a line

More information

Investigating Intercepts

Investigating Intercepts Unit: 0 Lesson: 01 1. Can more than one line have the same slope? If more than one line has the same slope, what makes the lines different? a. Graph the following set of equations on the same set of aes.

More information

MATH 150 Pre-Calculus

MATH 150 Pre-Calculus MATH 150 Pre-Calculus Fall, 2014, WEEK 5 JoungDong Kim Week 5: 3B, 3C Chapter 3B. Graphs of Equations Draw the graph x+y = 6. Then every point on the graph satisfies the equation x+y = 6. Note. The graph

More information

Plotting 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 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 information

Review Journal 6 Assigned Work: Page 146, All questions

Review 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 information

(a) Find the equation of the line that is parallel to this line and passes through the point.

(a) Find the equation of the line that is parallel to this line and passes through the point. 1. Consider the line. (a) Find the equation of the line that is parallel to this line and passes through the point. (b) Find the equation of the line that is perpendicular to this line and passes through

More information

Exploring the Pythagorean Theorem

Exploring the Pythagorean Theorem Exploring the Pythagorean Theorem Lesson 11 Mathematics Objectives Students will analyze relationships to develop the Pythagorean Theorem. Students will find missing sides in right triangles using the

More information

Name: 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. 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 information

2.3 Quick Graphs of Linear Equations

2.3 Quick Graphs of Linear Equations 2.3 Quick Graphs of Linear Equations Algebra III Mr. Niedert Algebra III 2.3 Quick Graphs of Linear Equations Mr. Niedert 1 / 11 Forms of a Line Slope-Intercept Form The slope-intercept form of a linear

More information

Math 154 :: Elementary Algebra

Math 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 information

CC Geometry H Aim #3: How do we rotate points 90 degrees on the coordinate plane? Do Now:

CC Geometry H Aim #3: How do we rotate points 90 degrees on the coordinate plane? Do Now: CC Geometry H Aim #3: How do we rotate points 90 degrees on the coordinate plane? Do Now: 1. a. Write the equation of the line that has a slope of m = and passes through the point (0, 3). Graph this equation

More information

5. Determine the slope of a line that is perpendicular to the line through W( 9, 7) and X(6, 10). a. c. 15

5. Determine the slope of a line that is perpendicular to the line through W( 9, 7) and X(6, 10). a. c. 15 Math 101 Chapter 6 Review Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Determine the slope of this line segment. A x 0 B. Determine the slope of the

More information

Educator s Guide to Graphing y = mx + b

Educator s Guide to Graphing y = mx + b Educator s Guide to Graphing y = mx + b Overview: Using an ipad and Sketchpad Explorer, students will graph a linear equation using the y intercept and slope. Grades and Subject Areas: High School Algebra

More information

Chapter 7, Part 1B Equations & Functions

Chapter 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 information

Sect Linear Equations in Two Variables

Sect 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 information

Lesson 16: The Computation of the Slope of a Non Vertical Line

Lesson 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 information

Lesson 7 Slope-Intercept Formula

Lesson 7 Slope-Intercept Formula Lesson 7 Slope-Intercept Formula Terms Two new words that describe what we've been doing in graphing lines are slope and intercept. The slope is referred to as "m" (a mountain has slope and starts with

More information

4.5 Equations of Parallel and Perpendicular Lines

4.5 Equations of Parallel and Perpendicular Lines Name Class Date.5 Equations of Parallel and Perpendicular Lines Essential Question: How can ou find the equation of a line that is parallel or perpendicular to a given line? Resource Locker Eplore Eploring

More information

3.1 Start Thinking. 3.1 Warm Up. 3.1 Cumulative Review Warm Up

3.1 Start Thinking. 3.1 Warm Up. 3.1 Cumulative Review Warm Up 3.1 Start Thinking Sketch two perpendicular lines that intersect at point. Plot one point on each line that is not. all these points and. onnect and to make. What type of figure do points,, and make? ould

More information

CHAPTER 3. Parallel & Perpendicular lines

CHAPTER 3. Parallel & Perpendicular lines CHAPTER 3 Parallel & Perpendicular lines 3.1- Identify Pairs of Lines and Angles Parallel Lines: two lines are parallel if they do not intersect and are coplaner Skew lines: Two lines are skew if they

More information

Geometry. Unit 3 Parallel and Perpendicular Lines. Name:

Geometry. Unit 3 Parallel and Perpendicular Lines. Name: Geometry Unit 3 Parallel and Perpendicular Lines Name: 1 Geometry Chapter 3 Parallel and Perpendicular Lines ***In order to get full credit for your assignments they must me done on time and you must SHOW

More information

Lesson 1b Linear Equations

Lesson 1b Linear Equations In the first lesson we looked at the concepts and rules of a Function. The first Function that we are going to investigate is the Linear Function. This is a good place to start because with Linear Functions,

More information

Chapter 3 Parallel and Perpendicular Lines Geometry. 4. For, how many perpendicular lines pass through point V? What line is this?

Chapter 3 Parallel and Perpendicular Lines Geometry. 4. For, how many perpendicular lines pass through point V? What line is this? Chapter 3 Parallel and Perpendicular Lines Geometry Name For 1-5, use the figure below. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them. 1. Find two pairs

More information

Homework 5 - Section 3.3 #5

Homework 5 - Section 3.3 #5 Homework 5 - Section. #5 Intermediate Algebra / MAT 15 Fall 01 possible master (Prof. Fleischner) Student Name/ID: 1. Rewrite the equation in A + B = C form. Use integers for A, B, and C. + 5 = +. Rewrite

More information

Cumulative Review : MAT-032 (Algebra B) 2013

Cumulative 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 information

Parallel and Perpendicular Lines on the Coordinate Plane

Parallel and Perpendicular Lines on the Coordinate Plane Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane 1.5 Learning Goals Key Term In this lesson, you will: Determine whether lines are parallel. Identify and write the

More information

1. Graph y = 2x 3. SOLUTION: The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept.

1. Graph y = 2x 3. SOLUTION: The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept. 1. Graph y = 2x 3. The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept. Plot the y-intercept (0, 3). The slope is. From (0, 3), move up 2 units and right 1

More information

PREREQUISITE/PRE-CALCULUS REVIEW

PREREQUISITE/PRE-CALCULUS REVIEW PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which

More information

Since each element is paired with unique element in the range, it is a function.

Since 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 information

Slopes of Lines Notes What is slope?

Slopes of Lines Notes What is slope? Slopes of Lines Notes What is slope? Find the slope of each line. 1 Find the slope of each line. Find the slope of the line containing the given points. 6, 2!!"#! 3, 5 4, 2!!"#! 4, 3 Find the slope of

More information

Mathematics Success Grade 8

Mathematics Success Grade 8 Mathematics Success Grade 8 T429 [OBJECTIVE] The student will solve systems of equations by graphing. [PREREQUISITE SKILLS] solving equations [MATERIALS] Student pages S207 S220 Rulers [ESSENTIAL QUESTIONS]

More information

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

5.1 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 information

Graphs of linear equations will be perfectly straight lines. Why would we say that A and B are not both zero?

Graphs 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 information

Algebra 1 Online:

Algebra 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 information