Section Graphs and Lines
|
|
- Laurence Farmer
- 5 years ago
- Views:
Transcription
1 Section Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity with the concepts. Understanding of certain skills is assumed, such as basic operations with integers, fractions and decimals, comparisons of numbers, and converting percents to decimals. Throughout the course, you will have access to a calculator, so you should make sure you are familiar with the way your calculator works. See Appendix A for more information on calculators. We ll start the prerequisite review with some reminders about graphing in the coordinate plane. The coordinate plane consists of two number lines, drawn perpendicular to one another. This divides the plane into four sections, which we call quadrants. The horizontal number line is called the x axis; the vertical number line is called the y axis. The axes intersect at a point that is called the origin. The coordinate plane allows us to assign a unique name to each point in the plane. We write that name using an ordered pair ( x, y ), where x tells how far to move to the left or right from the origin and y tells how far to move up or down. The origin 4,3 will be four units to is named by the ordered pair ( 0,0 ). The ordered pair ( ) the left of the origin and then up three units, while the ordered pair ( 3, 4) will be three units to the right of the origin and then down four units. Math 1314 Page 1 of 11 Section 1.1
2 You will need to be able to graph lines in the coordinate plane. A line is determined by two points, so you will need to know two points that lie on a line in order to graph it. We establish a relationship between the x and y coordinates of the points that lie on a line using an equation that contains both x and y. The general form for this equation is Ax + By = C. When you are given an equation in this form, you can graph the line by Choosing some values for x Computing the corresponding values for y Forming ordered pairs ( x, y ) and graphing them Drawing a line that passes through the ordered pairs You must find a minimum of two points; if you find more points, you may find it easier to sketch the graph. Example 1: Graph the line: 2x y = 3 Solution: Start by selecting some values for x. You may want to organize these in a table of values. Keep the numbers small. For this equation, we ll choose 1, 0, and 1. Next, compute the values for y. x y Math 1314 Page 2 of 11 Section 1.1
3 If x = 1, then we have 2( 1) y = 3 2 y = 3 y = 5 y = 5 If x = 0, then we have 2(0) y = 3 0 y = 3 y = 3 y = 3 If x = 1, then we have 2(1) y = 3 2 y = 3 y = 1 y = 1 Now fill these values into the y column of the table of values. x y We now have three ordered pairs, ( 1, 5 ), ( 0, 3 ) and ( 1, 1). Graph these in the coordinate plane. Math 1314 Page 3 of 11 Section 1.1
4 You can see that the three points lie along a line. Finally, draw a line through the three points and label the graph. Since you only need two points to graph a line, you may find that it s easy to find the two points where the graph crosses the axes and then use those to graph the line. These two points are called the intercepts. The x intercept is the point ( x,0) where the graph crosses the x axis, and the y intercept is the point ( 0, y ) where the graph crosses the y axis. To find the x intercept, let y = 0and solve for x. To find the y intercept, let x = 0and solve for y. Math 1314 Page 4 of 11 Section 1.1
5 Example 2: Find the x and y intercepts and use them to graph the line: 2x + 3y = 6. Solution: To find the x intercept, let y = 0and solve for x. 2x + 3(0) = 6 2x = 6 x = 3 To find the y intercept, let x = 0and solve for y. 2(0) + 3y = 6 3y = 6 y = 2 This gives you two points, ( 3,0) and ( 0,2 ). Graph them and draw the line. You may be familiar with the form y = mx + b. This is called the slope-intercept form, and it can be useful in graphing. In this equation, m represents the slope and b gives the y coordinate of the y intercept. The slope is the rate at which the y values are changing with respect to x. A slope of 1, then, states that, for each unit 2 change in x, the change in y is 1 rise unit. More often, we state that slope is 2 run, where the numerator tells how many units to move up or down and the Math 1314 Page 5 of 11 Section 1.1
6 denominator tells how many units to move left or right. So if the slope is 1 2, then you d rise 1 unit and run 2 units. Before the slope can help you graph a line, you ll need a starting point, and with the slope-intercept form, that point is the y intercept. It is the point ( 0,b ), so you can find the y intercept just by looking at the equation. Start by graphing the y intercept, then use the rise and run to obtain one or two more points. Once you can see the pattern for the line, draw in the line. Example 3: Graph the line 2 y = x 1. 3 Solution: Start by finding the information from the equation. The slope is 2 3, so rise is 2 and run is 3. The y intercept is the point ( 0, 1). Plot the point in the coordinate plane. Then use rise and run to find some more points. Now draw in the line and label it. Math 1314 Page 6 of 11 Section 1.1
7 If the slope that you are given is negative, then you will make either the rise or the run negative, but not both. (Remember that a negative number divided by a negative number is a positive number, so if you make both negative, the resulting slope is positive!) Also, if the slope you are given is an integer, you can write it as a fraction with a denominator of 1, so that you can determine both rise and run. Example 4: Graph y = 3x + 2. Solution: The slope is -3 and the y intercept is ( 0,2 ). Think of the slope as 3 m =, so rise is -3 and run is 1. Since the rise is negative, you ll move down 1 rather than up. Graph the y intercept, then use the rise and run to graph some more points. Now draw the line and label the graph. Math 1314 Page 7 of 11 Section 1.1
8 You will need to be able to write the equation of the line if you are given information about the line. You will usually be asked to write equations in slope-intercept form. To accomplish this, you will need to know the slope and one point that lies on the graph of the line. Example 5: Write an equation of the line that has slope -3 and passes through the point ( 0, 7). Solution: In this problem, you are given the slope, and the point that you are given is the y intercept. So you know both m and b. The equation of the line is y = 3x 7. Example 6: Write an equation of the line that has slope 2 and passes through the point ( 1, 4). Solution: In this problem, you are given the slope and a point that is not the y intercept. You will need to find b. From the ordered pair, you know that x = 1and y = 4. You are given the slope, so you know that m = 2. Substitute these numbers into y = mx + b and then solve for b. So you ll have 4 = 2(1) + b and b = 6. You now know both m and b, so you can write the equation: y = 2x 6. Math 1314 Page 8 of 11 Section 1.1
9 You may be given two points and asked to find the equation of the line. As noted earlier, to write an equation of the line, you need the slope and a point to write an equation of the line. You can find the slope of a line if you are given two points ( ) and (, ) x y x y that 1, y2 y1 lie on the line. The formula for slope is m =. Once you know the slope, x x 2 1 you can use the slope and either one of the given points to write the equation of the line. Example 7: Write an equation of the line that passes through the points ( 2,5 ) and ( 1,4 ). Solution: Start by finding the slope. Two points are given, so you know that x = 2, y = 5, x = 1, and y = 4. Substitute these numbers into the slope formula m = = = Now find b using the slope and either of the two given points. We ll use ( 2,5 ). 1 5 = ( 2) + b = + b = b = b = b 3 Now that you know both m and b, you can write the equation: 1 13 y = x Math 1314 Page 9 of 11 Section 1.1
10 The equation of a horizontal line is y Example 8: Graph y = 2. Solution: The equation is written in the form y = k ; the equation of a vertical line is x = j. = k, so the graph will be a horizontal line. A horizontal line will cross the y axis. In this case, the line will cross the y axis where 2 0, 2. Plot the point and draw the line. y =, which is the point ( ) Example 9: Graph x = 1. Solution: The equation is written in the form x = j, so the graph will be a vertical line. A vertical line will cross the x axis. In this case, the line will cross the x axis where 1 1,0. Plot the point and draw the line. x =, which is the point ( ) Math 1314 Page 10 of 11 Section 1.1
11 Example 10: Write an equation of the horizontal line that passes through the point ( 1,3 ). Solution: The equation of a horizontal line has the form y = k. Find the y coordinate of the point that is given. The y value of every point on the horizontal line through this point will be the same, 3. The equation is 3 y =. Example 11: Write an equation of the vertical line that passes through the point ( 4,7). Solution: The equation of a vertical line has the form x = j. Find the x coordinate of the point that is given. The x value of every point on the vertical line through this point will be the same, -4. The equation is 4 x =. Math 1314 Page 11 of 11 Section 1.1
UNIT 4 NOTES. 4-1 and 4-2 Coordinate Plane
UNIT 4 NOTES 4-1 and 4-2 Coordinate Plane y Ordered pairs on a graph have several names. (X coordinate, Y coordinate) (Domain, Range) (Input,Output) Plot these points and label them: a. (3,-4) b. (-5,2)
More informationMath-2. Lesson 3-1. Equations of Lines
Math-2 Lesson 3-1 Equations of Lines How can an equation make a line? y = x + 1 x -4-3 -2-1 0 1 2 3 Fill in the rest of the table rule x + 1 f(x) -4 + 1-3 -3 + 1-2 -2 + 1-1 -1 + 1 0 0 + 1 1 1 + 1 2 2 +
More informationGraphing Linear Equations
Graphing Linear Equations Question 1: What is a rectangular coordinate system? Answer 1: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system,
More information.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3)
Co-ordinate Geometry Co-ordinates Every point has two co-ordinates. (3, 2) x co-ordinate y co-ordinate Plot the following points on the plane..(3, 2) A (4, 1) D (2, 5) G (6, 3) B (3, 3) E ( 4, 4) H (6,
More informationSec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.
Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical
More informationWriting and Graphing Linear Equations. Linear equations can be used to represent relationships.
Writing and Graphing Linear Equations Linear equations can be used to represent relationships. Linear equation An equation whose solutions form a straight line on a coordinate plane. Collinear Points that
More informationSNAP Centre Workshop. Graphing Lines
SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.
More informationThe Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
More informationCollege Prep Algebra II Summer Packet
Name: College Prep Algebra II Summer Packet This packet is an optional review which is highly recommended before entering CP Algebra II. It provides practice for necessary Algebra I topics. Remember: When
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationPractice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise
Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,
More informationReview for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
More informationTopic. Section 4.1 (3, 4)
Topic.. California Standards: 6.0: Students graph a linear equation and compute the x- and y-intercepts (e.g., graph x + 6y = ). They are also able to sketch the region defined by linear inequality (e.g.,
More informationYou should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
More informationslope rise run Definition of Slope
The Slope of a Line Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the
More informationIntro. To Graphing Linear Equations
Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate (the ordinate).
More informationSection 3.1 Objective 1: Plot Points in the Rectangular Coordinate System Video Length 12:35
Section 3.1 Video Guide The Rectangular Coordinate System and Equations in Two Variables Objectives: 1. Plot Points in the Rectangular Coordinate System 2. Determine If an Ordered Pair Satisfies an Equation
More informationFLC Ch 3. Ex 1 Plot the points Ex 2 Give the coordinates of each point shown. Sec 3.2: Solutions and Graphs of Linear Equations
Math 100 Elementary Algebra Sec 3.1: The Rectangular Coordinate System x-axis and y-axis origin ordered pair x-coordinate y-coordinate quadrants (I, II, III, and IV) Rectangular/Cartesian Coordinate System
More information3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines
3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x + 2 3. y = x 3 4. y = 4 x + 3 3 November 2, 2015 3.4 Proofs with Perpendicular Lines Geometry 3.5 Equations of Parallel and Perpendicular Lines Day
More informationgraphing_9.1.notebook March 15, 2019
1 2 3 Writing the equation of a line in slope intercept form. In order to write an equation in y = mx + b form you will need the slope "m" and the y intercept "b". We will subsitute the values for m and
More informationList of Topics for Analytic Geometry Unit Test
List of Topics for Analytic Geometry Unit Test 1. Finding Slope 2. Rule of 4 (4 forms of a line) Graph, Table of Values, Description, Equation 3. Find the Equations- Vertical and Horizontal Lines 4. Standard
More informationSection 2.2 Graphs of Linear Functions
Section. Graphs of Linear Functions Section. Graphs of Linear Functions When we are working with a new function, it is useful to know as much as we can about the function: its graph, where the function
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationAlgebra 1 Semester 2 Final Review
Team Awesome 011 Name: Date: Period: Algebra 1 Semester Final Review 1. Given y mx b what does m represent? What does b represent?. What axis is generally used for x?. What axis is generally used for y?
More informationGRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.
GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),
More informationNotes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal
A, B, C are Notes Lesson 3 4 Standard Form of an Equation: Integers Ax + By = C Sometimes it is preferred that A is Positive All lines in the plane can be written in standard form. Oblique Coordinate Horizontal
More informationSection 7D Systems of Linear Equations
Section 7D Systems of Linear Equations Companies often look at more than one equation of a line when analyzing how their business is doing. For example a company might look at a cost equation and a profit
More informationTest Name: Chapter 3 Review
Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x - 8y = 18 Note: Each column
More informationName: NOTES 5: LINEAR EQUATIONS AND THEIR GRAPHS. Date: Period: Mrs. Nguyen s Initial: LESSON 5.1 RATE OF CHANGE AND SLOPE. A. Finding rates of change
NOTES : LINEAR EQUATIONS AND THEIR GRAPHS Name: Date: Period: Mrs. Nguen s Initial: LESSON. RATE OF CHANGE AND SLOPE A. Finding rates of change vertical change Rate of change = = change in x The rate of
More information3-6 Lines in the Coordinate Plane
3-6 Lines in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 1. m = 2, x = 3, and
More informationMath 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
More informationThe x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
More informationDid You Find a Parking Space?
Lesson.4 Skills Practice Name Date Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane Vocabulary Complete the sentence. 1. The point-slope form of the equation of the
More informationCHAPTER. Graphs of Linear Equations. 3.1 Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications
Graphs of Linear Equations CHAPTER 3 3.1 Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications Slide 2 3.1 Introduction to Graphing OBJECTIVES a Plot
More information1.6 Modeling with Equations
1.6 Modeling with Equations Steps to Modeling Problems with Equations 1. Identify the variable you want to solve for. 2. Express all unknown quantities in terms of this variable. 3. Set up the model by
More informationAlgebra I Notes Linear Equations and Inequalities in Two Variables Unit 04c
Big Idea: Describe the similarities and differences between equations and inequalities including solutions and graphs. Skill: graph linear equations and find possible solutions to those equations using
More informationRational Numbers on the Coordinate Plane. 6.NS.C.6c
Rational Numbers on the Coordinate Plane 6.NS.C.6c Copy all slides into your composition notebook. Lesson 14 Ordered Pairs Objective: I can use ordered pairs to locate points on the coordinate plane. Guiding
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationRational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
More informationLesson 19: The Graph of a Linear Equation in Two Variables is a Line
Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Classwork Exercises Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b),
More informationSection 1.1 The Distance and Midpoint Formulas
Section 1.1 The Distance and Midpoint Formulas 1 y axis origin x axis 2 Plot the points: ( 3, 5), (0,7), ( 6,0), (6,4) 3 Distance Formula y x 4 Finding the Distance Between Two Points Find the distance
More informationSection 7.6 Graphs of the Sine and Cosine Functions
Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.
More informationIn math, the rate of change is called the slope and is often described by the ratio rise
Chapter 3 Equations of Lines Sec. Slope The idea of slope is used quite often in our lives, however outside of school, it goes by different names. People involved in home construction might talk about
More informationAdvanced Algebra. Equation of a Circle
Advanced Algebra Equation of a Circle Task on Entry Plotting Equations Using the table and axis below, plot the graph for - x 2 + y 2 = 25 x -5-4 -3 0 3 4 5 y 1 4 y 2-4 3 2 + y 2 = 25 9 + y 2 = 25 y 2
More informationSection 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
More informationGraphs and Linear Functions
Graphs and Linear Functions A -dimensional graph is a visual representation of a relationship between two variables given by an equation or an inequality. Graphs help us solve algebraic problems by analysing
More informationAlgebra Unit 2: Linear Functions Notes. Slope Notes. 4 Types of Slope. Slope from a Formula
Undefined Slope Notes Types of Slope Zero Slope Slope can be described in several ways: Steepness of a line Rate of change rate of increase or decrease Rise Run Change (difference) in y over change (difference)
More informationExample 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
More informationBuilding Concepts: Moving from Proportional Relationships to Linear Equations
Lesson Overview In this TI-Nspire lesson, students use previous experience with proportional relationships of the form y = kx to consider relationships of the form y = mx and eventually y = mx + b. Proportional
More informationMath 154 Elementary Algebra. Equations of Lines 4.4
Math Elementary Algebra Caspers Name Date Equations of Lines. For each graph, solve each equation for y (if necessary), then write down the slope and y-intercept.. y x. y x - - - - - - - - - - - - - -
More informationf( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving
The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already
More information3.1. 3x 4y = 12 3(0) 4y = 12. 3x 4y = 12 3x 4(0) = y = x 0 = 12. 4y = 12 y = 3. 3x = 12 x = 4. The Rectangular Coordinate System
3. The Rectangular Coordinate System Interpret a line graph. Objectives Interpret a line graph. Plot ordered pairs. 3 Find ordered pairs that satisfy a given equation. 4 Graph lines. 5 Find x- and y-intercepts.
More informationChapter 4: Solving Linear Equations Study Guide
4.1: Plot Points in the Coordinate Plane Chapter 4: Solving Linear Equations Study Guide - Identify/graph ordered pairs Ex: Write the coordinates of - Identify the 4 quadrants point graphed and identify
More informationSection 2.1 Graphs. The Coordinate Plane
Section 2.1 Graphs The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of numbers to form
More informationOverview for Families
unit: Graphing Equations Mathematical strand: Algebra The following pages will help you to understand the mathematics that your child is currently studying as well as the type of problems (s)he will solve
More informationRational Numbers and the Coordinate Plane
Rational Numbers and the Coordinate Plane LAUNCH (8 MIN) Before How can you use the numbers placed on the grid to figure out the scale that is used? Can you tell what the signs of the x- and y-coordinates
More informationChapter 1 Section 1 Solving Linear Equations in One Variable
Chapter Section Solving Linear Equations in One Variable A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers with a 0. Linear equations
More informationTangent line problems
You will find lots of practice problems and homework problems that simply ask you to differentiate. The following examples are to illustrate some of the types of tangent line problems that you may come
More informationForms of Linear Equations
6. 1-6.3 Forms of Linear Equations Name Sec 6.1 Writing Linear Equations in Slope-Intercept Form *Recall that slope intercept form looks like y = mx + b, where m = slope and b = y=intercept 1) Writing
More informationSection 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or
More informationBell Ringer Write each phrase as a mathematical expression. Thinking with Mathematical Models
Bell Ringer Write each phrase as a mathematical expression. 1. the sum of nine and eight 2. the sum of nine and a number 3. nine increased by a number x 4. fourteen decreased by a number p 5. the product
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationConnexions module: m Linear Equations. Rupinder Sekhon
Connexions module: m18901 1 Linear Equations Rupinder Sekhon This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License 3.0 Abstract This chapter covers
More information4.1 The Coordinate Plane
4. The Coordinate Plane Goal Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin -ais -ais Ordered pair -coordinate -coordinate Quadrant Scatter plot Copright McDougal Littell, Chapter
More informationGeometry Unit 5 Geometric and Algebraic Connections. Table of Contents
Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing
More informationVertical Line Test a relationship is a function, if NO vertical line intersects the graph more than once
Algebra 2 Chapter 2 Domain input values, X (x, y) Range output values, Y (x, y) Function For each input, there is exactly one output Example: Vertical Line Test a relationship is a function, if NO vertical
More informationPoints and lines. x x 1 + y 1. y = mx + b
Points and lines Point is the fundamental element of the picture representation. It is nothing but the position in a plan defined as either pairs or triplets of number depending on whether the data are
More informationCHAPTER 5: LINEAR EQUATIONS AND THEIR GRAPHS Notes#26: Section 5-1: Rate of Change and Slope
Name: Date: Period: CHAPTER : LINEAR EQUATIONS AND THEIR GRAPHS Notes#: Section -: Rate of Change and Slope A. Finding rates of change vertical change Rate of change change in x The rate of change is constant
More informationUNIT 3B CREATING AND GRAPHING EQUATIONS Lesson 4: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing multiple equations on a graphing calculator graphing quadratic equations graphing linear equations Introduction A system
More informationDid you ever think that a four hundred year-old spider may be why we study linear relationships today?
Show Me: Determine if a Function is Linear M8221 Did you ever think that a four hundred year-old spider may be why we study linear relationships today? Supposedly, while lying in bed Rene Descartes noticed
More informationGraphing Linear Equations
Graphing Linear Equations A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. What am I learning today? How to graph a linear
More informationMath 3 Coordinate Geometry part 1 Unit November 3, 2016
Reviewing the basics The number line A number line is a visual representation of all real numbers. Each of the images below are examples of number lines. The top left one includes only positive whole numbers,
More information0,0 is referred to as the end point.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Chapter 2: Radical Functions 2.1 Radical Functions and Transformations (Day 1) For the function y x, the radicand, x, must
More informationUnit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
More informationExploring Slope. We use the letter m to represent slope. It is the ratio of the rise to the run.
Math 7 Exploring Slope Slope measures the steepness of a line. If you take any two points on a line, the change in y (vertical change) is called the rise and the change in x (horizontal change) is called
More informationUnit 1 Algebraic Functions and Graphs
Algebra 2 Unit 1 Algebraic Functions and Graphs Name: Unit 1 Day 1: Function Notation Today we are: Using Function Notation We are successful when: We can Use function notation to evaluate a function This
More informationChapter 11 GRAPHS OF LINEAR EQUATIONS
Chapter 11 GRAPHS OF LINEAR EQUATIONS 11.1 Graphs and Applications of Linear Equations Learning Objectives A Plot points associated with ordered pairs of numbers; determine the quadrant in which a point
More informationEach point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
More informationReview Exercise. 1. Determine vector and parametric equations of the plane that contains the
Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, 42. 2. In question 1, there are a variety of different answers possible,
More information3, 10,( 2, 4) Name. CP Algebra II Midterm Review Packet Unit 1: Linear Equations and Inequalities. Solve each equation. 3.
Name CP Algebra II Midterm Review Packet 018-019 Unit 1: Linear Equations and Inequalities Solve each equation. 1. x. x 4( x 5) 6x. 8x 5(x 1) 5 4. ( k ) k 4 5. x 4 x 6 6. V lhw for h 7. x y b for x z Find
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationWRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #1313
WRITING AND GRAPHING LINEAR EQUATIONS ON A FLAT SURFACE #11 SLOPE is a number that indicates the steepness (or flatness) of a line, as well as its direction (up or down) left to right. SLOPE is determined
More information2 and 6 4 and 8 1 and 5 3 and 7
Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different
More informationLinear Functions. College Algebra
Linear Functions College Algebra Linear Function A linear function is a function whose graph is a straight line. Linear functions can be written in the slope-intercept form of a line: f(x) = mx + b where
More informationSketching Straight Lines (Linear Relationships)
Sketching Straight Lines (Linear Relationships) The slope of the line is m = y x = y 2 y 1 = rise run. Horizontal lines have the form y = b and have slope m = 0. Vertical lines have the form x = a and
More information5.1 Introduction to the Graphs of Polynomials
Math 3201 5.1 Introduction to the Graphs of Polynomials In Math 1201/2201, we examined three types of polynomial functions: Constant Function - horizontal line such as y = 2 Linear Function - sloped line,
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationASMT 31: due Weds
Please do NOT use a calculator unless problem is marked with an asterisk. (*) ASMT 31: due Weds 10 4 17 SHOW work for all problems, including calculator-permitted ones! TEST 3 on Friday review Les 3.6
More informationAbout Graphing Lines
About Graphing Lines TABLE OF CONTENTS About Graphing Lines... 1 What is a LINE SEGMENT?... 1 Ordered Pairs... 1 Cartesian Co-ordinate System... 1 Ordered Pairs... 2 Line Segments... 2 Slope of a Line
More informationFour Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope
Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope https://www.youtube.com/watch?v=avs6c6_kvxm Direct Variation
More informationo Checkpoint Plot the points in the same coordinate plane.
The ~ al Coordinate Plane Plot points in a coordinate plane. VOCABULARY Coordinate plane Origin x-axls -axis Ordered pair x-coordinate -coordinate Quadrant Scatter plot ~. Lesson 4.. Algebra. Concepts
More informationCore Mathematics 1 Graphs of Functions
Regent College Maths Department Core Mathematics 1 Graphs of Functions Graphs of Functions September 2011 C1 Note Graphs of functions; sketching curves defined by simple equations. Here are some curves
More informationAlgebra I Notes Unit Six: Graphing Linear Equations and Inequalities in Two Variables, Absolute Value Functions
Sllabus Objective.4 The student will graph linear equations and find possible solutions to those equations using coordinate geometr. Coordinate Plane a plane formed b two real number lines (axes) that
More information2.1 Transforming Linear Functions
2.1 Transforming Linear Functions Before we begin looking at transforming linear functions, let s take a moment to review how to graph linear equations using slope intercept form. This will help us because
More informationLesson 18: There is Only One Line Passing Through a Given Point with a Given
Lesson 18: There is Only One Line Passing Through a Given Point with a Given Student Outcomes Students graph equations in the form of using information about slope and intercept. Students know that if
More information8 th Grade Math Reference Sheet
8 th Grade Math Reference Sheet Number Sense DECIMALS NS 1 To change a DECIMAL FRACTION, use the place value of the decimal as the denominator of the fraction; simplify if. 1. Line up decimal points 2.
More informationRevision Topic 11: Straight Line Graphs
Revision Topic : Straight Line Graphs The simplest way to draw a straight line graph is to produce a table of values. Example: Draw the lines y = x and y = 6 x. Table of values for y = x x y - - - - =
More informationBeginning and Intermediate Algebra Chapter 2: Graphing
Beginning and Intermediate Algebra Chapter 2: Graphing An open source (CC-BY) textbook by Tyler Wallace 1 ? Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0
More informationREVIEW, pages
REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in
More informationCourse of study- Algebra Introduction: Algebra 1-2 is a course offered in the Mathematics Department. The course will be primarily taken by
Course of study- Algebra 1-2 1. Introduction: Algebra 1-2 is a course offered in the Mathematics Department. The course will be primarily taken by students in Grades 9 and 10, but since all students must
More information