Straight Line Graphs

11 Key excerpts on "Straight Line Graphs"

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  eBook - PDF

  Teaching and Learning Algebra

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Chapter 6 Functions and Graphs Harnessing this new power [of computer technology] within mathematics and school mathematics is the challenge for the 21st century. (RS/JMC, 1997, p. 6) STRAIGHT-LINE GRAPHS Straight-line graphs were discussed in Chapter 3 as one of a number of ways of introducing algebraic ideas and symbols. They are particularly attractive in this respect because they provide a ready link between numbers, symbols and pictures. An equation provides a way of encapsulating the patterns in the co-ordinates of a set of points that lie on a straight line by acting as a unique label which highlights key properties. Although a graph is an abstract representation it has a visual appeal and looks interesting, particularly when a family of related graphs is depicted. Students need to understand the links between the equation, the table of values or set of co-ordinates and the graph, and to be able to move fluently between these different representa-tions. In Chapter 3 it was suggested that introductory work on straight-line graphs should be confined to positive whole numbers and should begin by looking at a set of points on a straight line, using the pattern in the numbers to determine the equation of the line. This builds on the idea of representing the terms of a linear sequence algebraically and makes clear from the start where the equation comes from and what it means. Text books often start with equations and show students how to produce a table of values and then plot the corresponding lines. Whilst this may seem simpler as it is a more routine task, it starts from something that is unfamiliar, namely the equation, which can set up an immediate barrier because it looks strange and new and seems to have appeared for no apparent reason. Co-ordinates and their graphical representation should already be familiar and therefore provide a more reassuring start to a new idea.

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  Bird's Basic Engineering Mathematics

  • John Bird(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  Chapter 17

  Straight Line Graphs

  Why it is important to understand: Straight Line Graphs

  Graphs have a wide range of applications in engineering and in physical sciences because of their inherent simplicity. A graph can be used to represent almost any physical situation involving discrete objects and the relationship among them. If two quantities are directly proportional and one is plotted against the other, a straight line is produced. Examples include an applied force on the end of a spring plotted against spring extension, the speed of a flywheel plotted against time, and strain in a wire plotted against stress (Hooke’s law). In engineering, the straight line graph is the most basic graph to draw and evaluate.

  At the end of this chapter you should be able to:

  • understand rectangular axes, scales and co-ordinates
  • plot given co-ordinates and draw the best straight line graph
  • determine the gradient of a straight line graph
  • estimate the vertical axis intercept
  • state the equation of a straight line graph
  • plot Straight Line Graphs involving practical engineering examples

  17.1. Introduction to graphs

  A graph is a visual representation of information, showing how one quantity varies with another related quantity.

  We often see graphs in newspapers or in business reports, in travel brochures and government publications. For example, a graph of the share price (in pence) over a 6 month period for a drinks company, Fizzy Pops, is shown in Fig. 17.1 .

  Figure 17.1

  Generally, we see that the share price increases to a high of 400 p in June, but dips down to around 280 p in August before recovering slightly in September. A graph should convey information more quickly to the reader than if the same information was explained in words.

  When this chapter is completed you should be able to draw up a table of values, plot co-ordinates, determine the gradient and state the equation of a straight line graph. Some typical practical examples are included in which straight lines are used.

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  Functions and Change

  A Modeling Approach to College Algebra

  • Bruce Crauder, Benny Evans, Alan Noell, , Bruce Crauder, Benny Evans, Alan Noell(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  one.lfnine.lfthree.lf STRAIGHT LINES in the form of city streets, directions, boundaries, and many other things are among the most obvious mathematical objects that we experience in daily life. Historically, mathematicians made extensive studies of the geometry of straight lines several centuries before these lines were associated y y with a linear formula . Today it is understood that it is the combination of pictures and formulas that make lines so useful and so easy to handle. Robert Brenner/PhotoEdit—All right Techniques for modeling linear data can be applied to the analysis of tuition increases at universities such as Columbia. See Exercise six.lf on page two.lftwo.lfseven.lf. The Geometry ofuni00A0Lines Linear Functions Modeling Data with Linear Functions Linear Regression Systems of Equations Chapter Summary Chapter Review Exercises A Further Look uni25CF The Correlation Coefficient uni25CF Parallel and Perpendicular Lines uni25CF Secant Lines Student resources are availableuni00A0on the website www.cengagebrain.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 one.lfnine.lffour.lf CHAPTER three.lf | S T RAIG HT LINE S AND LINEAR FUNC T ION S The Geometry of Lines uni25A0 Characterizations of Straight Lines One way in which straight lines are often characterized is that they are determined by two points. For example, to describe a straight ramp, it is only necessary to give the loca -tions of the ends of the ramp. In Figure 3.1, we have depicted a ramp with one end atop a 4-foot-high retaining wall and the other on the ground 10 feet away. It is often convenient to represent such lines on coordinate axes. If we choose the horizontal axis to be ground level and let the vertical axis follow the retaining wall, we get the picture in Figure 3.2. The ends of the ramp in Figure 3.1 match the points in Figure 3.2 where the line crosses the horizontal and vertical axes.

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  Essential Maths

  for Business and Management

  • Clare Morris(Author)
  • 2007(Publication Date)
  • Bloomsbury Academic

    (Publisher)

  For instance, x + y = 7 represents a straight line, but it needs to be rearranged (by subtracting x from both sides) into the form y = 7 – x or y = – x + 7 before we can recognise it as having the form y = mx + c with m = –1 and c = 7. Similarly, y / x = 2 doesn’t look much like the standard equation, until you realise that it can be rearranged as y = 2 x , which has m = 2 and c = 0. Finally, a word about sketching, as distinct from plotting, graphs. Because we only need to know the position of two points on a straight line graph in order to be able to plot the graph, it is possible to get a good idea of the appearance of the graph without actu-ally using graph paper or drawing up an extensive table of values as we did above. It’s a good idea to get used to doing this. Very often what we need is an overall picture of the shape of a relationship, rather than an accurate plot from which precise values can be read. And of course, it is very much quicker to sketch than to plot accurately, which can be useful when it comes to examinations! Some business applications of linear graphs Example 1: A demand graph. You don’t need to have a PhD in management to realise that, with many commodities, the more you charge, the less you sell. Economists, and people concerned with setting pricing policies, are interested in exploring the relationship between price charged and quantity sold. For example, the manager of a corner shop might notice that when she charges 80p for a small pack of biscuits, she sells 200 packs per month, but when she puts the price up to 90p, sales go down to 160 packs per month. If we are prepared to accept that the relationship between the price and the quantity sold can be represented with reasonable accuracy by a straight line, then we can use this information to work out an equation for that relationship.

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  Painless Pre-Algebra

  • Barron's Educational Series, Amy Stahl(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  y-value will always be 2. Then plot the points, and draw the line through the points. This is a horizontal line.

  x	y
  −2	2
  0	2
  3	2

  Graph x = −1.

  Make a table, but this time the x-values always stay the same. The x-value has to be −1. Choose three different y-values to help you make the graph. Plot the points, and draw the line. This is a vertical line.

  x	y
  −1	−2
  −1	0
  −1	2

  CAUTION—Major Mistake Territory!

  A horizontal line is always written in the form y = b. A vertical line is always written in the form x = a.

  Examples:

  y = 3 → Horizontal line

  x = −2 → Vertical line

  BRAIN TICKLERSSet # 17

  Graph each line.

  1.x = 4

  2.x = −2

  3.y = 5

  4.y = −1

  (Answers are on pages 117 –118 .)

  Slope of a Line

  Have you ever tried to run up a steep hill? Or ride a bike all the way up a hill without walking? Or ski down a hill? The concept of “steepness” is one that can easily be understood in the real world.

  Examples of different “slopes” are easy to see. As a beginner skier, which hill would you prefer to ski down? A beginner skier starts on the bunny hill, which is less steep than a hill considered a black diamond!

  In the world of mathematics, the word slope describes the “steepness” of a line.

  There are four basic types of slope.

  Positive slope Slants upward, from left to right	Negative slope Slants downward, from left to right
  Zero slope A horizontal line, left to right Has zero slope (like a floor)	No slope/Undefined slope A vertical line, up and down Has undefined slope (like a wall)

  The slope of a line is described as a ratio. It is the comparison of the vertical change in the line compared to the horizontal change in the line. Here is the formal definition of slope.

  For a painless way to remember slope, use this saying:

  •Rise is the vertical distance between points. Rise involves the y-

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Can you use the slope found to predict the score of a person who studies 8 hours for the examination? If so, make the prediction. If not, explain why not. In the following problems, two lines are described using a pair of points on each line. Compute the slope of each line and use the value of the slopes to determine if the lines are parallel, perpendicular, or neither. 26. Line 1 passes through (1, 2) and (5, 2 2), line 2 through (3, 4) and (7, 8). 27. Line 1 passes through (0, 3) and (4, 0), line 2 through (0, 1) and (2, 3). 28. Line 1 passes through (0, 2 3) and (3, 1), line 2 through ( 2 3, 2 4) and (0, 0). 29. Line 1 passes through ( 2 2, 0) and (0, 2), line 2 through (2, 3) and (5, 0). 4.3 T HE E QUATION OF A S TRAIGHT L INE Undoubtedly, by this time we have developed a fairly good understanding of the notion of the slope of a straight line. We’ve also seen that we can interpret the slope in applications and that the interpretation, essentially, tells us the rate at which a quantity is changing (recall the concept of “per”). Every line in the plane can be represented by a particular type of equa-tion, referred to as a linear equation . The equation will have the property that it will contain x , y , or both, with each letter (when occurring) appearing to the power 1 only, and be such that it can be rewritten in the general form Ax 1 By 1 C 5 0. At first glance, it may not be clear why such equations can be associated with straight-line graphs in the plane, but this relationship will be made clear shortly. Prior to exploring the various forms a linear equation can take, we should take a moment to clarify the need for the previously mentioned general form. It will turn out that the same line can be represented by infinitely many expres-sions, each of which is equivalent to the other. When we say “equivalent,” we mean that the equation can be rewritten using the rules of algebra into another form that may appear different from the form with which we started.

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  eBook - ePub

  Let's Review Regents: Algebra I Revised Edition

  • Barron's Educational Series, Gary M. Rubinstein(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  x – 2|. One way to graph the solution set is to create a chart.

  x	|2x – 2|
  –2	|2(–2) – 2| = |–4 – 2| = |–6| = 6
  –1	|2(–1) – 2| = |–2 – 2| = |–4| = 4
  0	|2(0) – 2| = |0 – 2| = |–2| = 2
  1	|2(1) – 2| = |2 – 2| = |0| = 0
  2	|2(2) – 2| = |4 – 2| = |2| = 2
  3	|2(3) – 2| = |6 – 2| = |4| = 4
  4	|2(4) – 2| = |8 – 2| = |6| = 6

  When these five points are graphed, they create a V shape. This is typical for linear equations involving absolute values.

  Solving Systems of Linear Equations by Graphing

  Chapter 4 showed how to solve systems of linear equations with algebra. Systems of linear equations can also be solved by graphing.

  Math Facts

  The ordered pair that is the solution to a system of linear equations will be related to the point of intersection of the two lines that are the graphs of the solution sets of the two equations.

  In Chapter 4 , there was an example where you had to solve the system of equations.

  Using algebra, the solution was (6, 4). This system can also be solved by graphing the solution sets of each equation and locating the intersection point. Find the intercepts:

  So the intercepts for the top equation are (10, 0) and (0, 10).

  So, the intercepts for the bottom equation are (2, 0) and (0, –2).

  Plot the intercepts on graph paper and draw the lines carefully with a ruler.

  The lines intersect at the point (6, 4) so the solution to the system of equations is x = 6 and y = 4.

  Math Facts

  Solving with algebra is generally the faster and more accurate method for solving a system of linear equations. But this is a technique you need for systems of linear inequalities, coming later.

  Passage contains an image

  5.2 Calculating and Interpreting Slope

  Key Ideas

  The slope of a line is a number that measures how steep it is. A horizontal line has a slope of 0. A line with a positive slope goes up as it goes to the right. A line with a negative slope goes down as it goes to the right. The variable used for slope is the letter m

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  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  151 Introductory Mathematics| Hands-On Point-by-point plotting of straight-line graphs You can draw a straight-line graph by using point-by-point plotting (drawing a table of x - and y -values). You are already familiar with drawing a table. In the examples below the straight-line graphs in the standard form, y = mx + c , will be drawn on a Cartesian plane. Points are plotted as dots ( • ). Example 1 Given: f ( x ) = 2 x – 1. 1. Sketch the graph of f ( x ) = 2 x – 1 on a system of axes using the table method for the x -values {–2; –1; 0; 1; 2}. 2. Write down the gradient of the graph. 3. Write down the y -intercept of the graph. Solutions 1. Step 1 Substitute the given x -values into the equation in order to calculate the corresponding y -values at that specific point. f ( x ) = 2 x – 1 f (–2) = 2(–2) – 1 = –5 f (–1) = 2(–1) – 1 = –3 f (0) = 2(0) – 1 = –1 f (1) = 2(1) – 1 = 1 f (2) = 2(2) – 1 = 3 Step 2 Draw a table. x –2 –1 0 1 2 y –5 –3 –1 1 3 Step 3 Sketch the graph. • Decide on a proper scale for both axes. Use the highest and lowest x -values in the table to find the domain of the graph, for example 10 mm = 1 unit on the x -axis. Use the highest and lowest y -values in the table to find the range of the graph, for example 10 mm = 1 unit on the y -axis. • Draw the axes with a ruler. • Label the x -and y -axes. • Number the x - and y -axes according to the chosen scale. • Plot the pairs of values (coordinates) from the table on the set of axes. 152 Module 5 • Algebraic graphs • Check that all the points lie in a straight line. • Sketch the straight line that connects all the points with a ruler. • Label the straight line, for example y = 2 x – 1. If a point does not lie on the straight line, a mistake was made during calculations. Check the y -value calculation for that specific point. –3 –4 –2 –1 0 1 2 3 4 y = 2 x – 1 3 2 1 –1 –2 –3 –4 –5 y x 2.

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  N1 Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  151 N1 Mathematics| Hands-On! Point-by-point plotting of straight-line graphs You can draw a straight-line graph by using point-by-point plotting (drawing a table of x - and y -values). You are already familiar with drawing a table. In the examples below the straight-line graphs in the standard form, y = mx + c , will be drawn on a Cartesian plane. Points are plotted as dots ( • ). Example 1 Given: f ( x ) = 2 x – 1. 1. Sketch the graph of f ( x ) = 2 x – 1 on a system of axes using the table method for the x -values {–2; –1; 0; 1; 2}. 2. Write down the gradient of the graph. 3. Write down the y -intercept of the graph. Solutions 1. Step 1 Substitute the given x -values into the equation in order to calculate the corresponding y -values at that specific point. f ( x ) = 2 x – 1 f (–2) = 2(–2) – 1 = –5 f (–1) = 2(–1) – 1 = –3 f (0) = 2(0) – 1 = –1 f (1) = 2(1) – 1 = 1 f (2) = 2(2) – 1 = 3 Step 2 Draw a table. x –2 –1 0 1 2 y –5 –3 –1 1 3 Step 3 Sketch the graph. • Decide on a proper scale for both axes. Use the highest and lowest x -values in the table to find the domain of the graph, for example 10 mm = 1 unit on the x -axis. Use the highest and lowest y -values in the table to find the range of the graph, for example 10 mm = 1 unit on the y -axis. • Draw the axes with a ruler. • Label the x -and y -axes. • Number the x - and y -axes according to the chosen scale. • Plot the pairs of values (coordinates) from the table on the set of axes. 152 Module 5 • Algebraic graphs • Check that all the points lie in a straight line. • Sketch the straight line that connects all the points with a ruler. • Label the straight line, for example y = 2 x – 1. If a point does not lie on the straight line, a mistake was made during calculations. Check the y -value calculation for that specific point. –3 –4 –2 –1 0 1 2 3 4 y = 2 x – 1 3 2 1 –1 –2 –3 –4 –5 y x 2.

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  Precalculus

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2001(Publication Date)
  • Trade Paper Press

    (Publisher)

  m = −2. We can use the first formula:

  2. First let’s start by finding the slope

  We don’t know the y-intercept, so we have to use the second formula:

  3. We don’t know the y-intercept, so we’ll use the second formula:

  4. The points we’re given are (6,0) and (0,−3). We need to find the slope.

  5. We know this is a horizontal line because both points have the same y value. We also know that the equation of any horizontal line is y = a constant, y = 1 is the equation of this horizontal line.

  6. We know this is a vertical line because both points have the same x value. We also know that the equation of any vertical line is x = a constant, x = 5 is the equation of this vertical line.

  4 Graphs of Linear Functions

  A linear equation is a polynomial equation whose highest exponent is 1. It’s also called a first-degree equation. The graph of a linear equation is always a straight line. We need only two points to graph a straight line. The points we want to see on your graphs are the intercepts. You’ll find graphing linear equations to be very simple, especially since we reviewed how to find intercepts in section 1 of this chapter.

  Example 10:

  Graph 2x + 4y = 8.

  Solution:

  We’ll start by finding the x and the y intercepts; then we’ll plot the points and connect them with a straight line. Be sure to extend the line beyond those points and put arrows at the ends of the lines to show that the graph doesn’t end at these points. If you don’t put arrows at the end, it’s a line segment, which is only part of the graph. Write the equation of the line on your graph and label the intercepts.

  Example 11:

  Graph −2x + 4y = 8.

  Solution:

  Notice that this equation is almost identical to the equation in example 10. The only difference is the negative in front of the 2x. Let’s see what kind of an effect this has on our graph.

  The negative in front of the 2x reverses the line’s direction. When we write the equation −2x + 4y = 8 in y = mx + b form, it’s . Now it’s rising instead of falling. The slope of the line in example 9 is positive and its y-intercept is 2. When we write the equation 2x + 4y = 8 in y = mx + b form it’s . The slope of the line in example 8 is negative and its y

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  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  The picture itself is called the graph of the equation. x y 0 −1 5 9 4 7 2 3 Answers 4. (0, 1) and (2, 9) 4. Which of the following ordered pairs is a solution to y = 4x + 1? (0, 1), (3, 11), (2, 9) Chapter 3 Graphing Linear Equations in Two Variables 260 EXAMPLE 5 Graph the solution set for x + y = 5. Solution We know that an infinite number of ordered pairs are solutions to the equation x + y = 5. We can’t possibly list them all. What we can do is list a few of them and see if there is any pattern to their graphs. Some ordered pairs that are solutions to x + y = 5 are (0, 5), (2, 3), (3, 2), (5, 0). The graph of each is shown in Figure 1. Now, by passing a straight line through these points we can graph the solution set for the equation x + y = 5. The graph of the solution set for x + y = 5 is shown in Figure 2. Every ordered pair that satisfies x + y = 5 has its graph on the line, and any point on the line has coordinates that satisfy the equation. So there is a one-to-one correspondence between points on the line and solutions to the equation. To summarize, the set of all points of a graph that satisfy an equation is the equation's solution set. Here is the precise definition for a linear equation in two variables. DEFINITION linear equation in two variables Any equation that can be put in the form Ax + By = C, where A, B, and C are real numbers and A and B are not both 0, is called a linear equation in two variables. The graph of any equation of this form is a straight line (that is why these equations are called “linear”). The form Ax + By = C is called standard form. To graph a linear equation in two variables, as we did in Example 5, we simply graph its solution set; that is, we draw a line through all the points whose coordinates satisfy the equation. Here are the steps to follow: 5. Graph the equation x + y = 3.

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