Mathematics
Line Graphs
Line graphs are a type of graph that displays data as points connected by straight lines. They are commonly used to show how a particular variable changes over time or in relation to another variable. The x-axis represents the independent variable, while the y-axis represents the dependent variable.
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11 Key excerpts on "Line Graphs"
eBook - PDF- 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.
eBook - PDFEssential Maths
for Business and Management
- Clare Morris(Author)
- 2007(Publication Date)
- Bloomsbury Academic(Publisher)
There are a couple of additional points that are worth noting in relation to graphs. First, you may have spotted in the hyperbola example above that, although the number of boxes, n , has to be a whole number, we have plotted the graph as if n could take on any value. Thus, strictly speaking, only the points on the graph corresponding to whole-number values of n have any meaning. You’ll find that this happens quite often – the points are joined so that we can see the general trend of the graph, but you need to remember that not all values that you might read off from the graph actually make sense in practical terms. Second, not all graphs represent relationships that have a neat algebraic form. Very often, we use a graph just to give a pictorial representation of numerical information. This type of graph is particularly common when we want to show how a set of figures has moved over time. So, for example, a company that wishes to show how its share price, averaged out over a month, has varied over a two-year period, might choose to display the information in a graph like Figure 4.12. This is called a time-series graph. This plot shows very clearly the way that the share price has dropped and then picked up again to more than recover its value. It would be quite misleading (as well as almost impossible) to try to plot any kind of smooth curve through this set of points – the best way to show the overall trend is simply to join the individual points by straight lines. Nor do the values in between the monthly points have any meaning – the line joining them is merely there to emphasise the movement of the figures. 70 Essential Maths Figure 4.12 Time-series graph Some terminology and notation Throughout this chapter we’ve been referring to graphs as showing relationships – the relationship between number of items sold and revenue, the relationship between x and y , and so on. An alternative and more technical term that you may come across is func-tion .
eBook - ePub- John Bird(Author)
- 2021(Publication Date)
- Routledge(Publisher)
Chapter 17Straight Line Graphs
Why it is important to understand: Straight Line GraphsGraphs 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 .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.Figure 17.1When 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.
eBook - ePubStatistical Literacy at School
Growth and Goals
- Jane M. Watson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
Points, wherever found, are often considered as representing single entities. In fact this is true in a scattergram but the entity has two defining characteristics that determine its location in a setting where the trend over many entities is important. Following on from work with “people graphs” for very young children to report frequency by having them line up behind the characteristic with which they are associated, people graphs also can be very helpful with older students to help distinguish two variables and their relationship to the point represented. 23 Laying out the perpendicular axes, with the origin, on the floor or playground, students can stand on a point, facing the origin, and extend their arms at right angles to indicate the measurements on each axis that determine where they are standing on the scattergram. Anecdotally this has been found to be a very useful technique. 3.6 Graph Interpretation: the Case of Bar Charts Bar charts, perhaps as derivatives of pictographs, are the graphs met most frequently by students in the elementary school years. As such they provide links to other basic aspects of the mathematics curriculum, particularly for younger children. One-to-one correspondence, addition, and subtraction, for example, are involved in the basic interpretation of bar charts, and judicious questioning can also focus discussion on issues related to the building of proto-statistical intuitions. Consider, for example, the bar graph in Fig. 3.15 that records how children in class arrive at school. This particular version of the bar graph, which is often a basis for this theme, has moveable bars that disappear in a slit at the base line. 24 Asking students what they can tell from the graph allows for a display of the natural starting points for graph interpretation. Some students, often FIG
eBook - ePub- Lisa Healey(Author)
- 2021(Publication Date)
- Chemeketa Press(Publisher)
In this section, we will see that, even without using numbers, a graph is a mathematical tool that can describe a wide variety of relationships. For example, there is a relationship between outdoor temperatures over the course of a year and the retail sales of ice cream. We can describe this relationship in a general way using a qualitative graph. As you study this section, you will learn to:♦ Read and interpret qualitative graphs ♦ Identify independent and dependent variables ♦ Identify and interpret an intercept of a graph ♦ Identify increasing and decreasing curves ♦ Sketch qualitative graphsA. Reading a Qualitative Graph
Both qualitative and quantitative graphs can have two axes and show the relationship between two variables. We also read both types of graph from left to right — just like a sentence. The difference is thatquantitative graphshave numerical increments on the axes (scaling and tick marks), whilequalitative graphsonly illustrate the general relationship between two variables.Example 1Use the qualitative graph, Figure 1 , and the quantitative graph, Figure 2 , to answer the following questions.Figure 1. The sale of ice cream at Joe’s Café (a qualitative graph).Figure 2. The population of Portland, Oregon (a quantitative graph).1.What does the qualitative graph tell us about ice cream sales at Joe’s Café? Do we know how many servings were sold in June?2.What does the quantitative graph tell us about the population of Portland, Oregon? What was the population in 1930?Solutions1.Ice cream sales are lowest at the beginning and at the end of the year and highest during the middle months. We cannot tell from this graph exactly how many servings are sold in any given month.2.The population of Portland, Oregon, has been increasing since 1850, except for a slight decrease in the 1950s and 1970s. The population in 1930 was about 300,000.B. Independent and Dependent Variables
A qualitative graph is a visual description of the relationship between two variables. The graph tells a “story” about how one quantity is determined or influenced by another quantity. For example, the number of calories one consumes in a week determines the number of pounds one will lose (or gain) that week. Another way to say this is that the change in a person’s weight is dependent on the number of calories they consume.
- H. Davies, H.G. Davies, G.A. Hicks(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
Chapter 9Determinants and matrices
9.1 Representation of data
The relationship between two quantities in engineering or science can be expressed graphically, and usually by a formula or equation.(a) Representation of data with a formula or equation
A formula or equation is a convenient way of expressing the relationship between two quantities. In equations such as y = 3x 2 + 2x – 1, x is called the independent variable. y is the dependent variable, so called because its value depends upon the x value. Such equations produce pairs of (x , y ) values which can be used as Cartesian coordinates.In equations such as r = 2 cos θ , θ is the independent variable and r the dependent variable, the pairs of (r , θ ) values produced are called polar coordinates. Polar graphs are examined in Section 9.14 .(b) Graphical representation of two quantities related by an equation
For each value of x (or θ ) a corresponding value of y (or r ) is obtained from the equation. Each pair of values is used as the co-ordinates of a point on a plane. These points trace out a curve or straight line which is the graph representing the equation. The shape of the graph is a good indication of how one quantity depends on the other.9.2 Cartesian and polar coordinates
To produce graphs and engineering drawings either by hand or by computer it is necessary to have a method of locating the positions of points on paper or on screen. At least two numbers, called coordinates, are required to locate a point in a plane. Two systems are used:(a) Cartesian coordinates (x , y )
This is the most commonly used system. Two perpendicular datum lines are used, the horizontal line is called the x -axis, the vertical line is called the y -axis, as shown in Fig. 9.1 . The point of intersection of the two axes is called the origin O. Any point P is located by its perpendicular distance from the two axes.Fig. 9.1
eBook - PDFExperimental Methods for Science and Engineering Students
An Introduction to the Analysis and Presentation of Data
- Les Kirkup(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
3 Graphical Presentation of Data 3.1 Overview: The Importance of Graphs Our ability to absorb and process information when it is presented in the form of a picture is so good that it is natural to exploit this talent when analysing data obtained from an experiment. When data are presented pictorially, trends or fea- tures in the data can be detected that we would be unlikely to recognise if the data were given only in tabular form. This is especially true in situations where a set of data consists of hundreds or thousands of values, which is a common occurrence when a computer is used to assist in data gathering. Additionally, a pictorial representation of data in the form of a graph is an excellent way to summarise many of the important features of an experiment. A graph can indicate: (i) the quantities being studied (ii) the range of values obtained through measurement (iii) the uncertainty in each value (iv) the existence or absence of a trend in the data gathered (for example, plotted points may lie in a straight line or a curve, or may appear to be scattered randomly across the graph) (v) which plotted points do not follow the general trend exhibited by most of the data. x–y graphs (also known as scatter plots or Cartesian coordinate graphs) are used extensively in science and engineering to present experimental data, and it is those that we will concentrate on in this chapter. 3.2 Plotting Graphs An x–y graph possesses horizontal and vertical axes, referred to as the x and y axes, respectively. Each point plotted on the graph is specified by a pair of numbers termed the coordinates of the point. For example, point A in Figure 3.1 has the coordinates x = 20, y = 50. The coordinates of the point may be written in shorthand as (x,y), which in the case of point A on Figure 3.1 would be (20,50). To assist in the accurate plotting of data, graph paper may be used on which are drawn evenly spaced vertical and horizontal gridlines as shown in Figure 3.1.
eBook - PDF- Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
123 Macrovector/Shutterstock.com Chapter 4 Graphing Linear Equations in Two Variables CHAPTER CONTENTS Section 4.1 Properties of the Rectangular Coordinate System Section 4.2 Interpretations of Graphs Section 4.3 Graphing Linear Equations Section 4.4 Slope and Marginal Change Section 4.5 Equations of Lines in Statistics 124 CHAPTER 4 • Graphing Linear Equations in Two Variables SECTION 4.1 Properties of the Rectangular Coordinate System We mentioned earlier that what commonly comes to mind when one hears the word “statistics” is formulas. That served as motivation for us to examine the arithmetic and algebraic tools that are useful in statistics. Another topic that comes to mind when one hears the word “statistics” is charts and graphs. Besides being formula driven, statistics is quite visual. As a result, we now turn our attention to some of the graphical tools that prepare us for statistics. In Chapter 1, we saw how we can plot numbers that are in a single set of data using the real number line. The type of data in a single group is called univariate data . In studying statistics, we often encounter statistical data that occur in pairs. Such paired data are found in many situations. College entrance exam scores could be paired with grade-point averages, air temperature and the heat index could be paired, and the number of cars sold nationwide could be coupled with the year of sale. In each of these cases, there are two variables involved. Consequently, we call this paired data bivariate data . Objectives 1 Plot Points on a Rectangular Coordinate System 2 Read Ordered Pairs from a Graph 3 Use Scatterplots in Applications 4 Construct and Interpret Time Series Scatterplots ■ ■ A set of data consisting of observations on a single characteristic is called univariate data . ■ ■ A set of data in which the observations occur in pairs is called bivariate data .
eBook - PDFMathematics NQF2 SB
TVET FIRST
- M Van Rensburg, I Mapaling A Thorne(Authors)
- 2017(Publication Date)
- Macmillan(Publisher)
58 Module 3 Topic 2: Functions and algebra Graphs of functions Module 3 Learning Outcomes This module will show you how to do the following: • Unit 3.1: Generate graphs by means of point-by-point plotting using, or supported by, available technology. • Unit 3.2: Define functions. • Unit 3.2: Identify characteristics of functions. • Units 3.3 to 3.7: Generalise the effects of the parameters a and q on the generated graphs of functions. • Units 3.3 to 3.7: Use the generated graphs to make and test conjectures. • Units 3.3 to 3.7: Sketch graphs and find equations of graphs for certain functions. Unit 3.1: Introduction to graphs A graph is a useful way to represent data visually and it enables us to easily see the relationship between the variables we are considering. A graph is drawn on the Cartesian plane , which is also known as a coordinate plane. –5 –4 –3 –2 –1 1 2 3 4 5 5 4 3 2 1 –1 –2 –3 –4 –5 Quadrant I Quadrant II Quadrant IV Quadrant III y x 0 – y – x Figure 3.1: The Cartesian plane This plane consists of a horizontal and vertical number line, with a positive and negative section that cross each other at zero. This point of intersection is called the origin . When the two axes cross each other, they form four quadrants , as shown in Figure 3.1. These are numbered I, II, III and IV in an anti-clockwise direction. The independent variable (usually x ) is plotted on the horizontal axis and the dependent variable (usually y ) is plotted on the vertical axis. It is important that the units of each axis are spaced at equal distances and marked off according to a scale when plotting graphs to ensure accuracy.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
5 MODULE Algebraic graphs On completion of this module, you should be able to: 5.1 Represent ordered number pairs (coordinates) on a Cartesian plane 5.2 Perform function notation: write a linear equation of the form y = mx + c in the form f ( x ) = mx + c and a rectangular hyperbola of the form y = c x in the form f ( x ) = c x 5.3 Draw a linear graph in the form y = mx + c with the aid of a table 5.3.1 Gradient-intercept form of a straight line: y = mx + c 5.3.2 Sketching a straight-line graph in standard form y = mx + c 5.4 Determine the gradient and y -intercept from a graph. 5.5 Draw a rectangular hyperbola: 5.5.1 Sketch a rectangular hyperbola in the form y = c x or xy = c , where c ≠ 0 5.5.2 Explain the concepts direct and inverse relation . 140 Module 5 • Algebraic graphs Introduction A graph is a diagram showing either the relationship between some variable quantities or the connections that exist between a set of points. Information can be represented in graph form, for example: • in medicine and pharmacy to work out the correct strength of drugs • to analyse or predict future markets and opportunities • to work out whether or not you are the correct healthy weight for your body height, for example the BMI (body mass index) graph • to show how the population of a country increased over a certain period, as shown on the graphs below. 1950 2000 Population 1950 2000 Population The population increased steadily over the 50 years. The population growth started off slow, but then increased rapidly over the 50 years. Different types of graph Linear graph Rectangular hyperbola 3 2 y x y = – x + 3 3 2 1 3 –1 –3 xy = 1 or y = y x 1 x Bar graph Parabola 0 2 4 Blue Brown Eye colour Frequency Green y x 1 1 –1 y = x 2 y x 141 Introductory Mathematics| Hands-On Pre-knowledge • Real numbers can be represented on a number line. –3 –2 –1 0 1 2 3 x = 3 This number line shows x = 3.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
5 MODULE Algebraic graphs On completion of this module, you should be able to: 5.1 Represent ordered number pairs (coordinates) on a Cartesian plane 5.2 Perform function notation: write a linear equation of the form y = mx + c in the form f ( x ) = mx + c and a rectangular hyperbola of the form y = c x in the form f ( x ) = c x 5.3 Draw a linear graph in the form y = mx + c with the aid of a table 5.3.1 Gradient-intercept form of a straight line: y = mx + c 5.3.2 Sketching a straight-line graph in standard form y = mx + c 5.4 Determine the gradient and y -intercept from a graph. 5.5 Draw a rectangular hyperbola: 5.5.1 Sketch a rectangular hyperbola in the form y = c x or xy = c , where c ≠ 0 5.5.2 Explain the concepts direct and inverse relation . 140 Module 5 • Algebraic graphs Introduction A graph is a diagram showing either the relationship between some variable quantities or the connections that exist between a set of points. Information can be represented in graph form, for example: • in medicine and pharmacy to work out the correct strength of drugs • to analyse or predict future markets and opportunities • to work out whether or not you are the correct healthy weight for your body height, for example the BMI (body mass index) graph • to show how the population of a country increased over a certain period, as shown on the graphs below. 1950 2000 Population 1950 2000 Population The population increased steadily over the 50 years. The population growth started off slow, but then increased rapidly over the 50 years. Different types of graph Linear graph Rectangular hyperbola 3 2 y x y = – x + 3 3 2 1 3 –1 –3 xy = 1 or y = y x 1 x Bar graph Parabola 0 2 4 Blue Brown Eye colour Frequency Green y x 1 1 –1 y = x 2 y x 141 N1 Mathematics| Hands-On! Pre-knowledge • Real numbers can be represented on a number line. –3 –2 –1 0 1 2 3 x = 3 This number line shows x = 3.
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