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| Coordinate System | Some Applications of Coordinate System | |
Chapter 15 Introduction to Graphs (Concepts)
Welcome to this fascinating chapter dedicated to the Introduction to Graphs. While previous studies in Data Handling equipped us with tools like bar graphs, pie charts, and histograms for summarizing and comparing discrete datasets, this chapter significantly broadens our horizons by introducing methods to visually represent relationships between quantities, track changes over time, and understand the structure of coordinate geometry. Graphs provide an incredibly powerful visual language for interpreting data, identifying patterns, and communicating complex information concisely.
We begin by briefly acknowledging the graphical tools already familiar to us – bar graphs for comparing distinct categories, pie charts for showing proportional parts of a whole, and histograms for displaying the frequency distribution of continuous data grouped into intervals. However, to represent relationships between two variables or locate positions precisely, we need a more structured framework. This leads us to the cornerstone concept of this chapter: the Cartesian Plane (also known as the coordinate plane). Imagine two perpendicular number lines intersecting at a point.
- The horizontal number line is called the x-axis.
- The vertical number line is called the y-axis.
- Their point of intersection is termed the origin, denoted by the coordinates $(0, 0)$.
With the Cartesian plane established, our focus shifts to specific types of graphs used to illustrate relationships and trends. A particularly important type emphasized here is the Line Graph. Line graphs are exceptionally useful for displaying data that changes continuously over a period of time or varies with respect to another continuous variable. Points representing specific data values (e.g., temperature at different times, distance covered over several hours) are plotted on the coordinate plane, and these points are then connected sequentially by straight line segments. Reading and interpreting line graphs involves analyzing the slope of the line segments – an upward slope indicates an increase, a downward slope signifies a decrease, and a horizontal line represents a stable period (no change). We learn to extract specific information by locating points on the graph and reading their corresponding values on the axes.
A special, fundamental subtype of line graph is the Linear Graph. As the name suggests, a linear graph is one where all the plotted points lie perfectly on a single straight line. This indicates a consistent, linear relationship between the two variables being plotted. Often, this relates to concepts like direct proportion (where the graph is a straight line passing through the origin, representing $y=kx$). We will learn how to plot linear graphs by creating a table of values for simple linear equations (like $y = 2x + 1$), finding several coordinate pairs $(x, y)$ that satisfy the equation, plotting these points, and drawing the straight line that connects them. While other graph types like scatter plots (showing correlation between two datasets) might be briefly mentioned, the emphasis remains on line graphs and linear graphs. Developing proficiency in reading information presented graphically – such as in distance-time graphs, temperature charts, or sales figures over time – is a core objective, enhancing our skills in data visualization and interpretation.
Coordinate System
In this chapter, we will learn how to represent relationships between quantities visually using graphs. To do this effectively, we need a way to precisely locate points in a plane. The coordinate system provides a method for describing the position of any point in a plane using a pair of numbers.
The Cartesian Coordinate System
The most widely used system for locating points in a plane is the Cartesian coordinate system, named after the brilliant French mathematician and philosopher, René Descartes. This system uses two perpendicular number lines that intersect at their zero points.
- The horizontal number line is called the x-axis. By convention, the positive direction is to the right, and the negative direction is to the left.
- The vertical number line is called the y-axis. By convention, the positive direction is upwards, and the negative direction is downwards.
- The point where the x-axis and y-axis intersect is called the origin. The numerical value at the origin on both axes is zero. The coordinates of the origin are $(0, 0)$.
These two axes define a plane called the Cartesian plane or the coordinate plane. The x-axis and the y-axis divide the coordinate plane into four regions, called quadrants.
The quadrants are numbered in a counter-clockwise direction starting from the top-right region:
- Quadrant I: In this region, both the x-coordinate and the y-coordinate of any point are positive ($x > 0, y > 0$).
- Quadrant II: In this region, the x-coordinate is negative, and the y-coordinate is positive ($x < 0, y > 0$).
- Quadrant III: In this region, both the x-coordinate and the y-coordinate are negative ($x < 0, y < 0$).
- Quadrant IV: In this region, the x-coordinate is positive, and the y-coordinate is negative ($x > 0, y < 0$).
Points that lie exactly on the x-axis or the y-axis do not belong to any quadrant.
Coordinates of a Point
Every point in the Cartesian plane can be uniquely identified and located by a pair of numbers called its coordinates. The coordinates are written as an ordered pair in the format $(x, y)$.
- The first number in the pair, $x$, is called the x-coordinate or the abscissa. It represents the perpendicular distance of the point from the y-axis. A positive x-coordinate means the point is to the right of the y-axis, and a negative x-coordinate means it is to the left. The x-coordinate is measured along the x-axis.
- The second number in the pair, $y$, is called the y-coordinate or the ordinate. It represents the perpendicular distance of the point from the x-axis. A positive y-coordinate means the point is above the x-axis, and a negative y-coordinate means it is below. The y-coordinate is measured along the y-axis.
To plot a point with coordinates $(x, y)$ on the Cartesian plane:
- Start at the origin $(0,0)$.
- Move $x$ units horizontally along the x-axis. Move to the right if $x$ is positive, and to the left if $x$ is negative. If $x$ is zero, stay on the y-axis.
- From the position reached in step 2, move $y$ units vertically parallel to the y-axis. Move upwards if $y$ is positive, and downwards if $y$ is negative. If $y$ is zero, stay on the x-axis.
- The point you arrive at is the location of the point with coordinates $(x, y)$.
Example 1. Plot the points A(2, 3), B(-3, 1), C(0, -2), D(-2, -4), E(4, 0) on a graph sheet.
Answer:
First, draw the x-axis and y-axis as two perpendicular lines intersecting at the origin. Mark a suitable scale on both axes (e.g., each grid box represents 1 unit).
- A(2, 3): Start at (0,0). Move 2 units right along the x-axis. From there, move 3 units up parallel to the y-axis. Mark the point and label it A.
- B(-3, 1): Start at (0,0). Move 3 units left along the x-axis. From there, move 1 unit up parallel to the y-axis. Mark the point and label it B.
- C(0, -2): Start at (0,0). The x-coordinate is 0, so do not move horizontally. Move 2 units down along the y-axis. Mark the point and label it C. This point is on the y-axis.
- D(-2, -4): Start at (0,0). Move 2 units left along the x-axis. From there, move 4 units down parallel to the y-axis. Mark the point and label it D.
- E(4, 0): Start at (0,0). Move 4 units right along the x-axis. The y-coordinate is 0, so do not move vertically. Mark the point and label it E. This point is on the x-axis.
Coordinates of Points on the Axes
Points that lie on the coordinate axes have one of their coordinates equal to zero.
- Any point on the x-axis has its y-coordinate equal to 0. Its coordinates are of the form $(x, 0)$, where $x$ is the distance from the origin along the x-axis.
- Any point on the y-axis has its x-coordinate equal to 0. Its coordinates are of the form $(0, y)$, where $y$ is the distance from the origin along the y-axis.
- The origin is the point where the x-axis and y-axis intersect. Its coordinates are $(0, 0)$.
Example 2. State the quadrant in which each of the following points lies, or state if it lies on an axis:
(3, -1), (-2, 5), (0, 4), (-4, -3), (5, 2), (-6, 0), (0, 0)
Answer:
- (3, -1): The x-coordinate is positive (3), and the y-coordinate is negative (-1). A point with positive x and negative y lies in Quadrant IV.
- (-2, 5): The x-coordinate is negative (-2), and the y-coordinate is positive (5). A point with negative x and positive y lies in Quadrant II.
- (0, 4): The x-coordinate is 0. A point with an x-coordinate of 0 lies on the y-axis. So, it lies on the y-axis.
- (-4, -3): The x-coordinate is negative (-4), and the y-coordinate is negative (-3). A point with negative x and negative y lies in Quadrant III.
- (5, 2): The x-coordinate is positive (5), and the y-coordinate is positive (2). A point with positive x and positive y lies in Quadrant I.
- (-6, 0): The y-coordinate is 0. A point with a y-coordinate of 0 lies on the x-axis. So, it lies on the x-axis.
- (0, 0): Both coordinates are 0. This is the origin, which lies on both axes but is not in any quadrant.
Some Applications of Coordinate System
The Cartesian coordinate system provides a framework for precisely locating points in a plane. This ability to locate points is fundamental to drawing graphs. Graphs are visual tools used to represent and analyse relationships between quantities. They make it easier to understand patterns and trends in data or equations.
Types of Graphs
You have already encountered some types of graphs in previous chapters or classes, especially in Data Handling (Chapter 5). These include:
- Bar Graph: Uses bars to compare discrete categories.
- Double Bar Graph: Uses pairs of bars to compare two sets of data for the same categories.
- Histogram: Uses adjacent bars to represent the frequency distribution of continuous data grouped into class intervals.
In this chapter, we will focus on graphs that show the relationship between two quantities represented by coordinate points. Two important types of such graphs are line graphs and linear graphs.
- Line Graph: A graph that uses points connected by line segments to show how a quantity changes over time or across categories. It is useful for showing trends.
- Linear Graph: A special type of line graph where all the plotted points lie on a single straight line. This indicates a constant rate of change between the two quantities, representing a linear relationship.
Drawing a Line Graph
To draw a line graph, you are typically given pairs of values (data points) that can be represented as ordered pairs $(x, y)$. You plot these points on the coordinate plane and connect them with line segments.
Example 1. The following table shows the temperature of a city on different days of a week.
| Day | Temperature ($^\circ$C) |
|---|---|
| Monday | 35 |
| Tuesday | 38 |
| Wednesday | 36 |
| Thursday | 39 |
| Friday | 37 |
Draw a line graph to represent this data.
Answer:
Given data: Pairs of (Day, Temperature).
Step 1: Draw the horizontal and vertical axes on a graph sheet. Label the horizontal axis (x-axis) to represent 'Day' and the vertical axis (y-axis) to represent 'Temperature ($^\circ$C)'.
Step 2: Choose suitable scales for both axes. For the x-axis, mark the days at equal intervals. For the y-axis, the temperature ranges from 35$^\circ$C to 39$^\circ$C. A scale starting from a base value near the minimum temperature (e.g., 30$^\circ$C) with a kink at the origin, and then uniform intervals (e.g., 1 unit = 1$^\circ$C or 2$^\circ$C) upwards is suitable to make the graph clear. Let's use a kink and then intervals of 1$^\circ$C per unit above 30$^\circ$C.
Step 3: Plot the points corresponding to each pair of (Day, Temperature) values on the graph.
- Monday: (Monday, 35)
- Tuesday: (Tuesday, 38)
- Wednesday: (Wednesday, 36)
- Thursday: (Thursday, 39)
- Friday: (Friday, 37)
Step 4: Connect the plotted points with straight line segments in the order of the days.
Step 5: Give the graph a suitable title: "Daily Temperature of a City".
The line graph shows how the temperature changed from day to day.
Drawing a Linear Graph
A linear graph represents a relationship between two quantities where the graph is a straight line. This happens when the quantities are directly proportional or related by a simple linear equation (like $y = mx + c$). If you are given pairs of values that follow a linear pattern, plotting them will result in points that lie on a straight line.
To draw a linear graph:
- Draw the x-axis and y-axis.
- Choose suitable scales for both axes based on the range of values of the quantities.
- Plot the given pairs of values as points on the coordinate plane.
- Check if all the plotted points lie on the same straight line. If they do, draw a straight line passing through them. Extend the line beyond the points if appropriate for the context.
- Label axes and give a title.
Example 2. Plot the points A(0, 0), B(1, 2), C(2, 4), D(3, 6) on a graph sheet and check if they lie on a straight line. If they do, draw the line.
Answer:
Given points: A(0, 0), B(1, 2), C(2, 4), D(3, 6).
Step 1: Draw the x and y axes. Choose scales (e.g., 1 unit per box on both axes is fine here).
Step 2: Plot the given points on the coordinate plane based on their coordinates.
Step 3: Use a ruler to see if the points lie on a single straight line.
Yes, the points A, B, C, and D all lie on a straight line that passes through the origin.
Step 4: Draw a straight line passing through these points. This is the linear graph representing the relationship between the x and y coordinates of these points.
Observation: Notice the relationship between the x and y coordinates: for each point, the y-coordinate is twice the x-coordinate ($0 = 2 \times 0$, $2 = 2 \times 1$, $4 = 2 \times 2$, $6 = 2 \times 3$). The equation of this line is $y = 2x$, which is a linear equation.
Graphs for Real-World Situations
Graphs are very effective in visualising relationships between two quantities in real-world scenarios, especially those involving constant rates, which result in linear relationships.
Example 3. The following table shows the distance travelled by a car at a constant speed over different periods of time.
| Time (in hours) | Distance (in km) |
|---|---|
| 0 | 0 |
| 1 | 50 |
| 2 | 100 |
| 3 | 150 |
Draw a graph for this data. Is it a linear graph? Use the graph to find the distance covered in $1.5$ hours.
Answer:
Given data: Pairs of (Time, Distance).
Step 1: Draw horizontal and vertical axes. Label the horizontal axis as 'Time (in hours)' and the vertical axis as 'Distance (in km)'.
Step 2: Choose suitable scales. For the Time axis, use 1 unit = 1 hour. For the Distance axis, the values increase in steps of 50 km, so a scale of 1 unit = 50 km is suitable.
Step 3: Plot the points corresponding to the (Time, Distance) pairs:
- (0, 0)
- (1, 50)
- (2, 100)
- (3, 150)
Step 4: Check if the points lie on a straight line. Since the speed is constant, the distance is directly proportional to the time, which is a linear relationship. The points (0,0), (1,50), (2,100), (3,150) lie on a straight line.
Step 5: Draw a straight line passing through these points. This is the graph of Distance vs Time.
Yes, it is a linear graph because the points lie on a straight line.
Step 6: Use the graph to find the distance covered in 1.5 hours. Locate 1.5 on the Time axis (midway between 1 and 2). Draw a vertical line from 1.5 up to the graph line. From the point where the vertical line meets the graph line, draw a horizontal line to the Distance axis.
Looking at the Distance axis, the horizontal line meets it at 75 km (midway between 50 and 100 km, which is consistent with a constant speed of 50 km/h).
The distance covered in 1.5 hours is 75 km.
Check: Speed = $\frac{50 \text{ km}}{1 \text{ hour}} = 50$ km/h. Distance = Speed $\times$ Time $= 50 \text{ km/h} \times 1.5 \text{ hours} = 75$ km. The result from the graph is consistent with the calculation.