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Class 8 — Mathematics (NCERT)

Introduction to Graphs

Class 8

  • ✓By the end of this lesson students will be able to understand the purpose and utility of graphs.
  • ✓By the end of this lesson students will be able to interpret and draw bar graphs and line graphs.
  • ✓By the end of this lesson students will be able to understand the Cartesian coordinate system and plot points.
  • ✓By the end of this lesson students will be able to identify and draw linear graphs from given data or equations.

Key concepts

Data and its Representation

Data refers to a collection of facts, such as numbers, words, measurements, observations, or even just descriptions of things. To make sense of large amounts of data, we often represent it visually using graphs. Graphs help us to compare, analyse, and interpret data easily.

Bar Graph

A bar graph is a visual representation of data using bars of uniform width. The height or length of each bar is proportional to the value it represents. Bar graphs are useful for comparing different categories of data.

Line Graph

A line graph displays data that changes continuously over periods of time. It uses line segments to connect points representing the data. Line graphs are particularly useful for showing trends, such as changes in temperature over a day or sales over a month.

Coordinates

To locate a point on a plane, we use a system called the Cartesian coordinate system. It consists of two perpendicular number lines: the horizontal line is called the X-axis, and the vertical line is called the Y-axis. The point where these two axes intersect is called the origin (0,0). Any point on the plane is represented by an ordered pair (x, y), where 'x' is the x-coordinate (abscissa) and 'y' is the y-coordinate (ordinate). The x-coordinate tells us the horizontal distance from the y-axis, and the y-coordinate tells us the vertical distance from the x-axis.

(x, y)
Linear Graph

A linear graph is a line graph where all the points plotted lie on a single straight line. This indicates a constant rate of change or a linear relationship between the two quantities being plotted. For example, if the distance covered by a car increases uniformly with time, its distance-time graph will be a linear graph.

Key facts to remember

  • 1Graphs are visual representations of data, making it easier to understand and interpret.
  • 2A bar graph uses bars of uniform width, where the height of each bar represents its value, ideal for comparing categories.
  • 3A line graph uses line segments to connect points, showing trends or changes over time.
  • 4The Cartesian coordinate system uses two perpendicular axes (X-axis and Y-axis) to locate points on a plane.
  • 5The origin is the point (0,0) where the X-axis and Y-axis intersect.
  • 6An ordered pair (x, y) represents a point, where 'x' is the x-coordinate (abscissa) and 'y' is the y-coordinate (ordinate).
  • 7A linear graph is a line graph where all the plotted points lie on a single straight line, indicating a linear relationship.
  • 8Always label the axes and provide a title for any graph drawn.

Worked examples

Example 1

The following table shows the number of students in Class 8 of a school over five academic years. Draw a line graph for this data.\n\n| Year | Number of Students |\n|------|--------------------|\n| 2018 | 200 |\n| 2019 | 250 |\n| 2020 | 220 |\n| 2021 | 280 |\n| 2022 | 300 |

IStep 1: Draw two perpendicular axes on a graph paper. Label the horizontal axis as 'Year' and the vertical axis as 'Number of Students'.
IIStep 2: Choose an appropriate scale for both axes. For the 'Year' axis, mark the years 2018, 2019, 2020, 2021, 2022 at equal intervals. For the 'Number of Students' axis, a scale of 1 unit = 50 students would be suitable (e.g., 50, 100, 150, ...).
IIIStep 3: Plot the points corresponding to the given data. For example, for the year 2018, plot the point (2018, 200). Similarly, plot (2019, 250), (2020, 220), (2021, 280), and (2022, 300).
IVStep 4: Connect the plotted points with straight line segments. This will form the required line graph.

Answer

The line graph will show points (2018, 200), (2019, 250), (2020, 220), (2021, 280), (2022, 300) connected by line segments, illustrating the trend of student numbers over the years.

Always label your axes and provide a title for the graph for clarity.

Example 2

Plot the following points on a graph paper and identify the figure formed: A(2, 3), B(6, 3), C(6, 7), D(2, 7).

IStep 1: Draw the X-axis and Y-axis on a graph paper. Mark the origin (0,0) and choose a suitable scale (e.g., 1 unit = 1 cm).
IIStep 2: To plot point A(2, 3), start from the origin, move 2 units along the positive X-axis, then move 3 units vertically upwards parallel to the Y-axis. Mark this point as A.
IIIStep 3: Similarly, plot point B(6, 3) by moving 6 units along the positive X-axis and 3 units upwards. Mark it as B.
IVStep 4: Plot point C(6, 7) by moving 6 units along the positive X-axis and 7 units upwards. Mark it as C.
VStep 5: Plot point D(2, 7) by moving 2 units along the positive X-axis and 7 units upwards. Mark it as D.
VIStep 6: Join the points in order: A to B, B to C, C to D, and D to A. Observe the figure formed.

Answer

The figure formed by joining the points A(2, 3), B(6, 3), C(6, 7), and D(2, 7) is a rectangle.

The x-coordinate is always written first, followed by the y-coordinate in an ordered pair (x, y).

Example 3

Draw a graph for the equation y = x + 1 by taking integer values of x from -2 to 2.

IStep 1: Create a table of values for x and y using the given equation y = x + 1 for x = -2, -1, 0, 1, 2.\n\n| x | y = x + 1 | (x, y) |\n|-----|-----------|--------|\n| -2 | -2 + 1 = -1 | (-2, -1) |\n| -1 | -1 + 1 = 0 | (-1, 0) |\n| 0 | 0 + 1 = 1 | (0, 1) |\n| 1 | 1 + 1 = 2 | (1, 2) |\n| 2 | 2 + 1 = 3 | (2, 3) |
IIStep 2: Draw the X-axis and Y-axis on a graph paper, marking the origin and choosing a suitable scale for both positive and negative values.
IIIStep 3: Plot the ordered pairs obtained from the table: (-2, -1), (-1, 0), (0, 1), (1, 2), and (2, 3).
IVStep 4: Observe that all these points lie on a straight line. Draw a straight line passing through all these points. Extend the line in both directions with arrows to indicate that it continues indefinitely.

Answer

The graph for y = x + 1 is a straight line passing through the points (-2, -1), (-1, 0), (0, 1), (1, 2), and (2, 3). This is a linear graph.

For a linear equation, only two points are sufficient to draw the line, but plotting more points helps to ensure accuracy.

Common mistakes

  • ✗Confusing the x-coordinate and y-coordinate while plotting points (e.g., plotting (3,2) instead of (2,3)).
  • ✗Not choosing an appropriate scale for the axes, leading to a cramped or overly spread-out graph.
  • ✗Forgetting to label the axes or provide a title for the graph, making it difficult to understand.
  • ✗Drawing a line graph with bars or a bar graph with connected lines.
  • ✗Not using uniform width for bars in a bar graph or not maintaining equal intervals between points on an axis.

Exam tips

  • ★Always use a sharp pencil and ruler for drawing graphs to ensure neatness and accuracy.
  • ★Read the question carefully to determine the type of graph required (bar, line, or linear).
  • ★Practice plotting points accurately, as this is fundamental to all graph drawing.
  • ★When drawing a linear graph, ensure all points lie on a straight line. If they don't, recheck your calculations or plotting.

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