Introduction to Graphs
A graph is a visual representation of data or a relationship between two quantities. Graphs make it easy to understand trends, compare values, and identify patterns that are difficult to see in raw numbers.
In Class 8 NCERT Maths, you study different types of graphs used for representing data: bar graphs, pie charts, histograms, and line graphs. You also learn to plot points on the Cartesian plane using ordered pairs (x, y).
Graphs are used in every field — science, economics, geography, sports, and daily life. A temperature chart in a hospital, a sales report in a company, and a weather forecast all use graphs.
Understanding how to read, interpret, and draw graphs is a fundamental skill in mathematics. This chapter lays the foundation for coordinate geometry, which you will study in detail in Class 9.
What is Introduction to Graphs?
Definition: A graph is a diagram that shows the relationship between two or more quantities using lines, bars, points, or sectors.
The Cartesian Plane:
- A flat surface with two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
- The point where they meet is the origin (0, 0).
- Every point on the plane is represented by an ordered pair (x, y).
- The x-value is called the abscissa; the y-value is the ordinate.
Types of graphs:
- Bar graph — uses rectangular bars to compare quantities.
- Pie chart (Circle graph) — uses sectors of a circle to show proportions.
- Histogram — uses bars for grouped (continuous) data with no gaps.
- Line graph — uses points connected by lines to show change over time.
- Linear graph — a straight line representing a linear relationship y = mx + c.
Introduction to Graphs Formula
Plotting a point (x, y):
- Start at the origin (0, 0).
- Move x units along the x-axis (right if positive, left if negative).
- Move y units parallel to the y-axis (up if positive, down if negative).
- Mark the point.
Pie chart sector angle:
Sector angle = (Value / Total) × 360°
Frequency for histogram:
Class width = Upper limit − Lower limit
Scale: Choose an appropriate scale for the axes so that all data fits within the graph. Common scales: 1 cm = 10 units, 1 cm = 5 units, etc.
Derivation and Proof
Why do we use the Cartesian plane?
The Cartesian plane (named after René Descartes) provides a systematic way to represent every point in a plane using just two numbers. This connects algebra to geometry:
- An equation like y = 2x can be represented as a set of points (0,0), (1,2), (2,4), (3,6), ... on the plane.
- These points, when connected, form a straight line.
- This is the foundation of coordinate geometry.
Why different types of graphs?
- Bar graphs are best for comparing discrete categories (e.g., favourite sports of students).
- Histograms are for continuous data grouped into intervals (e.g., marks distribution).
- Pie charts show proportions of a whole (e.g., budget allocation).
- Line graphs show change over time (e.g., temperature throughout a day).
Types and Properties
1. Bar Graph:
- Equal-width bars with gaps between them.
- Height of each bar = value of that category.
- Used for discrete (non-continuous) data.
2. Histogram:
- Bars touch each other (no gaps) because data is continuous.
- Each bar represents a class interval.
- Height = frequency of that interval.
3. Pie Chart:
- A circle divided into sectors.
- Each sector’s angle is proportional to the value it represents.
- Total angle = 360°.
4. Line Graph:
- Points plotted and joined by straight lines.
- Shows trends over time (increase, decrease, or constant).
5. Linear Graph:
- A straight line on the Cartesian plane.
- Represents a linear equation y = mx + c.
- m = slope (gradient), c = y-intercept.
Solved Examples
Example 1: Example 1: Plotting points on the Cartesian plane
Problem: Plot the points A(2, 3), B(−1, 4), C(−3, −2), and D(4, −1) on the Cartesian plane. Identify the quadrant of each.
Solution:
- A(2, 3): x = 2 (right), y = 3 (up) → Quadrant I
- B(−1, 4): x = −1 (left), y = 4 (up) → Quadrant II
- C(−3, −2): x = −3 (left), y = −2 (down) → Quadrant III
- D(4, −1): x = 4 (right), y = −1 (down) → Quadrant IV
Answer: A is in Quadrant I, B in Quadrant II, C in Quadrant III, D in Quadrant IV.
Example 2: Example 2: Reading a bar graph
Problem: A bar graph shows the number of books read by 5 students: Amit = 12, Priya = 8, Ravi = 15, Sita = 10, Vikram = 6. Who read the most books? What is the total?
Solution:
- Most books: Ravi (15)
- Total = 12 + 8 + 15 + 10 + 6 = 51 books
Answer: Ravi read the most (15 books). Total = 51 books.
Example 3: Example 3: Pie chart sector angles
Problem: A student spends time as follows: Study = 6 hours, Play = 3 hours, Sleep = 9 hours, Other = 6 hours. Find the sector angles for a pie chart.
Solution:
- Total = 6 + 3 + 9 + 6 = 24 hours
- Study: (6/24) × 360° = 90°
- Play: (3/24) × 360° = 45°
- Sleep: (9/24) × 360° = 135°
- Other: (6/24) × 360° = 90°
- Check: 90 + 45 + 135 + 90 = 360° ✓
Answer: Study = 90°, Play = 45°, Sleep = 135°, Other = 90°.
Example 4: Example 4: Drawing a line graph
Problem: The temperature of a city at different times: 6 AM = 18°C, 9 AM = 22°C, 12 PM = 30°C, 3 PM = 33°C, 6 PM = 27°C, 9 PM = 21°C. Describe the trend.
Solution:
- Plot time on x-axis and temperature on y-axis.
- Points: (6, 18), (9, 22), (12, 30), (15, 33), (18, 27), (21, 21)
- Connect with lines.
Trend: Temperature rises from 6 AM to 3 PM (peak at 33°C), then falls from 3 PM to 9 PM.
Example 5: Example 5: Linear graph from equation
Problem: Draw the graph of y = 2x for x = 0, 1, 2, 3, 4.
Solution:
- x = 0: y = 0 → (0, 0)
- x = 1: y = 2 → (1, 2)
- x = 2: y = 4 → (2, 4)
- x = 3: y = 6 → (3, 6)
- x = 4: y = 8 → (4, 8)
These points lie on a straight line passing through the origin.
Answer: The graph of y = 2x is a straight line through (0, 0) with slope 2.
Example 6: Example 6: Identifying graph type
Problem: Which type of graph is most suitable for: (a) Comparing populations of 5 cities, (b) Showing how temperature changes during a day, (c) Showing the distribution of marks in a class?
Solution:
- (a) Comparing discrete categories → Bar graph
- (b) Change over time → Line graph
- (c) Grouped continuous data → Histogram
Example 7: Example 7: Reading coordinates from a graph
Problem: A point is 3 units to the right of the origin and 5 units above it. What are its coordinates?
Solution:
- 3 units right = x = 3
- 5 units up = y = 5
- Coordinates = (3, 5)
Answer: The point is (3, 5).
Example 8: Example 8: Points on the axes
Problem: Where do the points (0, 4), (5, 0), and (0, 0) lie?
Solution:
- (0, 4): x = 0, so it lies on the y-axis
- (5, 0): y = 0, so it lies on the x-axis
- (0, 0): it is the origin
Example 9: Example 9: Histogram class intervals
Problem: The marks of 30 students are grouped: 0–10 (4 students), 10–20 (7), 20–30 (10), 30–40 (6), 40–50 (3). Describe how to draw the histogram.
Solution:
- x-axis: Marks (class intervals); y-axis: Number of students (frequency).
- Draw bars: width = class width (10), heights = 4, 7, 10, 6, 3.
- Bars touch each other (no gaps) since data is continuous.
- The tallest bar (20–30) shows the most common marks range.
Example 10: Example 10: Distance-time graph
Problem: A car travels at constant speed. After 1 hour it has covered 60 km, after 2 hours 120 km, after 3 hours 180 km. Plot the distance-time graph. What does the graph look like?
Solution:
- Points: (0, 0), (1, 60), (2, 120), (3, 180)
- These lie on the line y = 60x.
- The graph is a straight line through the origin.
- The slope (60) represents the speed of the car in km/h.
Answer: The graph is a straight line with slope 60, representing uniform speed.
Real-World Applications
Science: Line graphs show how physical quantities change with time — temperature, pressure, speed, and growth of organisms.
Economics: Bar graphs and pie charts display budget allocations, GDP comparisons, trade data, and market share.
Geography: Climate data (rainfall, temperature) is represented using line graphs and bar graphs.
Sports: Performance statistics, scoring trends, and player comparisons use bar and line graphs.
Medicine: Patient temperature charts, heart rate monitors, and epidemic spread curves are all graphs.
Navigation: Maps use coordinate systems derived from the Cartesian plane to locate positions.
Key Points to Remember
- A graph visually represents data or a relationship between quantities.
- The Cartesian plane has an x-axis (horizontal) and y-axis (vertical) meeting at the origin (0, 0).
- Every point is represented as an ordered pair (x, y).
- The plane has four quadrants: I (+, +), II (−, +), III (−, −), IV (+, −).
- Bar graph: compares discrete categories using bars with gaps.
- Histogram: shows continuous grouped data using touching bars.
- Pie chart: shows proportions using sectors; total angle = 360°.
- Line graph: shows change over time using connected points.
- Linear graph: a straight line representing y = mx + c.
- Always label axes, use an appropriate scale, and give the graph a title.
Practice Problems
- Plot the points (3, 2), (−2, 4), (−1, −3), and (5, −2) on the Cartesian plane and identify their quadrants.
- A school has 200 students in sports: Cricket = 60, Football = 50, Badminton = 40, Tennis = 30, Swimming = 20. Find the sector angles for a pie chart.
- Draw the graph of y = x + 3 for x = −2, −1, 0, 1, 2, 3.
- The monthly rainfall (in mm) is: Jan = 20, Feb = 15, Mar = 10, Apr = 5, May = 30, Jun = 80. Draw a bar graph and identify the wettest month.
- Marks distribution: 0–20 (5), 20–40 (12), 40–60 (18), 60–80 (10), 80–100 (5). Draw a histogram.
- Write the coordinates of 5 points on the line y = 3x − 1.
- A point lies on the x-axis at a distance of 7 units from the origin on the positive side. What are its coordinates?
- Name the type of graph most suitable for showing how the population of India changed from 1950 to 2020.
Frequently Asked Questions
Q1. What is a graph in mathematics?
A graph is a visual representation of data or a mathematical relationship using points, lines, bars, or sectors on a coordinate system or chart.
Q2. What is the Cartesian plane?
The Cartesian plane is a flat surface with two perpendicular number lines — the horizontal x-axis and the vertical y-axis — intersecting at the origin (0, 0). Every point is identified by an ordered pair (x, y).
Q3. What is the difference between a bar graph and a histogram?
A bar graph has gaps between bars and represents discrete data. A histogram has no gaps between bars and represents continuous grouped data.
Q4. How do you calculate the sector angle for a pie chart?
Sector angle = (Value / Total) × 360°.
Q5. What are the four quadrants of the Cartesian plane?
Quadrant I: x > 0, y > 0. Quadrant II: x < 0, y > 0. Quadrant III: x < 0, y < 0. Quadrant IV: x > 0, y < 0.
Q6. Where does a point with coordinates (0, y) lie?
On the y-axis. Any point with x = 0 lies on the y-axis.
Q7. What is a linear graph?
A linear graph is a straight line on the Cartesian plane. It represents a linear equation of the form y = mx + c, where m is the slope and c is the y-intercept.
Q8. When should you use a line graph?
Use a line graph when you want to show how a quantity changes over time — for example, temperature over hours, stock prices over days, or weight over weeks.
Q9. What does the slope of a line graph represent?
The slope represents the rate of change. In a distance-time graph, the slope gives speed. In a cost-quantity graph, the slope gives the price per unit.
Q10. What is the origin?
The origin is the point (0, 0) where the x-axis and y-axis intersect. It is the reference point for all coordinates.
