1. Why Use Graphs?

• They provide a quick way to see patterns, trends, and relationships in data.
• Graphs make it easier to compare numbers and identify changes over time.
• They are used in fields such as science, business, health, economics, and sports to convey important information clearly.

2. Types of Graphs

A. Bar Graphs

Bar graphs use rectangular bars to represent values. The height or length of each bar shows the quantity of each category.

• Used for: Comparing quantities across different categories (e.g., sales by product, students in each class).
• Features: Bars may be vertical or horizontal; all bars are equally spaced and have the same width.

B. Pie Charts

Pie charts show data as sectors of a circle. Each sector represents a part of the whole.

• Used for: Showing proportions or percentages in a dataset (e.g., market share, survey results).
• Features: The entire circle is 100%; each slice represents a fraction of the total.

C. Line Graphs

Line graphs use points connected by straight lines to show how a quantity changes over time or in relation to another variable.

• Used for: Tracking changes, trends, or continuous data (e.g., temperature over days, distance vs. time).
• Features: Dots or markers are connected to highlight increases, decreases, or patterns.

D. Histograms

Histograms look like bar graphs but are used for continuous data, showing frequency distribution in intervals.

• Used for: Showing how data is distributed (e.g., test scores, heights).
• Features: Bars are adjacent with no gaps, each bar represents a class interval.

E. Cartesian (Coordinate) Graphs

Cartesian graphs (also called coordinate or xy-graphs) plot pairs of numbers as points on a grid.

• Used for: Showing the relationship between two variables; graphing equations; locating points.
• Features: Uses horizontal (x-axis) and vertical (y-axis) lines, with each point at (x, y).

3. Parts of a Graph

• Axes: The two lines (x-axis and y-axis) that form the graph’s framework.
• Origin: The point where the axes intersect, labeled (0, 0).
• Scale: The numbers marked on axes, showing the size or quantity each unit represents.
• Labels: Descriptions on the axes to indicate what is being measured.
• Title: The heading that explains what the graph is about.
• Key/Legend: Explains symbols, colors, or patterns used.

4. Plotting Points on the Cartesian Plane

How to Plot Points

1. Draw the x-axis (horizontal) and y-axis (vertical) intersecting at the origin (0,0).
2. Choose a suitable scale for both axes.
3. To plot a point (a, b), move a units along the x-axis, then b units up (if b is positive) or down (if b is negative) along the y-axis.
4. Mark the point clearly with a dot and label it.

Example

• Point (3, 2): Move 3 units right, 2 units up from the origin.
• Point (−2, 4): Move 2 units left, 4 units up.
• Point (0, −5): Stay at the origin, move 5 units down.

5. Reading and Interpreting Graphs

• Find values by tracing from the data point to the axes.
• Look for patterns: is the line rising, falling, or constant?
• Compare heights of bars or lengths of sectors for relative sizes.
• Check the title and labels to understand the context.
• Look for maximum, minimum, and points where changes happen.

Worked Example

A line graph shows the temperature at noon for 7 days:
Monday: 28°C, Tuesday: 29°C, Wednesday: 32°C, Thursday: 31°C, Friday: 30°C, Saturday: 33°C, Sunday: 34°C.
Q: On which day was the highest temperature?
A: Sunday (34°C)

6. Using Graphs to Solve Problems

• Graphs help answer questions like: How much did sales increase? When was the fastest change?
• In science, graphs show how one quantity changes with another (e.g., distance over time).
• In economics, price and demand curves show market trends.
• In health, a patient’s temperature or heart rate can be tracked and interpreted using graphs.

7. Real-Life Applications of Graphs

• Weather Reports: Daily temperature and rainfall graphs help forecast weather.
• Finance: Line graphs and bar charts track stock prices and budgets.
• Sports: Bar graphs compare scores or performance over games.
• Health: Growth charts for children’s height and weight.
• Travel: Distance vs. time graphs help analyze travel routes.

8. Common Misconceptions and Errors

• Not labeling axes or using unclear scales.
• Forgetting to start bars or lines from zero.
• Plotting points inaccurately by reversing x and y values.
• Reading graphs without considering the context or title.
• Confusing correlation (two things change together) with causation (one causes the other).

Tip: Always check units, labels, and scales for accuracy and understanding.

9. Practice Problems

Basic

1. Draw a bar graph for the following data: Red – 5, Blue – 3, Green – 7, Yellow – 2.
2. Plot the points (2, 4), (−3, 1), (0, −5), (4, 0) on the coordinate plane.
3. In a pie chart, 60% of students prefer basketball, 25% football, and 15% tennis. Show this using sectors of a circle.
4. Read the value for Wednesday if a line graph shows sales of 10, 15, 12, 18, 16 for Monday to Friday.

Intermediate

5. Given a histogram of test scores with intervals: 0–10 (3 students), 10–20 (8), 20–30 (10), 30–40 (6), explain which interval had the highest number of students.
6. If the point (x, y) lies on the x-axis, what is the value of y?
7. A distance vs. time graph is a straight line sloping upwards. What does this mean?
8. Draw a line graph for the monthly rainfall (in cm): Jan – 4, Feb – 3, Mar – 6, Apr – 8, May – 7.

Advanced

9. A pie chart shows time spent in a day: Sleeping – 8 hours, School – 6 hours, Homework – 2 hours, Leisure – 4 hours, Others – 4 hours. What angle will each sector make at the center?
10. Describe what kind of graph would be best for showing: (a) a city’s population growth over 10 years, (b) favorite fruits in a class, (c) speed of a car during a race.
11. On a coordinate plane, plot and join (−2, 3), (0, 5), (2, 7), (4, 9). What pattern do you observe?
12. Why is it important to choose an appropriate scale when drawing graphs? Give an example.

10. Solutions

1. Draw vertical bars for each color with heights: Red – 5, Blue – 3, Green – 7, Yellow – 2.
2. Locate each point carefully, moving along x, then y, from the origin.
3. Draw a circle, divide into 3 sectors: Basketball – 216°, Football – 90°, Tennis – 54°.
4. Wednesday’s sales: 12 units.
5. 20–30 interval had the highest number (10 students).
6. y = 0 for any point on the x-axis.
7. A straight line upwards means constant speed (distance increases steadily over time).
8. Plot points and join with straight lines, connecting each month in order.
9. Total time = 24 hours, 1 hour = 15°. Sleeping: 120°, School: 90°, Homework: 30°, Leisure: 60°, Others: 60°.
10. (a) Line graph or bar graph, (b) Bar graph or pie chart, (c) Line graph (to show change in speed over time).
11. The points lie on a straight line, suggesting a linear relationship.
12. An inappropriate scale can make trends unclear. For example, if rainfall data ranges from 1 to 8 cm, using a scale from 0 to 100 cm would flatten the graph and hide variations.

Conclusion

Graphs are an essential part of mathematics and daily life, transforming raw numbers into clear, visual stories. By learning to draw, read, and interpret different types of graphs, you will be able to communicate information effectively, solve problems quickly, and spot important trends and patterns. With regular practice, graphs will become a powerful tool in your academic and professional journey.