Class 8 Mathematics Notes Chapter 15 (Introduction to Graphs) – Mathematics Book
Alright class, let's focus on a very important tool for understanding data: Graphs. Chapter 15, 'Introduction to Graphs', might seem simple, but mastering these concepts is crucial for data interpretation sections in many government exams. Graphs help us visualize information, making complex data easier to understand, compare, and analyze.
Chapter 15: Introduction to Graphs - Detailed Notes for Exam Preparation
1. What is a Graph?
- A graph is a visual representation of data.
- It helps in understanding the relationship between two or more quantities.
- Makes comparison easier and reveals trends or patterns in data.
2. Types of Graphs (Recap & Relevance):
While this chapter focuses heavily on line graphs and the coordinate system, remember other types:
- Bar Graph: Used for comparing discrete categories or items. Bars represent quantities. Useful for showing 'how much' or 'how many'.
- Pie Chart (Circle Graph): Used to show parts of a whole (percentages or fractions). The entire circle represents 100% or the total amount.
- Histogram: Similar to a bar graph, but used for continuous data grouped into intervals (like age groups, marks range). There are no gaps between the bars.
- Line Graph: (Focus of this Chapter) Used to show how data changes continuously over a period of time or the relationship between two changing quantities. Points are plotted and connected by lines.
3. The Cartesian Coordinate System (The Foundation):
This is the system used for plotting points and drawing line graphs.
- Axes: Two perpendicular lines:
- Horizontal Axis (x-axis): Usually represents the independent variable (like time, quantity).
- Vertical Axis (y-axis): Usually represents the dependent variable (like temperature, distance, cost).
- Origin: The point where the x-axis and y-axis intersect. Its coordinates are (0, 0).
- Coordinates: A pair of numbers that locate a specific point on the graph. Written as an ordered pair (x, y).
- x-coordinate (Abscissa): The perpendicular distance of the point from the y-axis. It tells you how far to move left or right from the origin.
- y-coordinate (Ordinate): The perpendicular distance of the point from the x-axis. It tells you how far to move up or down from the origin.
- Plotting a Point: To plot (a, b):
- Start at the origin (0, 0).
- Move 'a' units along the x-axis (right if 'a' is positive, left if negative).
- From there, move 'b' units parallel to the y-axis (up if 'b' is positive, down if negative). Mark the point.
- Important Locations:
- Any point on the x-axis has its y-coordinate equal to 0. Format: (x, 0).
- Any point on the y-axis has its x-coordinate equal to 0. Format: (0, y).
- The origin is (0, 0).
4. Line Graphs:
- Purpose: To display data that changes continuously over time or shows the relationship between two variables.
- Construction:
- Draw the axes and choose appropriate scales. Label the axes clearly.
- Plot the points corresponding to the given data pairs (e.g., (time, temperature), (overs, runs)).
- Join the plotted points consecutively using line segments.
- Interpretation:
- Upward Slope: Indicates an increase in the y-variable as the x-variable increases.
- Downward Slope: Indicates a decrease in the y-variable as the x-variable increases.
- Horizontal Line: Indicates no change in the y-variable as the x-variable increases (constant value).
- Steepness: A steeper line indicates a faster rate of change.
5. Linear Graphs:
- A line graph in which all the plotted points lie on a single straight line is called a linear graph.
- It represents a linear relationship between the two variables. This means the change in the dependent variable is directly proportional to the change in the independent variable.
- How to check if points form a linear graph: Plot the points. If they can all be connected by a single straight line, it's a linear graph.
- Example: The relationship between the number of articles purchased and their total cost (assuming a constant price per article) will form a linear graph passing through the origin (if 0 articles cost ₹0). Simple Interest vs Time (for a fixed principal and rate) also forms a linear graph.
6. Applications (Common Exam Scenarios):
- Distance-Time Graphs:
- Time on x-axis, Distance on y-axis.
- Slope represents speed (steeper slope = higher speed).
- Horizontal line means the object is stationary (distance not changing).
- Temperature-Time Graphs: Shows temperature fluctuations over time.
- Cost-Quantity Graphs: Shows how total cost changes with the number of items. Often a linear graph.
- Principal-Interest Graphs: Shows how interest earned changes with the principal amount or time.
Key Takeaways for Exams:
- Be comfortable reading coordinates (x, y). Remember x comes first (horizontal), then y (vertical).
- Know the coordinates of points on the axes and the origin.
- Understand how to interpret the slope/direction of a line graph (increasing, decreasing, constant).
- Recognize a linear graph (a straight line).
- Be able to read specific values from a graph and answer questions based on trends.
Multiple Choice Questions (MCQs):
-
In the coordinate point (5, -3), what is the value of the abscissa?
A) -3
B) 5
C) 2
D) 8 -
A point lies on the y-axis. Which of the following represents its coordinates?
A) (x, 0)
B) (0, y)
C) (x, y)
D) (0, 0) -
What are the coordinates of the origin in the Cartesian plane?
A) (1, 1)
B) (0, 1)
C) (1, 0)
D) (0, 0) -
Which type of graph is most suitable for showing the change in temperature recorded every hour throughout a day?
A) Bar Graph
B) Pie Chart
C) Line Graph
D) Histogram -
If points P(1, 2), Q(3, 4), and R(5, 6) are plotted on a graph, what type of graph will they form when joined?
A) A non-linear curve
B) A linear graph (straight line)
C) A triangle
D) Cannot be determined -
On a distance-time graph, a horizontal line segment indicates:
A) Uniform speed
B) Non-uniform speed
C) The object is stationary
D) The object is returning to the start -
The point (-4, -6) lies in which quadrant? (Assuming standard quadrant numbering)
A) First Quadrant
B) Second Quadrant
C) Third Quadrant
D) Fourth Quadrant -
What does the y-coordinate (ordinate) of a point represent?
A) Perpendicular distance from the y-axis
B) Perpendicular distance from the x-axis
C) Distance from the origin along the x-axis
D) Distance from the origin along the y-axis -
A graph shows the number of cars sold by a company each month for a year. If the line segment connecting June and July slopes downwards, it indicates:
A) Sales increased from June to July
B) Sales decreased from June to July
C) Sales remained the same in June and July
D) The graph is incorrect -
To plot the point (7, 0), where would you move from the origin?
A) 7 units up along the y-axis
B) 7 units left along the x-axis
C) 7 units right along the x-axis
D) 7 units down along the y-axis
Answer Key for MCQs:
- B
- B
- D
- C
- B (Notice the pattern: y = x + 1 for all points)
- C
- C (Both x and y are negative)
- B
- B
- C
Study these notes carefully. Practice plotting points and interpreting different line graphs from your textbook examples. Good luck with your preparation!