Complete Solutions and Summary of Introduction to Graphs – NCERT Class 8 Mathematics Chapter 13
Comprehensive explanations, examples, and exercises on graphical presentation of data, line graphs, linear graphs, coordinates, data interpretation, and applications from NCERT Class 8 Mathematics Chapter 13.
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Introduction to Graphs
Chapter 13: Mathematics
Complete Study Guide with Interactive Learning
Chapter Overview
What You'll Learn
Introduction to Graphs
Understanding the purpose of graphs for visual representation of data, trends, and comparisons.
Line Graphs
Learning how line graphs show data that changes continuously over time, with examples like time-temperature graphs.
Applications
Exploring real-life applications such as quantity-cost relations and principal-interest graphs.
Linear Graphs
Identifying linear graphs in direct variation scenarios and how to plot them.
Historical Context
This chapter introduces graphs as visual tools for data representation, easier than tables for understanding trends. It covers line graphs for continuous changes, with examples from temperature records to performance analysis, emphasizing their role in everyday applications like cost and interest calculations.
Key Highlights
Graphs help in quick comprehension of numerical data. Line graphs connect points to show patterns, and linear graphs represent direct variations, passing through the origin. Applications include dependent and independent variables in scenarios like distance-time and quantity-cost.
Comprehensive Chapter Summary
1. Introduction to Graphs
Graphs are visual representations of data, used in newspapers, TV, magazines, and books to show numerical facts clearly. They are easier to understand than tables, especially for trends and comparisons. Data can be presented in tables, but graphs provide a quicker visual insight.
Expanded: The chapter explains that graphs help in visualizing patterns, such as increases or decreases over time, making complex data accessible. For instance, a graph can show how temperature varies, highlighting peaks and valleys without reading numbers.
2. Line Graphs
Definition and Purpose
A line graph displays data that changes continuously over time. Points are plotted on a grid and connected by line segments. Example: Time-temperature graph where x-axis is time and y-axis is temperature in \(^\circ\)C.
Expanded: In the example, Renu's temperature is recorded: 6 a.m. - 37\(^\circ\)C, 10 a.m. - 40\(^\circ\)C, etc. The graph shows a rise to 40\(^\circ\)C and then a fall, illustrating patterns like a 3\(^\circ\)C increase from 6 a.m. to 10 a.m.
Key Benefits
Reveals patterns, trends, and allows estimation (e.g., temperature at 8 a.m. >37\(^\circ\)C). Useful for performance analysis, like batsmen's runs or car distance.
Expanded: In Example 1, batsmen A and B's runs are compared; A has peaks and valleys, B is steadier. This helps judge consistency.
Construction
Plot points on square grid, connect with lines. Horizontal (x-axis): independent variable; Vertical (y-axis): dependent variable.
Expanded: For distance-time, x-axis time, y-axis distance. Flat lines indicate no change, like car stopping from 11 a.m. to 12 noon.
3. Examples of Line Graphs
Example 1: Performance Graph
Runs scored by batsmen A and B in 2007 matches. Dotted line for A shows inconsistency; solid for B shows steadiness.
Expanded: Both scored 60 in 4th match. A scored 0 twice, below 40 in 5 matches; B never below 40.
Example 2: Distance-Time Graph
Car from P to Q (350 km). Started at 8 a.m., stopped 11 a.m.-12 noon, reached at 2 p.m.
Expanded: Traveled 50 km first hour, 100 km second, 50 km third. Speed varied, shown by slope changes.
Other Graphs in Exercises
Temperature of patient, sales figures, plant heights, temperature forecast.
Expanded: Patient's temperature 38.5\(^\circ\)C at noon; sales greatest difference 2004-2005; plants same height week 2.
4. Some Applications
Dependent and Independent Variables
Electric bill depends on usage (independent: electricity used; dependent: bill amount). Graphs show relations.
Expanded: Petrol cost depends on litres; interest on deposit. Linear graphs for direct variation pass through origin.
5. Linear Graphs and Direct Variation
Example 3: Quantity and Cost
Petrol: 10L - ₹500, etc. Linear graph; estimate for 12L = ₹600.
Expanded: Direct variation: cost = k × quantity, k=50. Graph through origin.
Example 4: Principal and Interest
10% SI: ₹100 - ₹10, etc. Annual interest for ₹250=₹25; deposit for ₹70=₹700.
Expanded: Formula: SI = P × r × t / 100, t=1 year, r=10. Linear relation.
Example 5: Time and Distance
30 km/h: Time for 75 km=2.5 h; distance in 3.5 h=105 km.
Expanded: Distance = speed × time; linear graph.
6. Additional Formulas and Concepts
Simple Interest: \[ SI = \frac{P \times r \times t}{100} \]
Distance: \[ d = s \times t \]
Cost: \[ c = k \times q \] (direct variation)
Expanded: In graphs, slope indicates rate of change; steeper slope = faster change.
Key Concepts and Definitions
Graph
Visual representation of data for quick understanding of trends and comparisons.
Line Graph
Connects points to show continuous changes over time.
x-axis
Horizontal axis, usually independent variable (e.g., time).
y-axis
Vertical axis, usually dependent variable (e.g., temperature).
Linear Graph
Straight line graph, often for direct variation, passes through origin.
Direct Variation
One quantity increases proportionally with another, e.g., cost with quantity.
Important Facts and Figures
Questions and Answers from Chapter
Short Questions
Q1. What was the patient’s temperature at 1 p.m.?
Q2. When was the patient’s temperature 38.5°C?
Q3. What were the two times when the patient’s temperature was the same?
Q4. What was the temperature at 1.30 p.m.?
Q5. During which periods did the patient’s temperature show an upward trend?
Q6. What were the sales in 2002?
Q7. What were the sales in 2006?
Q8. How high was Plant A after 2 weeks?
Q9. How high was Plant B after 3 weeks?
Q10. During which week did Plant A grow most?
Q11. On which days was the forecast temperature the same as actual?
Q12. What is the scale taken for the time axis in courier graph?
Q13. How much time did the courier take for travel?
Q14. Did the courier stop on his way?
Q15. During which period did he ride fastest?
Medium Questions
Q1. What were the sales in 2003 and 2005?
Q2. Compute the difference between sales in 2002 and 2006.
Q3. In which year was the greatest difference in sales compared to previous?
Q4. How much did Plant A grow during the 3rd week?
Q5. How much did Plant B grow from end of 2nd to 3rd week?
Q6. During which week did Plant B grow least?
Q7. Were the two plants same height any week?
Q8. What was the maximum forecast temperature during the week?
Q9. What was the minimum actual temperature during the week?
Q10. On which day did actual temperature differ most from forecast?
Q11. How much distance did the car cover during 7.30 a.m. to 8 a.m.?
Q12. What was the time when car covered 100 km since start?
Q13. Does the interest graph pass through origin?
Q14. Use graph to find interest on ₹2500 for a year.
Q15. To get ₹280 interest per year, how much to deposit?
Long Questions
Q1. Can there be a time-temperature graph as follows? Justify. (Graph i: Rising line)
Q2. Can there be a time-temperature graph as follows? Justify. (Graph ii: Falling line)
Q3. Can there be a time-temperature graph as follows? Justify. (Graph iii: Horizontal line)
Q4. Can there be a time-temperature graph as follows? Justify. (Graph iv: Zigzag)
Q5. Draw graph for number of days a hillside city received snow: 2003-8, 2004-10, 2005-5, 2006-12.
Q6. Draw graph for population of men and women: Men 2003-12, etc.; Women 2003-11.3, etc.
Q7. Draw graph for cost of apples: 1-5, 2-10, 3-15, 4-20, 5-25.
Q8. Draw graph for car distances: 6am-40km, 7am-80km, 8am-120km, 9am-160km.
Q9. Draw graph for square perimeter: side 2-8, 3-12, 3.5-14, 5-20, 6-24.
Q10. Draw graph for square area: side 2-4, 3-9, 4-16, 5-25, 6-36.
Q11. Use graph to find petrol for ₹800 in Example 3.
Q12. Is Example 4 a direct variation?
Q13. Use graph in Example 5 for time to 75 km.
Q14. Use graph in Example 5 for distance in 3.5 hours.
Q15. Explain why graphs are easier than tables for trends.
Interactive Knowledge Quiz
Test your understanding of Introduction to Graphs
Quick Revision Notes
Graphs Intro
- Visual data
- Trends/comparisons
- Easier than tables
Line Graphs
- Continuous change
- Connect points
- x-time, y-value
Applications
- Dependent/independent
- Cost-quantity
- Interest-principal
Linear Graphs
- Direct variation
- Through origin
- Straight line
Exam Strategy Tips
- Plot accurately
- Interpret trends
- Estimate values
- Identify variables
- Justify graphs
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