Chapter Overview
Line Graph
Continuous Data
x-axis
Independent Variable
y-axis
Dependent Variable
Linear
Direct Variation
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.
Questions and Answers from Chapter
Short Questions
Q1. What was the patient’s temperature at 1 p.m.?
Answer: 36.5°C.
Q2. When was the patient’s temperature 38.5°C?
Answer: At noon.
Q3. What were the two times when the patient’s temperature was the same?
Answer: 12 noon and 1 p.m.
Q4. What was the temperature at 1.30 p.m.?
Answer: 36°C (estimated from graph).
Q5. During which periods did the patient’s temperature show an upward trend?
Answer: 9-10 a.m., 10-11 a.m.
Q6. What were the sales in 2002?
Answer: 4 (in ₹x million).
Q7. What were the sales in 2006?
Answer: 8 (in ₹x million).
Q8. How high was Plant A after 2 weeks?
Answer: 7 cm.
Q9. How high was Plant B after 3 weeks?
Answer: 12 cm.
Q10. During which week did Plant A grow most?
Answer: Week 3.
Q11. On which days was the forecast temperature the same as actual?
Answer: Tuesday, Friday, Sunday.
Q12. What is the scale taken for the time axis in courier graph?
Answer: 2 cm = 1 hour.
Q13. How much time did the courier take for travel?
Answer: 3.5 hours.
Q14. Did the courier stop on his way?
Answer: Yes.
Q15. During which period did he ride fastest?
Answer: 8-9 a.m.
Medium Questions
Q1. What were the sales in 2003 and 2005?
Answer: 2003: 7 (in ₹x million), 2005: 10 (in ₹x million). Difference highlights trend changes. (3 marks)
Q2. Compute the difference between sales in 2002 and 2006.
Answer: 4 (in ₹x million). Shows overall growth over years. (3 marks)
Q3. In which year was the greatest difference in sales compared to previous?
Answer: 2005 (difference of 3 from 2004). Indicates peak variation. (3 marks)
Q4. How much did Plant A grow during the 3rd week?
Answer: 5 cm. Shows growth rate comparison. (3 marks)
Q5. How much did Plant B grow from end of 2nd to 3rd week?
Answer: 3 cm. Highlights slower growth. (3 marks)
Q6. During which week did Plant B grow least?
Answer: Week 1 (1 cm). Compares weekly changes. (3 marks)
Q7. Were the two plants same height any week?
Answer: Yes, week 2 (7 cm). Shows intersection. (3 marks)
Q8. What was the maximum forecast temperature during the week?
Answer: 35°C. Identifies peak. (3 marks)
Q9. What was the minimum actual temperature during the week?
Answer: 15°C. Shows lowest point. (3 marks)
Q10. On which day did actual temperature differ most from forecast?
Answer: Thursday (5°C difference). Analyzes variation. (3 marks)
Q11. How much distance did the car cover during 7.30 a.m. to 8 a.m.?
Answer: 20 km. Estimates from graph. (3 marks)
Q12. What was the time when car covered 100 km since start?
Answer: 8 a.m. (estimated). (3 marks)
Q13. Does the interest graph pass through origin?
Answer: Yes, as zero deposit gives zero interest. (3 marks)
Q14. Use graph to find interest on ₹2500 for a year.
Answer: ₹200. Interpolation. (3 marks)
Q15. To get ₹280 interest per year, how much to deposit?
Answer: ₹3500. From graph. (3 marks)
Long Questions
Q1. Can there be a time-temperature graph as follows? Justify. (Graph i: Rising line)
Answer: Yes, it shows continuous increase in temperature over time, which is possible in scenarios like heating. The line connects points indicating steady rise, similar to the initial part of Renu's graph from 6 a.m. to 10 a.m. where temperature rose from 37°C to 40°C.
Q2. Can there be a time-temperature graph as follows? Justify. (Graph ii: Falling line)
Answer: Yes, it represents a continuous decrease, like cooling, as in Renu's graph from 10 a.m. to 6 p.m. dropping from 40°C to 35°C, showing a downward trend over periods.
Q3. Can there be a time-temperature graph as follows? Justify. (Graph iii: Horizontal line)
Answer: Yes, constant temperature over time, like the car stopping in Example 2 from 11 a.m. to 12 noon at 200 km, no change in dependent variable.
Q4. Can there be a time-temperature graph as follows? Justify. (Graph iv: Zigzag)
Answer: No, as temperature can't jump discontinuously; line graphs assume continuous change, not abrupt shifts without intermediate points.
Q5. Draw graph for number of days a hillside city received snow: 2003-8, 2004-10, 2005-5, 2006-12.
Answer: Plot years on x-axis, days on y-axis. Connect points: (2003,8), (2004,10), (2005,5), (2006,12). Shows variation; not linear as no direct proportion.
Q6. Draw graph for population of men and women: Men 2003-12, etc.; Women 2003-11.3, etc.
Answer: Two lines: men and women over years. Men steady increase; women fluctuate. Compare trends for gender analysis.
Q7. Draw graph for cost of apples: 1-5, 2-10, 3-15, 4-20, 5-25.
Answer: Linear through origin; direct variation, cost = 5 × number. Estimate for non-plotted values.
Q8. Draw graph for car distances: 6am-40km, 7am-80km, 8am-120km, 9am-160km.
Answer: Linear; constant speed 40 km/h. 7.30-8am: 20km; 100km at 8.30am (estimated).
Q9. Draw graph for square perimeter: side 2-8, 3-12, 3.5-14, 5-20, 6-24.
Answer: Linear; perimeter = 4 × side, through origin, direct variation.
Q10. Draw graph for square area: side 2-4, 3-9, 4-16, 5-25, 6-36.
Answer: Not linear; quadratic curve, area = side², doesn't pass through origin linearly.
Q11. Use graph to find petrol for ₹800 in Example 3.
Answer: 16 litres. From vertical ₹800 to graph, then horizontal to x-axis.
Q12. Is Example 4 a direct variation?
Answer: Yes, interest proportional to principal at fixed rate/time; linear through origin.
Q13. Use graph in Example 5 for time to 75 km.
Answer: 2.5 hours. Vertical from 75 km to graph, horizontal to time axis.
Q14. Use graph in Example 5 for distance in 3.5 hours.
Answer: 105 km. Horizontal from 3.5 h to graph, vertical to distance.
Q15. Explain why graphs are easier than tables for trends.
Answer: Visual patterns like rises/falls visible instantly; e.g., in temperature graph, peak at 10 a.m. clear without calculations.
