15. Introduction to Graphs

Exercise 15.1

1. The following graph shows the temperature of a patient in a hospital, recorded every hour.
(a) What was the patient’s temperature at 1 p.m. ?
(b) When was the patient’s temperature 38.5° C?

(c) The patient’s temperature was the same two times during the period given. What were these two times?
(d) What was the temperature at 1.30 p.m.? How did you arrive at your answer?
(e) During which periods did the patients’ temperature showed an upward trend?

Solution: (a) The patient’s temperature at 1 p.m. was 36.5°C .  

(b) The patient’s temperature was 38.5°C at 12 noon .

(c) The two times was at 1pm and 2 pm .

(d) The temperature at 1.30 p.m. was 36.5°C . The point between 1 pm and 2 pm on the x-axis is equidistant from the two points showing 1 pm and 2 pm . Therefore , it will represent 1.30 pm . Similarly, the point on the y-axis , between 36°C and 37°C . Therefore, it will represent 36.5°C .  

(e) During  9 am to 10 am , 10 am to 11 am and 2 pm to 3 pm , the patients’ temperature showed an upward trend .
2. The following line graph shows the yearly sales figures for a manufacturing company.
(a) What were the sales in (i) 2002 (ii) 2006?
(b) What were the sales in (i) 2003 (ii) 2005?
(c) Compute the difference between the sales in 2002 and 2006.
(d) In which year was there the greatest difference between the sales as compared to its previous year?

Solution:  (a)  (i) The sales in 2002 were Rs 4 crore .

(ii) The sales in 2006 were Rs 8 crore .

(b) (i) The sales in 2003 were Rs 7 crore .

(i) The sales in 2005 were Rs 10 crore .

(c) The difference between the sales in 2002 and 2006 is Rs (8 – 4) crore = Rs 4 crore .

(d)  2005

3. For an experiment in Botany, two different plants, plant A and plant B were grown under similar laboratory conditions. Their heights were measured at the end of each week for 3 weeks. The results are shown by the following graph.

(a) How high was Plant A after (i) 2 weeks (ii) 3 weeks?
(b) How high was Plant B after (i) 2 weeks (ii) 3 weeks?
(c) How much did Plant A grow during the 3rd week?
(d) How much did Plant B grow from the end of the 2nd week to the end of the 3rd week?
(e) During which week did Plant A grow most?
(f) During which week did Plant B grow least?
(g) Were the two plants of the same height during any week shown here? Specify.

Solution: (a) (i) The high of the plant A is 7 cm .

(ii) The high of the plant B is 9 cm .

(b) (i) The plant B growth after 2 weeks is 7 cm .

(ii)  The plant B growth after 3 weeks is 10 cm .

(c) The Plant A growth during the 3rd week is 2 cm .

(d) 3 cm

(e) Second week

(f) First week

(g) At the end of the 2nd week .

4. The following graph shows the temperature forecast and the actual temperature for each day of a week.
(a) On which days was the forecast temperature the same as the actual temperature?
(b) What was the maximum forecast temperature during the week?
(c) What was the minimum actual temperature during the week?
(d) On which day did the actual temperature differ the most from the forecast temperature?

Solution : (a) Tuesday  , Friday and Sunday .

(b) The maximum forecast temperature during the week was 35°C (i.e., Sunday).

(c) The minimum actual temperature during the week was 15°C ( i.e., Monday , Thursday and Friday) .

(d)  Thursday  (i.e., (22.5 – 15)°C = 7.5°C ) .

5. Use the tables below to draw linear graphs.
(a) The number of days a hill side city received snow in different years.

Year	2003	2004	2005	2005
Days	8	10	5	12

(b) Population (in thousands) of men and women in a village in different years.

Year	2003	2004	2005	2006	2007
Number of men	12	12.5	13	13.2	13.5
Number of women	11.3	11.9	13	13.6	12.8

Solution:  (a) We draw the linear graph :

         

(b) We draw the linear graph :

      

6. A courier-person cycles from a town to a neighbouring suburban area to deliver a parcel to a merchant. His distance from the town at different times is shown by the following graph.
(a) What is the scale taken for the time axis?
(b) How much time did the person take for the travel?
(c) How far is the place of the merchant from the town?
(d) Did the person stop on his way? Explain.
(e) During which period did he ride fastest?

 

Solution: (a) 4 Square units = 1 hour .

(b)  hours (i.e., Time = (1+1+1+  ) hours = 3  hours ) .

(c) 22 km (i.e., Total distance = 22 km ).

(d) Yes ; This is indicated by the horizontal part of the graph ( 10 am – 10.30 am ) .

(e) He has raided fastest between 8 am and 9 am .

7. Can there be a time-temperature graph as follows? Justify your answer.

Solution: (i) This figure can be a representative time-temperature graph, where the temperature increases as the time increase .

(ii) This figure can be represented time-temperature graph, where the temperature decreases as the time increase .

(iii) This figure cannot be a representative time-temperature graph .

(iv) This figure can be represented time-temperature graph, where the temperature reamains constant and the time is increasing .

Exercise 15.2

1. Plot the following points on a graph sheet. Verify if they lie on a line
(a) A(4, 0), B(4, 2), C(4, 6), D(4, 2.5)
(b) P(1, 1), Q(2, 2), R(3, 3), S(4, 4)
(c) K(2, 3), L(5, 3), M(5, 5), N(2, 5)

Solution: (a) A(4, 0), B(4, 2), C(4, 6), D(4, 2.5)

We plotting the points are A(4,0) , B(4,2) , C(4,6) and D(4,2.5) on the grapg paper and join them . Then we find a line .
(b) P(1, 1), Q(2, 2), R(3, 3), S(4, 4)

We plotting the points are P(1,1) , Q(2,2) , R(3,3) and S(4,4) on the grapg paper and join them . Then we find a line .
(c) K(2, 3), L(5, 3), M(5, 5), N(2, 5)

We plotting the points are K(2,3) , L(5,3) , M(5,5) and N(2,5) on the grapg paper and join them . Then we can not find a line .

2. Draw the line passing through (2, 3) and (3, 2). Find the coordinates of the points at which this line meets the x-axis and y-axis.

Solution:

We plotting the points are P(4,2) and Q(3,2) on the grapg paper and join them . Then we find a line . Therefore, the coordinates of the points meets at the x-axis is B (5 , 0) and y-axis is A(0 , 5).

3. Write the coordinates of the vertices of each of these adjoining figures.

Solution: Given figure :

The coordinate of the point A (2,0) .

The coordinate of the point B (2,3) .

The coordinate of the point C (0 ,3) .

The coordinate of the point P (4,3) .

The coordinate of the point Q (6,1) .

The coordinate of the point R (6,5) .

The coordinate of the point S (4,7) .

The coordinate of the point K (10,5) .

The coordinate of the point L (7,7) .

The coordinate of the point M (10,8) .
4. State whether True or False. Correct that are false.
(i) A point whose x coordinate is zero and y-coordinate is non-zero will lie on the y-axis.
(ii) A point whose y coordinate is zero and x-coordinate is 5 will lie on y-axis.
(iii) The coordinates of the origin are (0, 0).

Solution: (i) True  [Example : Here , x = 0 and y = 2 (any number) , then the coordinate is (0,2) . So, the point lie on y-axis]

(ii) False [because, the coordinate is (5,0) , then the point lie on x-axis]

(iii) True . [The point of intersection of the axes is called the origin .So,the coordinates of the origin are (0, 0). ]

Exercise 15.3

1. Draw the graphs for the following tables of values, with suitable scales on the axes.
(a) Cost of apples

Number of apples	1	2	3	4	5
Cost (in Rs)	5	10	15	20	25

Solution: (a) Here, x- axis represent number of apples and y-axis represent the cost (in Rs) .

We plotting the points are (1,5),(2,10),(3,15),(4,20) and (5,25) on the graph paper and join them .Then we find a line .

(b) Distance travelled by a car

Time (in hours)	6 a.m.	7 a.m.	8 a.m.	9 a.m.
Distances (in km)	40	80	120	160

i) How much distance did the car cover during the period 7.30 a.m. to 8 a.m
(ii) What was the time when the car had covered a distance of 100 km since it’s start?

Solution:  (b) Here, x- axis represent time (in hours) and y-axis represent distance (in km) .

We plotting the points are (6,40),(7,80),(8,120) and (9,160) on the graph paper and join them .Then we find a line .

(i) The distance = (100 - 80) km = 20 km

(ii) The time = 7.30 am

(c) Interest on deposits for a year.

Deposit (in Rs)	1000	2000	3000	4000	5000
Simple Intersect (in Rs)	80	160	240	320	400

(i) Does the graph pass through the origin?
(ii) Use the graph to find the interest on Rs 2500 for a year.
(iii) To get an interest of Rs 280 per year, how much money should be deposited?

Solution: (c) Here, x- axis represent side of square (in cm) and y-axis represent perimeter (in cm) .

We plotting the points are (1000,80),(2000,160),(3000,240),(4000,320) and (5000,400) on the graph paper and join them .Then we find a line .

(i) Yes , the graph passes through the origin .

(ii) Rs 200 .

(iii) Rs 3500 .

2. Draw a graph for the following.

(i)

Side of square (in cm)	2	3	3.5	5	6
Perimeter (in cm)	8	12	14	20	24

  Is it a linear graph?

(ii)  

Side of square (in cm)	2	3	4	5	6
Area  (in )	4	9	16	25	36

Is it a linear graph?

Solution: (i) Here, x- axis represent side of square (in cm) and y-axis represent perimeter (in cm) .

We plotting the points are (2,8),(3,12),(3.5,14),(5,20) and (6,24) on the graph paper and join them .Then we find line . Yes , it is a linear graph .

(ii) Here, x- axis represent side of square (in cm) and y-axis represent area (in cm²) .

We plotting the points are (2,4),(3,9),(4,16),(5,25) and (6,36) on the graph paper and join them .Then we find a curve .No , it is no a linear graph .