Introduction to Graphs — Questions and Answers
Welcome to HSLC Guru. On this page you will find complete, step-by-step solutions for Chapter 15, “Introduction to Graphs”, of ASSEB (Assam State School Education Board) Class 8 General Mathematics — every exercise, worked example and the group activity, explained with clearly labelled graphs.
Summary
In everyday life we use graphs to present numerical data simply and clearly. We see graphs in newspapers, magazines, television and advertisements. In this chapter we study the pictograph, bar graph, double bar graph, histogram, line graph and linear graph, and learn how to find the co-ordinates of a point.
A pictograph shows data using pictures or symbols. A bar graph compares data using equal-width rectangular bars; a double bar graph uses two bars for each item to compare two sets of data. A histogram is a bar graph in which the class intervals are continuous, so there is no gap between the bars.
A line graph shows how data change over a period of time by joining points with line segments. In the Cartesian plane two mutually perpendicular axes — the horizontal X-axis and the vertical Y-axis — are used to describe the position of a point as $(\text{abscissa},\ \text{ordinate})$. This system was introduced by the 17th-century French mathematician René Descartes, and so is called the Cartesian co-ordinate system.
Summary: This ASSEB Class 8 General Mathematics Chapter 15 Introduction to Graphs solution explains pictographs, bar graphs, double bar graphs, histograms, line graphs and linear graphs, and shows how to plot a point using its Cartesian coordinates — the abscissa (X-coordinate) and the ordinate (Y-coordinate) — on the X and Y axes about the origin. Every question of Exercise 15.1, 15.2 and 15.3, all worked examples and the group activity are solved with clearly labelled graphs.
Textbook Questions and Answers
Co-ordinates and Quadrants — Key Idea
In the Cartesian plane the point where the two axes meet is the origin (O), whose co-ordinate is $(0,0)$. For any point P with co-ordinate $(x, y)$, $x$ is the abscissa (X-coordinate) and $y$ is the ordinate (Y-coordinate). In the figure below, P is 4 units from the Y-axis and 5 units from the X-axis, so its co-ordinate is $(4,5)$.
The two axes divide the plane into four parts; each part is called a quadrant. In this chapter we deal only with points in the first quadrant $(+,+)$.
Worked Examples
Example 1: The body temperature of a sick girl was recorded every three hours. Draw and study the line graph of the data.
Time: 7 AM, 10 AM, 1 PM, 4 PM, 7 PM; Temperature (°C): 38, 40, 37, 37, 36.
Answer: From the graph: (i) it shows the body temperature over 12 hours of a day; (ii) the temperatures from 7 AM to 7 PM are shown; (iii) the temperature is maximum at 10 AM (40°C) and falls to 36°C by 7 PM; (iv) the temperature starts falling after 10 AM and stays unchanged (37°C) between 1 PM and 4 PM.
Example 2: Study the line graph of a company’s annual sale (in crores) for the last five years.
Year: 2015→2, 2016→4, 2017→4, 2018→8, 2019→6 (in crores of rupees).
Answer: (i) Years are shown along the X-axis and sale (in crores) along the Y-axis. (ii) The sale is lowest in 2015 and highest in 2018. (iii) The sale in 2016 and 2017 is the same. (iv) The sale in 2019 is less than in 2018. (v) Difference between the maximum and minimum sale $= (8-2) = $ Rs 6 crores.
Example 3: A graph shows the total runs scored by two batsmen P and Q in 10 different cricket matches in 2019 (P — solid line, Q — dashed line). Answer the questions from the graph.
Answer: (i) The graph shows the runs scored by batsmen P and Q in 10 matches played in 2019. (ii) The horizontal X-axis shows the number of matches and the vertical Y-axis shows the runs scored. (iii) The dashed line ($—$) represents batsman Q’s runs. (iv) Both batsmen scored equal runs in the 4th match. (v) Yes, batsman P failed to score a single run in the 5th and 10th matches. (vi) Overall, batsman Q performed better. (vii) Batsman P’s highest score was 120 runs and his lowest was 0.
Example 4: Plot the point $(4,2)$ on the graph paper. Is this point the same as $(2,4)$?
Answer: Taking the side of a small square = 1 unit, from the origin O move 4 units along the X-axis and then 2 units up (parallel to the Y-axis) to reach $(4,2)$. Similarly, for $(2,4)$ move 2 units along the X-axis and 4 units up. Hence $(4,2)$ and $(2,4)$ are two different points — changing the order of the co-ordinates changes the point.
Example 5: Complete the following table with the help of the graph. The points are A(4,5), B(8,10), C(14,3), D(17,8), E(22,6), F(23,0).
Answer: The abscissa is the distance from the Y-axis and the ordinate is the distance from the X-axis. Point F lies on the X-axis, so its ordinate (distance from the X-axis) is 0; hence F has abscissa 23 and co-ordinate $(23,0)$. The completed table is:
| Point | Abscissa | Ordinate | Distance from X-axis | Distance from Y-axis | Co-ordinate |
|---|---|---|---|---|---|
| A | 4 | 5 | 5 | 4 | (4, 5) |
| B | 8 | 10 | 10 | 8 | (8, 10) |
| C | 14 | 3 | 3 | 14 | (14, 3) |
| D | 17 | 8 | 8 | 17 | (17, 8) |
| E | 22 | 6 | 6 | 22 | (22, 6) |
| F | 23 | 0 | 0 | 23 | (23, 0) |
Example 6: Plot the points A(1,8), B(3,6), C(5,4), D(6,3), E(8,1) on the graph paper and verify whether they lie on one straight line.
Answer: Plotting and joining the points gives the line segment AE. For every point the sum of the abscissa and ordinate is $x+y=9$ (for example $1+8=9,\ 3+6=9,\ 5+4=9$), so the points lie on one straight line.
Example 7: The table gives the quantity of petrol and its cost. Draw a linear graph. Petrol (litre): 5, 10, 15, 20; Cost (Rs.): 350, 700, 1050, 1400.
Answer: Marking petrol along the X-axis and cost along the Y-axis, the points A(5, 350), B(10, 700), C(15, 1050), D(20, 1400) joined together give the straight line AD. The cost of one litre is $\frac{350}{5}=70$ rupees, so the cost is directly proportional to the quantity. Since 0 litre costs Rs 0, the graph passes through the origin (direct variation).
Example 8: Principal and its simple interest (at 5% per annum) are given. Draw a linear graph and complete the table. Principal: 1000, 2000, 3000, 4000, 6000; Simple interest: 50, 100, 150, 200, 300.
Answer: Interest = 5% of the principal $= \frac{P}{20}$. Joining the points gives the straight line OA through the origin. Using the graph (or formula): for principal 2500, interest $=\frac{2500}{20}=125$; for interest 250, principal $=250\times20=5000$; for principal 6500, interest $=\frac{6500}{20}=325$; for interest 400, principal $=400\times20=8000$.
| Principal (Rs.) | 2500 | 5000 | 6500 | 8000 |
|---|---|---|---|---|
| Simple interest (Rs.) | 125 | 250 | 325 | 400 |
Exercise 15.1
1. A car went from Lakhimpur to Guwahati. A line graph of distance covered and time taken is given below. Study the graph and answer the following.
(i) What data do the two axes represent?
Answer: The X-axis represents time (from 7 AM to 3 PM) and the Y-axis represents the distance covered (in kilometres).
(ii) What is the total distance covered and the total time taken by the car?
Answer: The total distance covered is 375 km and the total time taken is 8 hours (7 AM to 3 PM).
(iii) At what time did the car have maximum speed?
Answer: Between 12 noon and 1 PM the speed is maximum, since in this one hour the car covers $300-200 = 100$ km (the graph is steepest here).
(iv) Did the car stop on the road? If so, for how long?
Answer: Yes, the car stopped for 1 hour, from 11 AM to 12 noon — the distance stays at 200 km, so the graph is horizontal in this interval.
(v) At what time was the speed of the car equal?
Answer: The speed was equal (50 km/h) from 7 AM to 11 AM and again from 1 PM to 2 PM — the graph has the same slope in these parts.
2. The line graph of the electricity bills of the last six months of a family is given. Study it and answer the following.
Answer: (i) The graph represents the electricity (units) consumed by a family in the last six months. (ii) The Y-axis represents the units of electricity consumed. (iii) The maximum electricity (240 units) was consumed in June. (iv) The minimum electricity (60 units) was consumed in September. (v) Equal units (150 each) were consumed in April and August.
3. A line graph shows the different games liked by 100 students of a school. Study it and answer the following.
Answer: (i) Data for 7 games are given (cricket, kabaddi, race, jump, musical chair, badminton, football). (ii) Football is liked by the most students (25). (iii) Badminton is liked by the fewest students (5). (iv) Kabaddi and jump are liked by an equal number (10 each); race and musical chair are also liked by an equal number (15 each).
4. The number of children born in a week in a hospital is shown in the graph. Study it and answer the following.
Answer: (i) The X-axis represents the days of the week (Mon, Tue, Wed, Thu, Fri, Sat, Sun). (ii) The maximum number of children (15) were born on Saturday. (iii) The minimum number of children (3) were born on Thursday. (iv) Total number of children born in the week $= 10+7+12+3+8+15+5 = $ 60.
Group Activity
While doing a project, Medhashree and Meghashree grew two plants A and B in a laboratory. The weekly growth of both plants was measured. The data collected at the end of one month are shown in the graph.
(i) How high was plant A in (a) the 1st week, (b) the 2nd week, (c) the 3rd week?
Answer: Plant A — (a) 2 cm, (b) 5 cm, (c) 8 cm.
(ii) How high was plant B in (a) the 1st week, (b) the 2nd week, (c) the 4th week?
Answer: Plant B — (a) 1 cm, (b) 3 cm, (c) 9 cm.
(iii) Are the changes of growth of plants A and B in the 2nd and 4th weeks the same? If not, what are the changes?
Answer: No. In the 2nd week, A grew $5-2=3$ cm and B grew $3-1=2$ cm. In the 4th week, A grew $10-8=2$ cm and B grew $9-8=1$ cm. So in both weeks plant A grew 1 cm more than plant B.
(iv) In which week were the heights of the two plants equal?
Answer: In the 3rd week both plants were of equal height (8 cm).
(v) Which plant had the better growth at the end of one month?
Answer: Plant A had the better growth, since it reached 10 cm, which is more than plant B’s 9 cm.
Exercise 15.2
1. Find the abscissa and ordinate of the following points. (i) (0, 4) (ii) (5, 9) (iii) (7, 7) (iv) (5, 0)
Answer: (i) $(0,4)$: abscissa 0, ordinate 4. (ii) $(5,9)$: abscissa 5, ordinate 9. (iii) $(7,7)$: abscissa 7, ordinate 7. (iv) $(5,0)$: abscissa 5, ordinate 0.
2. Determine on which axis each of the following points lies. (i) (3, 0) (ii) (0, 8) (iii) (9, 0) (iv) (0, 10)
Answer: (i) $(3,0)$: ordinate is zero, so it lies on the X-axis. (ii) $(0,8)$: abscissa is zero, so it lies on the Y-axis. (iii) $(9,0)$: on the X-axis. (iv) $(0,10)$: on the Y-axis. (A point with ordinate 0 lies on the X-axis, and a point with abscissa 0 lies on the Y-axis.)
3. Plot the following points and examine whether they lie on the same straight line.
Answer: (i) A(2,2), B(3,3), C(5,5), D(6,6): all satisfy $y=x$, so they lie on one straight line. (ii) K(4,0), L(4,2), M(4,5), N(4,6): all have abscissa 4, so they lie on the vertical line $x=4$. (iii) P(1,2), Q(4,4), R(6,7): slope of PQ $=\frac{4-2}{4-1}=\frac{2}{3}$ and slope of QR $=\frac{7-4}{6-4}=\frac{3}{2}$; the slopes are unequal, so the points are not collinear. (iv) S(2,1), T(2,5), O(5,5), P(7,7): S and T have abscissa 2 (line $x=2$), but O(5,5) and P(7,7) do not lie on it, so the points are not collinear.
4. Plot the points $(2,6)$ and $(5,3)$. Find the co-ordinates of the two points where the line joining them meets the X-axis and the Y-axis.
Answer: The slope of the line through the two points is $\frac{3-6}{5-2}=\frac{-3}{3}=-1$, so its equation is $y-6=-1(x-2)$, i.e. $x+y=8$. On the X-axis $y=0$ gives $x=8$, so the point is $(8,0)$. On the Y-axis $x=0$ gives $y=8$, so the point is $(0,8)$.
5. Write the co-ordinates of the vertices of the geometrical figures drawn on the graph paper (for example, S(3,8)).
Answer: The co-ordinates of the vertices are — Parallelogram PQRS: P(1, 6), Q(5, 6), R(7, 8), S(3, 8). Triangle LMN: L(8, 5), M(12, 5), N(10, 8). Rectangle ABCD: A(1, 1), B(4, 1), C(4, 3), D(1, 3). Rectangle EFGH: E(6, 1), F(9, 1), G(9, 4), H(6, 4).
6. State whether the following statements are true or false. Correct the false ones.
Answer: (i) “A point whose co-ordinate is $(5,0)$ lies on the Y-axis.” — False. Correction: since its ordinate is 0, the point lies on the X-axis. (ii) “If the X-coordinate of a point is zero but the Y-coordinate is non-zero, the point lies on the Y-axis.” — True. (iii) “The co-ordinate of the origin is $(0,0)$.” — True.
Exercise 15.3
1. Using a proper scale, draw a linear graph with the data given in the following tables.
(a) Cost of eggs — Number of eggs: 1, 3, 5, 6; Cost (Rs.): 6, 18, 30, 36.
Answer: Marking the number of eggs along the X-axis and the cost along the Y-axis, the points $(1,6), (3,18), (5,30), (6,36)$ joined together give a straight line through the origin, because each egg costs Rs 6 ($\text{cost}=6\times\text{number of eggs}$) — a linear graph.
(b) Distance covered by a car — Time (hrs): 1, 2, 3, 4; Distance (km): 50, 100, 150, 200.
Answer: Here $\text{distance}=50\times\text{time}$, so the graph is a straight line through the origin. (i) Time needed to cover 300 km $=\frac{300}{50}=6$ hours. (ii) Distance covered in 5 hours $=50\times5=250$ km.
(c) Principal and interest — Principal (Rs.): 200, 500, 1000, 1500; Simple interest (Rs.): 20, 50, 100, 150.
Answer: Here interest $=$ 10% of the principal $\left(\frac{P}{10}\right)$, so the graph is a straight line through the origin. (i) Interest for Rs 400 $=\frac{400}{10}=$ Rs 40. (ii) Principal for an interest of Rs 120 $=120\times10=$ Rs 1200.
2. Multiples of 6 are given in the table. Draw a linear graph with these data. X: 1, 2, 3, 4, 5; Y: 6, 12, 18, 24, 30.
Answer: Marking the number along the X-axis and its multiple along the Y-axis, the points $(1,6),(2,12),(3,18),(4,24),(5,30)$ give a straight line through the origin, since $Y=6X$ — a linear graph.
3. Cubic values of some numbers are given. Draw a graph. Is this a linear graph? Numbers: 2, 3, 4, 5; Cubic number: 8, 27, 64, 125.
Answer: Plotting and joining $(2,8),(3,27),(4,64),(5,125)$ gives a curve, not a straight line. Because the relation $y=x^{3}$ is not linear (a number and its cube are not directly proportional), this is not a linear graph.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. What is the first value in the co-ordinate of a point called?
(a) ordinate (b) abscissa (c) origin (d) axis
Answer: (b) abscissa
2. Which graph is used to show data with continuous class intervals?
(a) pictograph (b) line graph (c) histogram (d) double bar graph
Answer: (c) histogram
3. What is the co-ordinate of the origin?
(a) $(1,1)$ (b) $(0,1)$ (c) $(1,0)$ (d) $(0,0)$
Answer: (d) $(0,0)$
4. On which axis does the point $(0,7)$ lie?
(a) X-axis (b) Y-axis (c) both (d) neither
Answer: (b) Y-axis
5. Who introduced the Cartesian co-ordinate system?
(a) Euclid (b) Pythagoras (c) René Descartes (d) Newton
Answer: (c) René Descartes
6. Which graph is most useful for comparing two sets of data?
(a) pictograph (b) double bar graph (c) linear graph (d) histogram
Answer: (b) double bar graph
7. Where does the point $(5,0)$ lie?
(a) on the X-axis (b) on the Y-axis (c) at the origin (d) in the first quadrant
Answer: (a) on the X-axis
8. Into how many quadrants do the two axes divide the plane?
(a) 2 (b) 3 (c) 4 (d) 6
Answer: (c) 4
9. What is the ordinate of the point $(4,5)$?
(a) 4 (b) 5 (c) 9 (d) 1
Answer: (b) 5
10. What is a graph that shows data using pictures or symbols called?
(a) bar graph (b) histogram (c) pictograph (d) line graph
Answer: (c) pictograph
Fill in the Blanks
Answer: 1. The Y-coordinate of a point is called its ordinate.
Answer: 2. The point where the X-axis and Y-axis meet is called the origin.
Answer: 3. In a histogram there is no gap between the bars.
Answer: 4. The abscissa of a point is its distance from the Y-axis.
Answer: 5. A line graph that is a complete straight line is called a linear graph.
True or False
Answer: 1. $(0,0)$ is the co-ordinate of the origin. — True.
Answer: 2. The point $(3,0)$ lies on the Y-axis. — False (it lies on the X-axis).
Answer: 3. In a bar graph all the bars have equal width. — True.
Answer: 4. Swapping the abscissa and ordinate of a co-ordinate gives the same point. — False.
Answer: 5. A line graph is used to show change over a period of time. — True.
Short Answer Questions
1. What is the difference between a pictograph and a histogram?
Answer: A pictograph shows data using pictures or symbols, whereas a histogram shows data using bars of continuous class intervals with no gap between the bars.
2. What do the abscissa and the ordinate mean?
Answer: The X-coordinate of a point is its abscissa and the Y-coordinate is its ordinate. For example, the point $(4,5)$ has abscissa 4 and ordinate 5.
3. Are $(2,5)$ and $(5,2)$ the same point? Explain.
Answer: No. In $(2,5)$ the abscissa is 2 and the ordinate is 5, but in $(5,2)$ the abscissa is 5 and the ordinate is 2. Since the order of the co-ordinates differs, they are two different points.
4. If a linear graph passes through the origin, what is the relation between the two quantities?
Answer: A linear graph through the origin shows direct variation — the two quantities are directly proportional ($y \propto x$).
Key Terms
| Term | Meaning |
|---|---|
| Graph | A pictorial representation of data |
| Pictograph | A graph that shows data using pictures or symbols |
| Bar graph | A graph that shows data using equal-width rectangular bars |
| Histogram | A bar graph with continuous class intervals |
| Line graph | Points joined together by line segments |
| Linear graph | A line graph that is a single straight line |
| Axis | One of the two mutually perpendicular reference lines |
| Origin | The point $(0,0)$ where the two axes meet |
| Abscissa | The X-coordinate of a point |
| Ordinate | The Y-coordinate of a point |
| Co-ordinate | The position (abscissa, ordinate) of a point |
| Quadrant | One of the four parts into which the axes divide the plane |
| Cartesian plane | The plane containing the X-axis and the Y-axis |