Class 8 Maths - Introduction to Graphs NCERT Solutions

Chapter 15: Introduction to Graphs (NCERT Solutions)

Exercise 15.1

Q1. 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?

(a) By observing the graph on the y-axis against 1 p.m. on the x-axis, the temperature was 36.5° C.

(b) The temperature 38.5° C is on the y-axis. The corresponding time on the x-axis is 12 noon.

(c) The graph shows a horizontal line between 1 p.m. and 2 p.m. So, the temperature was same at 1 p.m. and 2 p.m.

(d) The point between 1 p.m. and 2 p.m. on the x-axis is 1:30 p.m. Due to the horizontal line segment, the temperature is constant in this interval. Thus, the temperature at 1:30 p.m. was 36.5° C.

(e) Upward trends (graph going up from left to right) are observed from 9 a.m. to 10 a.m., 10 a.m. to 11 a.m., and 2 p.m. to 3 p.m.

Q2. 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?

(a) (i) Sales in 2002 were ₹ 4 crore. (ii) Sales in 2006 were ₹ 8 crore.

(b) (i) Sales in 2003 were ₹ 7 crore. (ii) Sales in 2005 were ₹ 10 crore.

(c) Sales in 2006 = ₹ 8 crore. Sales in 2002 = ₹ 4 crore. Difference = 8 - 4 = ₹ 4 crore.

(d) Difference between 2002 and 2003 = 7 - 4 = ₹ 3 crore.
Difference between 2003 and 2004 = 7 - 6 = ₹ 1 crore.
Difference between 2004 and 2005 = 10 - 6 = ₹ 4 crore.
Difference between 2005 and 2006 = 10 - 8 = ₹ 2 crore.
The greatest difference was in the year 2005.

Q3. 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.

(a) (i) Plant A after 2 weeks = 7 cm. (ii) Plant A after 3 weeks = 9 cm.

(b) (i) Plant B after 2 weeks = 7 cm. (ii) Plant B after 3 weeks = 10 cm.

(c) Growth of A during 3rd week = Height after 3 weeks - Height after 2 weeks = 9 - 7 = 2 cm.

(d) Growth of B during 3rd week = Height after 3 weeks - Height after 2 weeks = 10 - 7 = 3 cm.

(e) Growth of A: 1st week = 2 cm, 2nd week = 7-2 = 5 cm, 3rd week = 2 cm. It grew most during the 2nd week.

(f) Growth of B: 1st week = 1 cm, 2nd week = 7-1 = 6 cm, 3rd week = 3 cm. It grew least during the 1st week.

(g) Yes, they were of the same height at the end of the 2nd week (both were 7 cm).

Q4. 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?

(a) Both lines intersect on Tuesday, Friday, and Sunday.

(b) The highest point on the forecast line is 35° C (on Sunday).

(c) The lowest point on the actual temperature line is 15° C (on Thursday and Friday).

(d) By checking differences vertically, the maximum difference is on Thursday (Actual = 15° C, Forecast = 22.5° C. Difference = 7.5° C).

Exercise 15.2

Q1. 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)

(a) Yes, all points lie on a vertical line passing through x = 4. (Since all x-coordinates are 4). They form a line.

(b) Yes, all points lie on a straight line passing through the origin. (Since x-coordinate = y-coordinate for all). They form a line.

(c) No, these points do not lie on a single line. They form the vertices of a quadrilateral (specifically, a rectangle). They do not form a line.

Q2. 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.

Plot points (2, 3) and (3, 2). Draw a straight line passing through them. Extend the line to cut axes.
From the graph, we observe:
The line meets the x-axis at (5, 0).
The line meets the y-axis at (0, 5).

Q3. Write the coordinates of the vertices of each of these adjoining figures.
(Graph showing a rectangle OABC, parallelogram PQRS, triangle LMK)

Vertices of rectangle OABC: O(0, 0), A(2, 0), B(2, 3), C(0, 3).

Vertices of parallelogram PQRS: P(4, 3), Q(6, 1), R(6, 5), S(4, 7).

Vertices of triangle LMK: L(7, 7), M(10, 8), K(10, 5).

Q4. 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).

(i) True. Any point of the form (0, y) lies on the y-axis.

(ii) False. The point is (5, 0). Any point with y-coordinate zero lies on the x-axis.

(iii) True. Origin is the intersection of axes, (0, 0).

Exercise 15.3

Q1. 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 ₹): 5, 10, 15, 20, 25

Graph Construction:
X-axis: Number of apples (0, 1, 2, 3, 4, 5)
Y-axis: Cost in ₹ (0, 5, 10, 15, 20, 25)
Plot the points: (1, 5), (2, 10), (3, 15), (4, 20), (5, 25).
Join them to get a straight line passing through the origin.

(b) Distance travelled by a car:
Time (in hours): 6 a.m., 7 a.m., 8 a.m., 9 a.m.
Distance (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 its start?

Graph Construction:
X-axis: Time (with hours 6, 7, 8, 9).
Y-axis: Distance in km (0, 40, 80, 120, 160).
Plot points: (6, 40), (7, 80), (8, 120), (9, 160) and draw a line.

(i) From the graph, distance at 7.30 a.m. = 100 km. Distance at 8 a.m. = 120 km. Distance covered = 120 - 100 = 20 km.

(ii) Finding 100 km on the y-axis and tracing it to the x-axis, the corresponding time is 7.30 a.m.

Q2. 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 cm²): 4, 9, 16, 25, 36
Is it a linear graph?

(i) Plot points (2, 8), (3, 12), (3.5, 14), (5, 20), (6, 24).
When joined, they form a straight unbroken line.
Yes, it is a linear graph.

(ii) Plot points (2, 4), (3, 9), (4, 16), (5, 25), (6, 36).
When joined, they form a curve, not a straight line.
No, it is not a linear graph.

Class 8 Maths - Introduction to Graphs Practice Questions

Chapter 15: Introduction to Graphs (Practice Questions)

RD Sharma / Extra Practice Questions

Q1. What are the coordinates of the origin?

The coordinates of the origin are (0, 0), where the x-axis and y-axis intersect.

Q2. The point (5, 0) lies on which axis?

Since the y-coordinate is 0, the point lies on the x-axis.

Q3. The point (0, 7) lies on which axis?

Since the x-coordinate is 0, the point lies on the y-axis.

Q4. In the point (4, 9), what is the x-coordinate and what is the y-coordinate?

The x-coordinate (abscissa) is 4.
The y-coordinate (ordinate) is 9.

Q5. Which graph is best suited to show the temperature of a patient measured every hour?

A line graph is best suited to show continuous changes over periods of time, such as hourly temperature.

Q6. Name the graph which is an unbroken straight line.

A completely unbroken straight line graph is called a linear graph.

Q7. If a point lies on the x-axis, what is its y-coordinate?

If a point lies on the x-axis, its distance from the x-axis is zero, so its y-coordinate is 0.

Q8. A car travels at a uniform speed of 50 km/h. When we draw a distance-time graph, what kind of graph will it be?

Since the speed is uniform, distance increases steadily with time. The graph will be a linear graph (a straight line passing through the origin).

Q9. Plot the points A(2, 3), B(2, 6), C(2, 9). What do you observe?

Observation: All points have the same x-coordinate (which is 2). Therefore, they all lie on a vertical line parallel to the y-axis.

Q10. Plot the points P(1, 4), Q(5, 4), R(8, 4). What do you observe?

Observation: All points have the same y-coordinate (which is 4). Therefore, they all lie on a horizontal line parallel to the x-axis.

Q11. The cost of 1 pen is ₹ 10. The relation between number of pens (x) and total cost (y) is y = 10x. What is the value of y when x = 5?

Substitute x = 5 in the equation y = 10x.
y = 10 × 5 = 50.
The cost of 5 pens is ₹ 50.

Q12. What is the distance of the point (6, 8) from the y-axis?

The distance of a point from the y-axis is given by its x-coordinate.
The x-coordinate is 6. Therefore, the distance from the y-axis is 6 units.

Q13. In a distance-time graph, time is usually plotted on which axis?

Time is an independent variable. Independent variables are usually plotted on the x-axis (horizontal axis).

Q14. What are the coordinates of a point whose x-coordinate is 3 and which lies on the x-axis?

Since it lies on the x-axis, its y-coordinate is 0. The x-coordinate is given as 3.
So, the coordinates are (3, 0).

Q15. Write the coordinates of the vertices of a square of side 2 units, given that its one vertex is at the origin and two sides lie along the positive x and y axes.

One vertex is at origin: O(0, 0).
Second vertex on positive x-axis at distance 2: A(2, 0).
Third vertex on positive y-axis at distance 2: B(0, 2).
Fourth vertex forming the square: C(2, 2).

Q16. Find the perimeter of a rectangle whose vertices have the coordinates O(0,0), A(5,0), B(5,3) and C(0,3).

Length (distance from 0,0 to 5,0) = 5 units.
Breadth (distance from 5,0 to 5,3) = 3 units.
Perimeter = 2(Length + Breadth) = 2(5 + 3) = 2(8) = 16 units.

Q17. The graph between Principal and Simple Interest for a fixed rate of interest and time is a:

Since Simple Interest = (Principal × Rate × Time) / 100, if Rate and Time are fixed, SI is directly proportional to Principal.
Therefore, the graph is a linear graph (straight line passing through the origin).

Q18. State True or False: If a point's x-coordinate is zero and y-coordinate is non-zero, it lies on the y-axis.

True. Any point of the form (0, y) lies on the y-axis.

Q19. The two mutually perpendicular lines in a coordinate plane are called:

They are called coordinate axes (x-axis and y-axis).

Q20. From the point (3, 4), a perpendicular is drawn to x-axis. Find the coordinates of the point at which it meets the x-axis.

When a perpendicular is drawn from (3, 4) to the x-axis, the x-coordinate remains the same, but the y-coordinate becomes 0.
Therefore, it meets the x-axis at (3, 0).

Class 8 Maths - Introduction to Graphs Summary

Chapter 15: Introduction to Graphs (Concepts & Formulas)

1. Purpose of Graphs

The purpose of a graph is to show numerical facts in visual form so that they can be understood quickly, easily, and clearly. Thus, graphs are visual representations of data collected.

2. Types of Graphs

A Bar Graph: Used to show comparison among categories. It consists of two or more parallel vertical (or horizontal) bars (rectangles).
A Pie Graph (Circle Graph): Used to compare parts of a whole. The circle represents the whole.
A Histogram: A bar graph that shows data in intervals. It has adjacent bars over the intervals.
A Line Graph: Displays data that changes continuously over periods of time.
A Linear Graph: A line graph in which all the line segments form a single unbroken straight line.

3. Coordinates (Cartesian System)

To locate any point on a plane, we need a system. This system is called the Cartesian system. We use two number lines perpendicular to each other, one horizontal and one vertical.

X-axis: The horizontal number line.
Y-axis: The vertical number line.
Origin: The point of intersection of the x-axis and y-axis. It is denoted by O and its coordinates are (0, 0).
Coordinates of a point: A point is located by a pair of numbers, say (x, y).
- The first number 'x' represents the horizontal distance from the y-axis, called the x-coordinate or abscissa.
- The second number 'y' represents the vertical distance from the x-axis, called the y-coordinate or ordinate.

4. Some Applications

In everyday life, you might observe that the change in one quantity depends on the other. For example, if you read more books, you have to pay more money to the library. Here, money paid depends on the number of books read.

Independent Variable: The quantity which can change independently (e.g., number of books). Plotted on the x-axis.
Dependent Variable: The quantity whose value depends on the independent variable (e.g., money paid). Plotted on the y-axis.

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