graphical representation questions

Graphical Representation of Data

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables. Graphical representation of data is done through different mediums such as lines, plots, diagrams, etc. Let us learn more about this interesting concept of graphical representation of data, the different types, and solve a few examples.

Definition of Graphical Representation of Data

A graphical representation is a visual representation of data statistics-based results using graphs, plots, and charts. This kind of representation is more effective in understanding and comparing data than seen in a tabular form. Graphical representation helps to qualify, sort, and present data in a method that is simple to understand for a larger audience. Graphs enable in studying the cause and effect relationship between two variables through both time series and frequency distribution. The data that is obtained from different surveying is infused into a graphical representation by the use of some symbols, such as lines on a line graph, bars on a bar chart, or slices of a pie chart. This visual representation helps in clarity, comparison, and understanding of numerical data.

Representation of Data

The word data is from the Latin word Datum, which means something given. The numerical figures collected through a survey are called data and can be represented in two forms - tabular form and visual form through graphs. Once the data is collected through constant observations, it is arranged, summarized, and classified to finally represented in the form of a graph. There are two kinds of data - quantitative and qualitative. Quantitative data is more structured, continuous, and discrete with statistical data whereas qualitative is unstructured where the data cannot be analyzed.

Principles of Graphical Representation of Data

The principles of graphical representation are algebraic. In a graph, there are two lines known as Axis or Coordinate axis. These are the X-axis and Y-axis. The horizontal axis is the X-axis and the vertical axis is the Y-axis. They are perpendicular to each other and intersect at O or point of Origin. On the right side of the Origin, the Xaxis has a positive value and on the left side, it has a negative value. In the same way, the upper side of the Origin Y-axis has a positive value where the down one is with a negative value. When -axis and y-axis intersect each other at the origin it divides the plane into four parts which are called Quadrant I, Quadrant II, Quadrant III, Quadrant IV. This form of representation is seen in a frequency distribution that is represented in four methods, namely Histogram, Smoothed frequency graph, Pie diagram or Pie chart, Cumulative or ogive frequency graph, and Frequency Polygon.

Advantages and Disadvantages of Graphical Representation of Data

Listed below are some advantages and disadvantages of using a graphical representation of data:

• It improves the way of analyzing and learning as the graphical representation makes the data easy to understand.
• It can be used in almost all fields from mathematics to physics to psychology and so on.
• It is easy to understand for its visual impacts.
• It shows the whole and huge data in an instance.
• It is mainly used in statistics to determine the mean, median, and mode for different data

The main disadvantage of graphical representation of data is that it takes a lot of effort as well as resources to find the most appropriate data and then represent it graphically.

Rules of Graphical Representation of Data

While presenting data graphically, there are certain rules that need to be followed. They are listed below:

• Suitable Title: The title of the graph should be appropriate that indicate the subject of the presentation.
• Measurement Unit: The measurement unit in the graph should be mentioned.
• Proper Scale: A proper scale needs to be chosen to represent the data accurately.
• Index: For better understanding, index the appropriate colors, shades, lines, designs in the graphs.
• Data Sources: Data should be included wherever it is necessary at the bottom of the graph.
• Simple: The construction of a graph should be easily understood.
• Neat: The graph should be visually neat in terms of size and font to read the data accurately.

Uses of Graphical Representation of Data

The main use of a graphical representation of data is understanding and identifying the trends and patterns of the data. It helps in analyzing large quantities, comparing two or more data, making predictions, and building a firm decision. The visual display of data also helps in avoiding confusion and overlapping of any information. Graphs like line graphs and bar graphs, display two or more data clearly for easy comparison. This is important in communicating our findings to others and our understanding and analysis of the data.

Types of Graphical Representation of Data

Data is represented in different types of graphs such as plots, pies, diagrams, etc. They are as follows,

Related Topics

Listed below are a few interesting topics that are related to the graphical representation of data, take a look.

• x and y graph
• Frequency Polygon
• Cumulative Frequency

Examples on Graphical Representation of Data

Example 1 : A pie chart is divided into 3 parts with the angles measuring as 2x, 8x, and 10x respectively. Find the value of x in degrees.

We know, the sum of all angles in a pie chart would give 360º as result. ⇒ 2x + 8x + 10x = 360º ⇒ 20 x = 360º ⇒ x = 360º/20 ⇒ x = 18º Therefore, the value of x is 18º.

Example 2: Ben is trying to read the plot given below. His teacher has given him stem and leaf plot worksheets. Can you help him answer the questions? i) What is the mode of the plot? ii) What is the mean of the plot? iii) Find the range.

Solution: i) Mode is the number that appears often in the data. Leaf 4 occurs twice on the plot against stem 5.

Hence, mode = 54

ii) The sum of all data values is 12 + 14 + 21 + 25 + 28 + 32 + 34 + 36 + 50 + 53 + 54 + 54 + 62 + 65 + 67 + 83 + 88 + 89 + 91 = 958

To find the mean, we have to divide the sum by the total number of values.

Mean = Sum of all data values ÷ 19 = 958 ÷ 19 = 50.42

iii) Range = the highest value - the lowest value = 91 - 12 = 79

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Practice Questions on Graphical Representation of Data

Faqs on graphical representation of data, what is graphical representation.

Graphical representation is a form of visually displaying data through various methods like graphs, diagrams, charts, and plots. It helps in sorting, visualizing, and presenting data in a clear manner through different types of graphs. Statistics mainly use graphical representation to show data.

What are the Different Types of Graphical Representation?

The different types of graphical representation of data are:

• Stem and leaf plot
• Scatter diagrams
• Frequency Distribution

Is the Graphical Representation of Numerical Data?

Yes, these graphical representations are numerical data that has been accumulated through various surveys and observations. The method of presenting these numerical data is called a chart. There are different kinds of charts such as a pie chart, bar graph, line graph, etc, that help in clearly showcasing the data.

What is the Use of Graphical Representation of Data?

Graphical representation of data is useful in clarifying, interpreting, and analyzing data plotting points and drawing line segments , surfaces, and other geometric forms or symbols.

What are the Ways to Represent Data?

Tables, charts, and graphs are all ways of representing data, and they can be used for two broad purposes. The first is to support the collection, organization, and analysis of data as part of the process of a scientific study.

What is the Objective of Graphical Representation of Data?

The main objective of representing data graphically is to display information visually that helps in understanding the information efficiently, clearly, and accurately. This is important to communicate the findings as well as analyze the data.

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2: Graphical Representations of Data

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In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.

• 2.1: Introduction In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. Our emphasis will be on histograms and box plots.
• 2.2: Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs A stem-and-leaf plot is a way to plot data and look at the distribution, where all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends.  A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories.
• 2.3: Histograms, Frequency Polygons, and Time Series Graphs A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat.
• 2.4: Using Excel to Create Graphs Using technology to create graphs will make the graphs faster to create, more precise, and give the ability to use larger amounts of data. This section focuses on using Excel to create graphs.
• 2.5: Graphs that Deceive It's common to see graphs displayed in a misleading manner in social media and other instances. This could be done purposefully to make a point, or it could be accidental. Either way, it's important to recognize these instances to ensure you are not misled.
• 2.E: Graphical Representations of Data (Exercises) These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.

Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

• RD Sharma Solutions
• Chapter 23 Graphical Representation Of Statistical Data

RD Sharma Solutions for Class 9 Maths Chapter 23 Graphical Representation of Statistical Data

Rd sharma solutions class 9 maths chapter 23 – free pdf download.

RD Sharma Solutions for Class 9 Maths Chapter 23 Graphical Representation of Statistical Data are given here, which consists of questions and answers related to statistics. In this chapter, students will learn the graphical representation of data. The main objective of providing exercise-wise solutions in PDF is to help students speed up their exam preparation. The solutions are designed based on the latest CBSE Syllabus and exam pattern. Understanding the concepts before solving the exercise-wise problems is very important to obtain a better score in the annual exam. The students can download solutions that are available in PDFs easily for free.

Graphical representation is a better way to analyse any numerical data. A graph is a type of chart in which the statistical data are represented (In the form of lines or curves). Graphs help us to understand the relationship between variables and measure the position or values of one variable when the other variable is changed by a certain value. The RD Sharma Solutions Class 9 given here explains the solutions in a descriptive manner for the questions in Chapter 23. Click on the links below to access all the solutions.

• RD Sharma Class 9 Solutions Maths Chapter 1 Number System
• RD Sharma Class 9 Solutions Maths Chapter 2 Exponents of Real Numbers
• RD Sharma Class 9 Solutions Maths Chapter 3 Rationalisation
• RD Sharma Class 9 Solutions Maths Chapter 4 Algebraic Identities
• RD Sharma Class 9 Solutions Maths Chapter 5 Factorization of Algebraic Expressions
• RD Sharma Class 9 Solutions Maths Chapter 6 Factorization of Polynomials
• RD Sharma Class 9 Solutions Maths Chapter 7 Introduction to Euclids Geometry
• RD Sharma Class 9 Solutions Maths Chapter 8 Lines and Angles
• RD Sharma Class 9 Solutions Maths Chapter 9 Triangle and Its Angles
• RD Sharma Class 9 Solutions Maths Chapter 10 Congruent Triangles
• RD Sharma Class 9 Solutions Maths Chapter 11 Coordinate Geometry
• RD Sharma Class 9 Solutions Maths Chapter 12 Heron’s Formula
• RD Sharma Class 9 Solutions Maths Chapter 13 Linear Equation in Two Variables
• RD Sharma Class 9 Solutions Maths Chapter 14 Quadrilaterals
• RD Sharma Class 9 Solutions Maths Chapter 15 Areas Of Parallelograms and Triangles
• RD Sharma Class 9 Solutions Maths Chapter 16 Circles
• RD Sharma Class 9 Solutions Maths Chapter 17 Constructions
• RD Sharma Class 9 Solutions Maths Chapter 18 Surface Area and Volume of A Cuboid and Cube
• RD Sharma Class 9 Solutions Maths Chapter 19 Surface Area and Volume of Right Circular Cylinder>
• RD Sharma Class 9 Solutions Maths Chapter 20 Surface Area and Volume of Right Circular Cone
• RD Sharma Class 9 Solutions Maths Chapter 21 Surface Area and Volume of Sphere
• RD Sharma Class 9 Solutions Maths Chapter 22 Tabular Representation of Statistical Data
• RD Sharma Class 9 Solutions Maths Chapter 23 Graphical Representation of Statistical Data
• RD Sharma Class 9 Solutions Maths Chapter 24 Measures of Central Tendency
• RD Sharma Class 9 Solutions Maths Chapter 25 Probability
• Exercise 23.1 Chapter 23 Graphical Representation of Statistical Data
• Exercise 23.2 Chapter 23 Graphical Representation of Statistical Data
• Exercise 23.3 Chapter 23 Graphical Representation of Statistical Data

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Access Answers to RD Sharma Solutions for Class 9 Maths Chapter 23 Graphical Representation of Statistical Data

Exercise 23.1 page no: 23.7.

Question 1: The following table shows the daily production of T.V. sets in an industry for 7 days of a week.

Represent the above information by a pictograph.

The given information can be represented using a pictograph as below:

Question 2: The following table shows the number of Maruti cars sold by five dealers in a particular month:

Question 3: The population of Delhi State in different census years is as given below:

Represent the above information with the help of a bar graph.

Let us consider the horizontal and vertical axes represent the years and population in lakhs, respectively.

The heights of the rectangles are proportional to the population in lakhs.

Question 4: Read the bar graph shown below and answer the following questions:

(i) What is the information given by the bar graph?

(ii) How many tickets from Assam State Lottery were sold by the agent?

(iii) Of which state was the maximum number of tickets sold?

(iv) State whether true or false.

The maximum number of tickets sold is three times the minimum number of tickets sold.

(v) Of which state was the minimum number of tickets sold?

(i) Bar graph represents the number of tickets of different state lotteries sold by an agent on a day.

(ii) Number of tickets of Assam State Lottery were sold by the agent = 40.

(iii) The maximum number of tickets were sold is 100 in the state of Haryana.

(iv) The maximum number of tickets were sold is 100 in the state of Haryana. The minimum number of tickets were sold is 20 in the state of Rajasthan.

It is clear that 100 is equal to 5 times of 20.

Hence, the statement is false.

(v) The minimum number of tickets were sold is 20 in the state of Rajasthan.

Question 5: Study the bar graph representing the number of persons in various age groups in a town shown in the figure. Observe the bar graph and answer the following questions:

(i) What is the percentage of the youngest age-group persons over those in the oldest age group?

(ii) What is the total population of the town?

(iii) What is the number of persons in the age-group 60-65?

(iv) How many persons are more in the age-group 10-15 than in the age group 30-35?

(v) What is the age-group of exactly 1200 persons living in the town?

(vi) What is the total number of persons living in the town in the age-group 50-55?

(vii) What is the total number of persons living in the town in the age-groups 10-15 and 60-65?

(vii) Whether the population in general increases, decreases or remains constant with the increase in the age-group.

(i) Youngest age-group is 10-15 years

The number of persons belonging to this group = 1400

The oldest age-group is 70-75 years and

The number of persons belonging to this group = 300

The percentage of youngest age-group persons over those in the oldest group is as below:

1400/300 × 100 = 1400/3

(ii) Population of the town = 300 + 800 + 900 + 1000 + 1100 + 1200 + 1400 = 6700

(iii) Number of persons in the age group 60 – 65 = 800.

(iv) Number of persons in the age group 10 – 15 = 1400

The number of persons in the age group 30-35 = 1100.

Hence the number of more persons in the age group 10 – 15 than the group 30-35 is 1400 — 1100 = 300.

(v) Age group of 1200 persons living in the town is 20 – 25.

(vi) The total number of persons living in the town in the age-group 50 – 55 is 900.

(vii) The total number of persons living in the town in the age-groups 10 -15 and 60 – 65 is 1400 + 800 = 2200.

(viii) We have observed that the height of the bars decreases as the age-group increases. Hence, the population decreases with the increases in the age-group.

Exercise 23.2 Page No: 23.23

Question 1: Explain the reading and interpretation of bar graphs.

Solution: A bar graph consists of a sequence of vertical or horizontal bar lines or rectangles. Bar lines may be either horizontal or vertical. We can easily collect the information and conclude various observations from a given bar graph which is referred to as the interpretation of the bar graph.

Question 2: Read the following bar graph and answer the following questions:

(i) What information is given by the bar graph?

(ii) In which year is the export minimum?

(iii) In which year is the import maximum?

(iv) In which year the difference between the values of export and import is maximum?

(i) The bar graph represents the import and export (in 100 Crores of rupees) from 1982-83 to 1986-87.

(ii) 1982-83

(iii) 1986-87

(iv) 1986-87

Question 3: The following bar graph shows the results of an annual examination in a secondary school. Read the bar graph given below, and choose the correct alternative in each of the following:

(i)The pair of classes in which the results of boys and girls are inversely proportional are:

(a) VI, VIII (b) VI, IX (c) VII, IX (d) VIII, X

(ii) The class having the lowest failure rate of girls is:

(a) VI (b) X (c) IX (d) VIII

(iii) The class having the lowest pass rate of students is:

(a) VI (b) VII (c) VIII (d) IX

(i) Option (b) is correct.

(ii) Option (a) is correct.

(iii) Option (b) is correct.

The sum of the heights of the bars for boys and girls in class VII = 95 + 40 = 135 (which is minimum)

Question 4: The following data gives the number (in thousands) of applicants registered with an Employment Exchange during 1995-2000:

Construct a bar graph to represent the above data.

Let us consider that the horizontal and vertical axes represent the years and the number of applicants registered in thousands, respectively.

Question 5: The production of saleable steel in some of the steel plants of our country during 1999 is given below:

Construct a bar graph to represent the above data on a graph paper by using the scale 1 big division = 20 thousand tonnes.

Let us consider that the horizontal and vertical axes represent the plants and the production in thousand tonnes, respectively.

Question 6: The following table gives the route length (in thousand kilometres) of the Indian Railways in some of the years:

Represent the above data with the help of a bar graph.

Let us consider that the horizontal and vertical axes represent the years and the route lengths in thousand km, respectively.

Exercise 23.3 Page No: 23.41

Question 1: Construct a histogram for the following data:

Let us consider that the horizontal and vertical axes represent the monthly school fees and the number of schools, respectively. Construct rectangles with class-intervals as bases and respective frequencies as heights as below.

Question 2: The distribution of heights (in cm) of 96 children is given below. Construct a histogram and a frequency polygon on the same axes.

Let us consider that the horizontal and vertical axes represent the height (in cm) and the number of children, respectively. Construct rectangles with class-intervals as bases and respective frequencies as heights as below.

Question 3: The time taken in seconds to solve a problem by each of 25 pupils is as follows:

16, 20, 26, 27, 28, 30, 33, 37, 38, 40, 42, 43, 46, 46, 46, 48, 49, 50, 53, 58, 59, 60, 64, 52, 20

(a) Construct a frequency distribution for these data using a class interval of 10 seconds.

(b) Draw a histogram to represent the frequency distribution.

Arrange raw data into ascending order:

16, 20, 20, 26, 27, 28, 30, 33, 37, 38, 40, 42, 43, 46, 46, 46, 48, 49, 50, 52, 53, 58, 59, 60, 64

(a) Frequency distribution for the given data, using a class interval of 10 seconds.

Consider horizontal and vertical axes represent the seconds and frequency, respectively. Frequencies are the heights of rectangles.

Question 4: Draw, in the same diagram, a histogram and a frequency polygon to represent the following data which shows the monthly cost of living index of a city in a period of 2 years:

Consider the horizontal axis as the cost of living (in Rs.), and the vertical axis represents the number of months.

Histogram and a frequency polygon:

In Chapter 23 of Class 9 RD Sharma Solutions , students will study important concepts as listed below:

• Graphical representation of statistical data introduction
• Frequency polygon and its algorithm and many more

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SAT Mathematics : Graphical Representation of Functions

Study concepts, example questions & explanations for sat mathematics, all sat mathematics resources, example questions, example question #1 : graphing quadratics & polynomials.

Example Question #2 : Graphing Quadratics & Polynomials

Example Question #3 : Graphing Quadratics & Polynomials

Complete the square to calculate the maximum or minimum point of the given function.

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

where when multiplied out,

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term. 

Complete the square for this particular function is as follows.

First identify the middle term coefficient.

Now divide the middle term coefficient by two.

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

When simplified the new function is,

Example Question #4 : Graphing Quadratics & Polynomials

From here write the function with the perfect square.

Example Question #5 : Graphing Quadratics & Polynomials

First factor out a negative one.

Now identify the middle term coefficient.

Example Question #6 : Graphing Quadratics & Polynomials

In which quadrant of the xy-plane does the vertex of the following function lie?

The vertex lies on one of the axes, not in a quadrant.

Quadrant III

Quadrant II

Start by factoring out the common 3 to see that:

Note: You could also find the vertex by simply leveraging the two formulas below:

Example Question #7 : Graphing Quadratics & Polynomials

This equation isn’t at all impossible to factor, but you may not notice that at first glance (or on a future problem you may try to factor and get stuck). If you see this problem on the calculator section, it’s likely most efficient to simply graph it on your calculator and count the number of times that the graph touches the y-axis. Your graph will look like:

Example Question #8 : Graphing Quadratics & Polynomials

Which of the following represents the maximum or minimum point of the given function?

Middle Term Coefficient = 2

From here, substitute the the  x  value into the original function.

Therefore the minimum value occurs at the point  ( − 1 , − 5 ) .

Example Question #9 : Graphing Quadratics & Polynomials

Which of the following is an equivalent form of the function f above for which the minimum value of f appears as a constant or coefficient?

By either graphing the function, utilizing the formula for the vertex, or completing the square, we can see that the minimum value of the function is -49. In order to comply with the parameters of the question stem, this minimum must visually appear in the answer option, so 

is our only feasible option. That said, we can prove this process by "completing the square."

Here, if we foil 

we arrive at

to complete the square, we want to add, and subsequently subtract 4 as shown below

which simplifies to

and can be rewritten as 

our equivalent equation in vertex form!

Example Question #10 : Graphing Quadratics & Polynomials

none of the above

II and III 

To tackle this question, we'll want to see the number sense behind factoring by inspection. In the quadratic construction 

we're looking for two numbers that multiply to give us our constant, "c," and sum to give us the coefficient "b."

In this case, those two numbers are 9 and -4, meaning we can reconstruct the function to read

since we're looking for our x intercepts or "zeros," we want the case where our output or "y" = 0, so we can set the righthand side of the function equal to zero as such

in order for this equation to hold true, either of our parentheticals will need to equal zero, meaning that if

then x could be equal to -9 or 4. So,  I and III are zeros of the function. 

*Note - if given this question on a calculator-friendly section, we could graph and identify zeros as well*

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• School Guide
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• NCERT Soln. Class 9 Maths
• RD Sharma Soln. Class 9
• Math Formulas Class 9
• CBSE Class 9 Maths Revision Notes

Chapter 1: Number System

• Number System in Maths
• Natural Numbers | Definition, Examples, Properties
• Whole Numbers - Definition, Properties and Examples
• Rational Number: Definition, Examples, Worksheet
• Irrational Numbers: Definition, Examples, Symbol, Properties
• Real Numbers
• Decimal Expansion of Real Numbers
• Decimal Expansions of Rational Numbers
• Representation of Rational Numbers on the Number Line | Class 8 Maths
• Represent √3 on the number line
• Operations on Real Numbers
• Rationalization of Denominators
• Laws of Exponents for Real Numbers

Chapter 2: Polynomials

• Polynomials in One Variable - Polynomials | Class 9 Maths
• Polynomial Formula
• Types of Polynomials
• Zeros of Polynomial
• Factorization of Polynomial
• Remainder Theorem
• Factor Theorem
• Algebraic Identities

Chapter 3: Coordinate Geometry

• Coordinate Geometry
• Cartesian Coordinate System
• Cartesian Plane

Chapter 4: Linear equations in two variables

• Linear Equations in One Variable
• Linear Equation in Two Variables
• Graph of Linear Equations in Two Variables
• Graphical Methods of Solving Pair of Linear Equations in Two Variables
• Equations of Lines Parallel to the x-axis and y-axis

Chapter 5: Introduction to Euclid's Geometry

• Euclidean Geometry
• Equivalent Version of Euclid’s Fifth Postulate

Chapter 6: Lines and Angles

• Lines and Angles
• Types of Angles
• Pairs of Angles - Lines & Angles
• Transversal Lines
• Angle Sum Property of a Triangle

Chapter 7: Triangles

• Triangles in Geometry
• Congruence of Triangles |SSS, SAS, ASA, and RHS Rules
• Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 Maths
• Triangle Inequality Theorem, Proof & Applications

Chapter 8: Quadrilateral

• Angle Sum Property of a Quadrilateral
• Quadrilateral - Definition, Properties, Types, Formulas, Examples
• Introduction to Parallelogram: Properties, Types, and Theorem
• Rhombus: Definition, Properties, Formula, Examples
• Trapezium in Maths | Formulas, Properties & Examples
• Square in Maths - Area, Perimeter, Examples & Applications
• Kite - Quadrilaterals
• Properties of Parallelograms
• Mid Point Theorem

Chapter 9: Areas of Parallelograms and Triangles

• Area of Triangle | Formula and Examples
• Area of Parallelogram (Definition, Formulas & Examples)
• Figures on the Same Base and between the Same Parallels

Chapter 10: Circles

• Circles in Maths
• Radius of Circle
• Tangent to a Circle
• What is the longest chord of a Circle?
• Circumference of Circle - Definition, Perimeter Formula, and Examples
• Angle subtended by an arc at the centre of a circle
• What is Cyclic Quadrilateral
• Theorem - The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths

Chapter 11: Construction

• Basic Constructions - Angle Bisector, Perpendicular Bisector, Angle of 60°
• Construction of Triangles

Chapter 12: Heron's Formula

• Area of Equilateral Triangle
• Area of Isosceles Triangle
• Heron's Formula
• Applications of Heron's Formula
• Area of Quadrilateral
• Area of Polygons

Chapter 13: Surface Areas and Volumes

• Surface Area of Cuboid
• Volume of Cuboid | Formula and Examples
• Surface Area of Cube
• Volume of a Cube
• Surface Area of Cylinder
• Volume of a Cylinder: Formula, Definition and Examples
• Surface Area of Cone
• Volume of Cone: Formula, Derivation and Examples
• Surface Area of Sphere: Formula, Derivation and Solved Examples
• Volume of a Sphere
• Surface Area of a Hemisphere
• Volume of Hemisphere

Chapter 14: Statistics

• Collection and Presentation of Data

Graphical Representation of Data

• Bar Graphs and Histograms
• Central Tendency
• Mean, Median and Mode

Chapter 15: Probability

• Experimental Probability
• Empirical Probability
• CBSE Class 9 Maths Formulas
• NCERT Solutions for Class 9 Maths: Chapter Wise PDF 2024
• RD Sharma Class 9 Solutions

Graphical Representation of Data: In today’s world of the internet and connectivity, there is a lot of data available, and some or other method is needed for looking at large data, the patterns, and trends in it.

There is an entire branch in mathematics dedicated to dealing with collecting, analyzing, interpreting, and presenting numerical data in visual form in such a way that it becomes easy to understand and the data becomes easy to compare as well, the branch is known as Statistics .

The branch is widely spread and has a plethora of real-life applications such as Business Analytics, demography, Astro statistics, and so on. In this article, we have provided everything about the graphical representation of data, including its types, rules, advantages, etc.

Table of Content

• What is Graphical Representation?

Types of Graphical Representations

Graphical representations used in maths, principles of graphical representations, advantages and disadvantages of using graphical system, general rules for graphical representation of data, solved examples on graphical representation of data, what is graphical representation.

Graphics Representation is a way of representing any data in picturized form. It helps a reader to understand the large set of data very easily as it gives us various data patterns in visualized form.

There are two ways of representing data,

• Pictorial Representation through graphs.

They say, “A picture is worth a thousand words”.  It’s always better to represent data in a graphical format. Even in Practical Evidence and Surveys, scientists have found that the restoration and understanding of any information is better when it is available in the form of visuals as Human beings process data better in visual form than any other form.

Does it increase the ability 2 times or 3 times? The answer is it increases the Power of understanding 60,000 times for a normal Human being, the fact is amusing and true at the same time.

Comparison between different items is best shown with graphs, it becomes easier to compare the crux of the data about different items. Let’s look at all the different types of graphical representations briefly: 

Line Graphs

A line graph is used to show how the value of a particular variable changes with time. We plot this graph by connecting the points at different values of the variable. It can be useful for analyzing the trends in the data and predicting further trends. 

A bar graph is a type of graphical representation of the data in which bars of uniform width are drawn with equal spacing between them on one axis (x-axis usually), depicting the variable. The values of the variables are represented by the height of the bars. 

Histograms 

This is similar to bar graphs, but it is based frequency of numerical values rather than their actual values. The data is organized into intervals and the bars represent the frequency of the values in that range. That is, it counts how many values of the data lie in a particular range. 

It is a plot that displays data as points and checkmarks above a number line, showing the frequency of the point. 

Stem and Leaf Plot 

This is a type of plot in which each value is split into a “leaf”(in most cases, it is the last digit) and “stem”(the other remaining digits). For example: the number 42 is split into leaf (2) and stem (4).  

Box and Whisker Plot 

These plots divide the data into four parts to show their summary. They are more concerned about the spread, average, and median of the data. 

It is a type of graph which represents the data in form of a circular graph. The circle is divided such that each portion represents a proportion of the whole. 

Graphs in Math are used to study the relationships between two or more variables that are changing. Statistical data can be summarized in a better way using graphs. There are basically two lines of thoughts of making graphs in maths: 

• Value-Based or Time Series Graphs

Frequency Based

Value-based or time series graphs .

These graphs allow us to study the change of a variable with respect to another variable within a given interval of time. The variables can be anything. Time Series graphs study the change of variable with time. They study the trends, periodic behavior, and patterns in the series. We are more concerned with the values of the variables here rather than the frequency of those values. 

Example: Line Graph

These kinds of graphs are more concerned with the distribution of data. How many values lie between a particular range of the variables, and which range has the maximum frequency of the values. They are used to judge a spread and average and sometimes median of a variable under study. 

Example: Frequency Polygon, Histograms.

All types of graphical representations require some rule/principles which are to be followed. These are some algebraic principles. When we plot a graph, there is an origin, and we have our two axes. These two axes divide the plane into four parts called quadrants. The horizontal one is usually called the x-axis and the other one is called the y-axis. The origin is the point where these two axes intersect.

The thing we need to keep in mind about the values of the variable on the x-axis is that positive values need to be on the right side of the origin and negative values should be on the left side of the origin. Similarly, for the variable on the y-axis, we need to make sure that the positive values of this variable should be above the x-axis and negative values of this variable must be below the y-axis. 

• It gives us a summary of the data which is easier to look at and analyze.
• It saves time.
• We can compare and study more than one variable at a time.

Disadvantages

It usually takes only one aspect of the data and ignores the other. For example, A bar graph does not represent the mean, median, and other statistics of the data. 

We should keep in mind some things while plotting and designing these graphs. The goal should be a better and clear picture of the data. Following things should be kept in mind while plotting the above graphs: 

• Whenever possible, the data source must be mentioned for the viewer.
• Always choose the proper colors and font sizes. They should be chosen to keep in mind that the graphs should look neat.
• The measurement Unit should be mentioned in the top right corner of the graph.
• The proper scale should be chosen while making the graph, it should be chosen such that the graph looks accurate.
• Last but not the least, a suitable title should be chosen.

Frequency Polygon

A frequency polygon is a graph that is constructed by joining the midpoint of the intervals. The height of the interval or the bin represents the frequency of the values that lie in that interval. 

• Diagrammatic and Graphic Presentation of Data
• What are the different ways of Data Representation?

Question 1: What are different types of frequency-based plots? 

Types of frequency based plots:  Histogram Frequency Polygon Box Plots

Question 2: A company with an advertising budget of Rs 10,00,00,000 has planned the following expenditure in the different advertising channels such as TV Advertisement, Radio, Facebook, Instagram, and Printed media. The table represents the money spent on different channels. 

Draw a bar graph for the following data. 

• Put each of the channels on the x-axis
• The height of the bars is decided by the value of each channel.

Question 3: Draw a line plot for the following data 

• Put each of the x-axis row value on the x-axis
• joint the value corresponding to the each value of the x-axis.

Question 4: Make a frequency plot of the following data: 

• Draw the class intervals on the x-axis and frequencies on the y-axis.
• Calculate the mid point of each class interval.

Now join the mid points of the intervals and their corresponding frequencies on the graph. 

This graph shows both the histogram and frequency polygon for the given distribution.

Graphical Representation of Data – FAQs

What are the advantages of using graphs to represent data.

Graphs offer visualization, clarity, and easy comparison of data, aiding in outlier identification and predictive analysis.

What are the common types of graphs used for data representation?

Common graph types include bar, line, pie, histogram, and scatter plots, each suited for different data representations and analysis purposes.

How do you choose the most appropriate type of graph for your data?

Select a graph type based on data type, analysis objective, and audience familiarity to effectively convey information and insights.

How do you create effective labels and titles for graphs?

Use descriptive titles, clear axis labels with units, and legends to ensure the graph communicates information clearly and concisely.

How do you interpret graphs to extract meaningful insights from data?

Interpret graphs by examining trends, identifying outliers, comparing data across categories, and considering the broader context to draw meaningful insights and conclusions.

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MCQ on Graphical Representation of Data

Graphical representation of data uses charts, graphs, and diagrams to visually present information and patterns. It enhances understanding, aids in data analysis, and simplifies complex data, making it accessible to a wider audience. Explore our interactive quiz or MCQ on Graphical Representation of Data to test your comprehension of various graphs, charts, and their applications.

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50 Essential Graph Data Structure Interview Questions

Graphs are non-linear data structures that consist of vertices (or nodes) connected by edges. They can represent complex real-world systems like network topology, social networks, or web pages, and therefore often feature in coding interviews. Candidates’ understanding of this structure, its traversal algorithms like Depth-First Search and Breadth-First Search , along with knowledge of pathfinding or graph theory algorithms , play a crucial role during a tech interview process.

Introduction to Graphs

What is a graph .

A graph is a data structure that represents a collection of interconnected nodes through a set of edges .

This abstract structure is highly versatile and finds applications in various domains, from social network analysis to computer networking.

Core Components

A graph consists of two main components:

• Nodes : Also called vertices , these are the fundamental units that hold data.
• Edges : These are the connections between nodes, and they can be either directed or undirected .

Visual Representation

Graph Representations

There are several ways to represent graphs in computer memory, with the most common ones being adjacency matrix , adjacency list , and edge list .

Adjacency Matrix

In an adjacency matrix, a 2D Boolean array indicates the edges between nodes. A value of True at index [i][j] means that an edge exists between nodes i and j .

Here is the Python code:

Adjacency List

An adjacency list represents each node as a list, and the indices of the list are the nodes. Each node’s list contains the nodes that it is directly connected to.

An edge list is a simple list of tuples, where each tuple represents an edge between two nodes.

What are some common Types and Categories of Graphs ?

Graphs serve as adaptable data structures for various computational tasks and real-world applications. Let’s look at their diverse types.

Types of Graphs

• Undirected : Edges lack direction, allowing free traversal between connected nodes. Mathematically, ( u , v ) (u,v) as an edge implies ( v , u ) (v,u) as well.
• Directed (Digraph) : Edges have a set direction, restricting traversal accordingly. An edge ( u , v ) (u,v) doesn’t guarantee ( v , u ) (v,u) .

Weight Considerations

• Weighted : Each edge has a numerical “weight” or “cost.”
• Unweighted : All edges are equal in weight, typically considered as 1.

Presence of Cycles

• Cyclic : Contains at least one cycle or closed path.
• Acyclic : Lacks cycles entirely.

Edge Density

• Dense : High edge-to-vertex ratio, nearing the maximum possible connections.
• Sparse : Low edge-to-vertex ratio, closer to the minimum.

Connectivity

• Connected : Every vertex is reachable from any other vertex.
• Disconnected : Some vertices are unreachable from others.

Edge Uniqueness

• Multigraph : Allows duplicate edges between vertices.
• Simple : Limits vertices to a single connecting edge.

What is the difference between a Tree and a Graph ?

Graphs and trees are both nonlinear data structures, but there are fundamental distinctions between them.

Key Distinctions

• Uniqueness : Trees have a single root, while graphs may not have such a concept.
• Topology : Trees are hierarchical , while graphs can exhibit various structures.
• Focus : Graphs center on relationships between individual nodes, whereas trees emphasize the relationship between nodes and a common root.

Graphs: Versatile and Unstructured

• Elements : Composed of vertices/nodes (denoted as V) and edges (E) representing relationships. Multiple edges and loops are possible.
• Directionality : Edges can be directed or undirected.
• Connectivity : May be disconnected , with sets of vertices that aren’t reachable from others.
• Loops : Can contain cycles.

Trees: Hierarchical and Organized

• Elements : Consist of nodes with parent-child relationships.
• Directionality : Edges are strictly parent-to-child.
• Connectivity : Every node is accessible from the unique root node.
• Loops : Cycles are not allowed.

How can you determine the Minimum number of edges for a graph to remain connected?

To ensure a graph remains connected , it must have a minimum number of edges determined by the number of vertices. This is known as the edge connectivity of the graph.

Edge Connectivity Formula

The minimum number of edges required for a graph to remain connected is given by:

Edge Connectivity = max ⁡ ( δ ( G ) , 1 ) \text{{Edge Connectivity}} = \max(\delta(G),1)

• δ ( G ) \delta(G) is the minimum degree of a vertex in G G .
• The maximum function ensures that the graph remains connected even if all vertices have a degree of 1 or 0.

For example, a graph with a minimum vertex degree of 3 or more requires at least 3 edges to stay connected.

Define Euler Path and Euler Circuit in the context of graph theory.

In graph theory , an Euler Path and an Euler Circuit serve as methods to visit all edges (links) exactly once, with the distinction that an Euler Circuit also visits all vertices once.

Euler Path and Euler Circuit Definitions

A graph has an Euler Path if it contains exactly two vertices of odd degree.

A graph has an Euler Circuit if every vertex has even degree.

Degree specifies the number of edges adjacent to a vertex.

Key Concepts

• Starting Vertex : In an Euler Path, the unique starting and ending vertices are the two with odd degrees.
• Reachability : In both Euler Path and Circuit, every edge must be reachable from the starting vertex.
• Direction-Consistency : While an Euler Path is directionally open-ended, an Euler Circuit is directionally closed.

Visual Representation: Euler Path and Circuit

Compare Adjacency Lists and Adjacency Matrices for graph representation.

Graphs can be represented in various ways, but Adjacency Matrix and Adjacency List are the most commonly used data structures. Each method offers distinct advantages and trade-offs, which we’ll explore below.

Space Complexity

• Adjacency Matrix : Requires a N × N N \times N matrix, resulting in O ( N 2 ) O(N^2) space complexity.
• Adjacency List : Utilizes a list or array for each node’s neighbors, leading to O ( N + E ) O(N + E) space complexity, where E E is the number of edges.

Time Complexity for Edge Look-Up

• Adjacency Matrix : Constant time, O ( 1 ) O(1) , as the presence of an edge is directly accessible.
• Adjacency List : Up to O ( k ) O(k) , where k k is the degree of the vertex, as the list of neighbors may need to be traversed.

Time Complexity for Traversal

• Adjacency Matrix : Requires O ( N 2 ) O(N^2) time to iterate through all potential edges.
• Adjacency List : Takes O ( N + E ) O(N + E) time, often faster for sparse graphs.

Time Complexity for Edge Manipulation

• Adjacency Matrix : O ( 1 ) O(1) for both addition and removal, as it involves updating a single cell.
• Adjacency List : O ( k ) O(k) for addition or removal, where k k is the degree of the vertex involved.

Time Complexity for Vertex Manipulation

• Adjacency Matrix : O ( N 2 ) O(N^2) as resizing the matrix is needed.
• Adjacency List : O ( 1 ) O(1) as it involves updating a list or array.

Code Example: Adjacency Matrix & Adjacency List

What is an incidence matrix , and when would you use it.

An incidence matrix is a binary graph representation that maps vertices to edges. It’s especially useful for directed and multigraphs . The matrix contains 0 0 s and 1 1 s, with positions corresponding to “vertex connected to edge” relationships.

Matrix Structure

• Columns : Represent edges
• Rows : Represent vertices
• Cells : Indicate whether a vertex is connected to an edge

Each unique row-edge pair depicts an incidence of a vertex in an edge, relating to the graph’s structure differently based on the graph type.

Example: Incidence Matrix for a Directed Graph

Example: Incidence Matrix for an Undirected Multigraph

Applications of Incidence Matrices

• Algorithm Efficiency : Certain matrix operations can be faster than graph traversals.
• Graph Comparisons : It enables direct graph-to-matrix or matrix-to-matrix comparisons.
• Database Storage : A way to represent graphs in databases amongst others.
• Graph Transformations : Useful in transformations like line graphs and dual graphs.

Discuss Edge List as a graph representation and its use cases.

Edge list is a straightforward way to represent graphs. It’s apt for dense graphs and offers a quick way to query edge information.

• Edge Storage : The list contains tuples (a, b) to denote an edge between nodes a a and b b .
• Edge Direction : The edges can be directed or undirected.
• Edge Duplicates : Multiple occurrences signal multigraph. Absence ensures simple graph.

Visual Example

Code Example: Edge List

Here is the Python 3 code:

Explain how to save space while storing a graph using Compressed Sparse Row (CSR).

In Compressed Sparse Row format, the graph is represented by three linked arrays. This streamlined approach can significantly reduce memory use and is especially beneficial for sparse graphs .

Let’s go through the data structures and the detailed process.

Data Structures

• Indptr Array (IA) : A list of indices where each row starts in the adj_indices array. It’s of length n_vertices + 1 .
• Adjacency Index Array (AA) : The column indices for each edge based on their position in the indptr array.
• Edge Data : The actual edge data. This array’s length matches the number of non-zero elements.

Code Example: CSR Graph Representation

Graph traversal algorithms, explain the breadth-first search (bfs) traversing method..

Breadth-First Search (BFS) is a graph traversal technique that systematically explores a graph level by level. It uses a queue to keep track of nodes to visit next and a list to record visited nodes, avoiding redundancy.

Key Components

• Queue : Maintains nodes in line for exploration.
• Visited List : Records nodes that have already been explored.

Algorithm Steps

• Initialize : Choose a starting node, mark it as visited, and enqueue it.
• Explore : Keep iterating as long as the queue is not empty. In each iteration, dequeue a node, visit it, and enqueue its unexplored neighbors.
• Terminate : Stop when the queue is empty.

Complexity Analysis

Time Complexity : O ( V + E ) O(V + E) where V V is the number of vertices in the graph and E E is the number of edges. This is because each vertex and each edge will be explored only once.

Space Complexity : O ( V ) O(V) since, in the worst case, all of the vertices can be inside the queue.

Code Example: Breadth-First Search

Explain the depth-first search (dfs) algorithm..

Depth-First Search (DFS) is a graph traversal algorithm that’s simpler and often faster than its breadth-first counterpart (BFS). While it might not explore all vertices , DFS is still fundamental to numerous graph algorithms.

• Initialize : Select a starting vertex, mark it as visited, and put it on a stack.
• Remove the top vertex from the stack.
• Explore its unvisited neighbors and add them to the stack.
• Finish : When the stack is empty, the algorithm ends, and all reachable vertices are visited.

Code Example: Depth-First Search

What are the key differences between bfs and dfs .

BFS and DFS are both essential graph traversal algorithms with distinct characteristics in strategy, memory requirements, and use-cases.

Core Differences

• Search Strategy : BFS moves level-by-level, while DFS goes deep into each branch before backtracking.
• Data Structures : BFS uses a Queue, whereas DFS uses a Stack or recursion.
• Space Complexity : BFS requires more memory as it may need to store an entire level ( O ( ∣ V ∣ ) O(|V|) ), whereas DFS usually uses less ( O ( log ⁡ n ) O(\log n) on average).
• Optimality : BFS guarantees the shortest path; DFS does not.

Code Example: BFS & DFS

Implement a method to check if there is a path between two vertices in a graph., problem statement.

Given an undirected graph, the task is to determine whether or not there is a path between two specified vertices.

The problem can be solved using Depth-First Search (DFS) .

• Start from the source vertex.
• For each adjacent vertex, if not visited, recursively perform DFS.
• If the destination vertex is found, return True . Otherwise, backtrack and explore other paths.
• Time Complexity : O ( V + E ) O(V + E) V V is the number of vertices, and E E is the number of edges.
• Space Complexity : O ( V ) O(V) For the stack used in recursive DFS calls.

Implementation

Solve the problem of printing all paths from a source to destination in a directed graph with bfs or dfs..

Given a directed graph and two vertices s r c src and d e s t dest , the objective is to print all paths from s r c src to d e s t dest .

Recursive Depth-First Search (DFS) Algorithm in Graphs: DFS is used because it can identify all the paths in a graph from source to destination. This is done by employing a backtracking mechanism to ensure that all unique paths are found.

To deal with cycles , a list of visited nodes is crucial. By utilizing this list, the algorithm can avoid revisiting and getting stuck in an infinite loop.

Time Complexity : O ( V + E ) O(V + E)

• V V is the number of vertices and E E is the number of edges.
• We’re essentially visiting every node and edge once.

Space Complexity : O ( V ) O(V)

• In the worst-case scenario, the entire graph can be visited, which would require space proportional to the number of vertices.

Graph Properties and Types

What is a bipartite graph how to detect one.

A bipartite graph is one where the vertices can be divided into two distinct sets, U U and V V , such that every edge connects a vertex from U U to one in V V . The graph is denoted as G = ( U , V , E ) G = (U, V, E) , where E E represents the set of edges.

Detecting a Bipartite Graph

You can check if a graph is bipartite using several methods:

Breadth-First Search (BFS)

BFS is often used for this purpose. The algorithm colors vertices alternately so that no adjacent vertices have the same color.

Code Example: Bipartite Graph using BFS

Cycle detection.

A graph is not bipartite if it contains an odd cycle. Algorithms like DFS or Floyd’s cycle-detection algorithm can help identify such cycles.

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• Linear Search
• Binary Search
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• Bubble Sort
• Selection Sort
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• Basics of Greedy Algorithms

Graph Representation

• Breadth First Search
• Depth First Search
• Minimum Spanning Tree
• Shortest Path Algorithms
• Flood-fill Algorithm
• Articulation Points and Bridges
• Biconnected Components
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• Topological Sort
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• Minimum Cost Maximum Flow
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• Introduction to Dynamic Programming 1
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• Dynamic Programming and Bit Masking

Graphs are mathematical structures that represent pairwise relationships between objects. A graph is a flow structure that represents the relationship between various objects. It can be visualized by using the following two basic components:

Nodes: These are the most important components in any graph. Nodes are entities whose relationships are expressed using edges. If a graph comprises 2 nodes $$A$$ and $$B$$ and an undirected edge between them, then it expresses a bi-directional relationship between the nodes and edge.

Edges: Edges are the components that are used to represent the relationships between various nodes in a graph. An edge between two nodes expresses a one-way or two-way relationship between the nodes.

Types of nodes

Root node: The root node is the ancestor of all other nodes in a graph. It does not have any ancestor. Each graph consists of exactly one root node. Generally, you must start traversing a graph from the root node.

Leaf nodes: In a graph, leaf nodes represent the nodes that do not have any successors. These nodes only have ancestor nodes. They can have any number of incoming edges but they will not have any outgoing edges.

Types of graphs

• Undirected: An undirected graph is a graph in which all the edges are bi-directional i.e. the edges do not point in any specific direction.

• Directed: A directed graph is a graph in which all the edges are uni-directional i.e. the edges point in a single direction.

Weighted: In a weighted graph, each edge is assigned a weight or cost. Consider a graph of 4 nodes as in the diagram below. As you can see each edge has a weight/cost assigned to it. If you want to go from vertex 1 to vertex 3, you can take one of the following 3 paths:

• 1 -> 2 -> 3
• 1 -> 4 -> 3

Therefore the total cost of each path will be as follows: - The total cost of 1 -> 2 -> 3 will be (1 + 2) i.e. 3 units - The total cost of 1 -> 3 will be 1 unit - The total cost of 1 -> 4 -> 3 will be (3 + 2) i.e. 5 units

Cyclic: A graph is cyclic if the graph comprises a path that starts from a vertex and ends at the same vertex. That path is called a cycle. An acyclic graph is a graph that has no cycle.

A tree is an undirected graph in which any two vertices are connected by only one path. A tree is an acyclic graph and has N - 1 edges where N is the number of vertices. Each node in a graph may have one or multiple parent nodes. However, in a tree, each node (except the root node) comprises exactly one parent node.

Note : A root node has no parent.

A tree cannot contain any cycles or self loops, however, the same does not apply to graphs.

Graph representation

You can represent a graph in many ways. The two most common ways of representing a graph is as follows:

Adjacency matrix

An adjacency matrix is a VxV binary matrix A . Element $$A_{i,j}$$ is 1 if there is an edge from vertex i to vertex j else $$A_{i,j}$$ is 0.

Note : A binary matrix is a matrix in which the cells can have only one of two possible values - either a 0 or 1.

The adjacency matrix can also be modified for the weighted graph in which instead of storing 0 or 1 in $$A_{i,j}$$ , the weight or cost of the edge will be stored.

In an undirected graph, if $$A_{i,j}$$ = 1, then $$A_{j,i}$$ = 1. In a directed graph, if $$A_{i,j}$$ = 1, then $$A_{j,i}$$ may or may not be 1.

Adjacency matrix provides constant time access (O(1) ) to determine if there is an edge between two nodes. Space complexity of the adjacency matrix is O($$V^2$$) .

The adjacency matrix of the following graph is: i/j : 1 2 3 4 1 : 0 1 0 1 2 : 1 0 1 0 3 : 0 1 0 1 4 : 1 0 1 0

The adjacency matrix of the following graph is:

i/j : 1 2 3 4 1 : 0 1 0 0 2 : 0 0 0 1 3 : 1 0 0 1 4 : 0 1 0 0

Consider the directed graph given above. Let's create this graph using an adjacency matrix and then show all the edges that exist in the graph.

Input file 4 $$\hspace{2cm}$$ // nodes 5 $$\hspace{2cm}$$//edges 1 2 $$\hspace{1.5cm}$$ //showing edge from node 1 to node 2 2 4 $$\hspace{1.5cm}$$ //showing edge from node 2 to node 4 3 1 $$\hspace{1.5cm}$$ //showing edge from node 3 to node 1 3 4 $$\hspace{1.5cm}$$ //showing edge from node 3 to node 4 4 2 $$\hspace{1.5cm}$$ //showing edge from node 4 to node 2

There is an edge between 3 and 4.

There is no edge between 2 and 3.

Adjacency list

The other way to represent a graph is by using an adjacency list. An adjacency list is an array A of separate lists. Each element of the array A i is a list, which contains all the vertices that are adjacent to vertex i.

For a weighted graph, the weight or cost of the edge is stored along with the vertex in the list using pairs. In an undirected graph, if vertex j is in list $$A_{i}$$ then vertex i will be in list $$A_{j}$$ .

The space complexity of adjacency list is O(V + E) because in an adjacency list information is stored only for those edges that actually exist in the graph. In a lot of cases, where a matrix is sparse using an adjacency matrix may not be very useful. This is because using an adjacency matrix will take up a lot of space where most of the elements will be 0, anyway. In such cases, using an adjacency list is better.

Note: A sparse matrix is a matrix in which most of the elements are zero, whereas a dense matrix is a matrix in which most of the elements are non-zero.

Consider the same undirected graph from an adjacency matrix. The adjacency list of the graph is as follows:

Consider the same directed graph from an adjacency matrix. The adjacency list of the graph is as follows:

A1 → 2 A2 → 4 A3 → 1 → 4 A4 → 2

Consider the directed graph given above. The code for this graph is as follows:

4 $$\hspace{2cm}$$ // nodes 5 $$\hspace{2cm}$$ //edges 1 2 $$\hspace{1.5cm}$$ //showing edge from node 1 to node 2 2 4 $$\hspace{1.5cm}$$ //showing edge from node 2 to node 4 3 1 $$\hspace{1.5cm}$$ //showing edge from node 3 to node 1 3 4 $$\hspace{1.5cm}$$ //showing edge from node 3 to node 4 4 2 $$\hspace{1.5cm}$$ //showing edge from node 4 to node 2

• Adjacency list of node 1: 2
• Adjacency list of node 2: 4
• Adjacency list of node 3: 1 --> 4
• Adjacency list of node 4: 2

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• Graphical Representation of Motion

Graphical Representation makes it simpler for us to understand data. When analyzing motion, graphs representing values of various parameters of motion make it simpler to solve problems. Let us understand the concept of motion and the other entities related to it using the graphical method.

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Using a graph for a pictorial representation of two sets of data is called a graphical representation of data . One entity is represented on the x-axis of the graph while the other is represented on the y-axis. Out of the two entities, one is a dependent set of variables while the other is independent an independent set of variables.

We use line graphs to describe the motion of an object. This graph shows the dependency of a physical quantity speed or distance on another quantity, for example, time.

Browse more Topics under Motion

• Introduction to Motion and its Parameters
• Equations of Motion
• Uniform Circular Motion

Distance Time Graph

The distance-time graph determines the change in the position of the object. The speed of the object as well can be determined using the line graph. Here the time lies on the x-axis while the distance on the y-axis. Remember, the line graph of uniform motion is always a straight line .

Why? Because as the definition goes, uniform motion is when an object covers the equal amount of distance at equal intervals of time.  Hence the straight line. While the graph of a non-uniform motion is a curved graph.

Velocity and  Time Graph

A velocity-time graph is also a straight line. Here the time is on the x-axis while the velocity is on the y-axis. The product of time and velocity gives the displacement of an object moving at a uniform speed. The velocity of time and graph of a velocity that changes uniformly is a straight line. We can use this graph to calculate the acceleration of the object.

Acceleration =(Change in velocity)/time

For calculating acceleration draw a perpendicular on the x-axis from the graph point as shown in the figure. Here the acceleration will be equal to the slope of the velocity-time graph. Distance travelled will be equal to the area of the triangle, Therefore,

Distance traveled= (Base × Height)/2

Just like in the distance-time graph, when the velocity is non-uniform the velocity-time graph is a curved line.

Solved Examples for You

Question: The graph shows position as a function of time for an object moving along a straight line. During which time(s) is the object at rest?

• 0.5 seconds
• From 1 to 2 seconds
• 2.5 seconds

Solution: Option B. Slope of the curve under the position-time graph gives the instantaneous velocity of the object. The slope of the curve is zero only in the time interval 1 < t < 2 s. Thus the object is at rest (or velocity is zero) only from 1 to 2 s. Hence option B is correct.

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Problems Based on Graphical Representation of Motion for Class 9 Science

• Last modified on: 2 years ago
• Reading Time: 6 Minutes

Graphical Representation of Motion

A graph represents the relation between two variable quantities in pictorial form. It is plotted between two variable quantities. The quantity that is varied our choice is called independent variable. The other quantity, which varies as a result of this change, is called dependent variable. For example, in distance-time graph, time is independent variable and distance is dependent variable. Similarly, in velocity-time graph, time is independent variable and velocity is dependent variable

Problems Based on Graphical Representation of Motion

Q.1. The velocity time graph of an ascending passenger lift is given below. What is the acceleration of the lift: (i) during the first two seconds (ii) between 2nd and 10th second (iii) during the last two seconds.

Q.2. The figure is the distance-time graph of an object. Do you think it represents a real situation? If so, why? If not, why not?

Q.3. A body is moving uniformly with a velocity of 5m/s. Find graphically the distance travelled by it in 5 seconds.

Q.4. Study the speed-time graph of a body shown in below figure and answer the following questions: (a) What type of motion is represented by OA? (b) What type of motion is represented by AB? (c) What type of motion is represented by BC? (d) Calculate the acceleration of the body. (e) Calculate the retardation of the body. (f) Calculate the distance travelled by the body from A to B.

Q.5. In the above question, calculate (i) distance travelled from O to A (ii) distance travelled from B to C. (iii) total distance travelled by the body in 16 sec.

Q.6. A car is moving on a straight road with uniform acceleration. The following table gives the speed of the car at various instants of time:

Draw the speed time graph choosing a convenient scale. Determine from it (i) the acceleration of the car (ii) the distance travelled by the car in 50 sec.

Q.7. The figure is the distance-time graph of an object. Do you think it represents a real situation? If so, why? If not, why not?

Q.8. The graph in below figure shows the positions of a body at different times. Calculate the speed of the body as it moves from (i) A to B (ii) B to C and (iii) C to D.

Q.9. A car is moving on a straight road with uniform acceleration. The speed of the car varies with time as follows:

Draw the speed time graph choosing a convenient scale. Determine from it (i) the acceleration of the car (ii) the distance travelled by the car in 10 sec.

Q.10. The graph given below is the velocity-time graph for a moving body. Find (i) velocity of the body at point C (ii) acceleration acting on the body between A and B (iii) acceleration acting on the body between B and C.

Numericals on Force and Laws of Motion for Class 9

Last modified on:8 months agoReading Time:14Minutes Numericals on Force and Laws of Motion for Class 9 Here you will find numericals on force and laws of motion for class 9. Problem: Find the force acting on a rocket and its acceleration if its velocity is 600 m s–1 at t = 60 s and 1000 ms–1…

[PDF] Download Physics Numerical for Class 9

Last modified on:8 months agoReading Time:10Minutes[PDF] Download Physics Numerical for Class 9 Here you will find Physics Numericals for Class 9. Each and every topic numericals covered in this article. Class 9 Physics Numericals Problems Based on Work and Energy for Class 9 Science Numericals for Class 9 Science Chapter 12 Sound Problems Based on Class…

Extra Questions for Class 9 Science Chapter 12 Sound

Last modified on:1 year agoReading Time:10MinutesExtra Questions for Class 9 Science Chapter 12 Sound Very Short Answer Type Show Answer 1. Sound is a form of energy or the medium by which we can communicate with each other. 2. The to and fro motion of an object is called vibration. 3. Voice box or larynx 4.…

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• 1. Multiple Choice Edit 2 minutes 1 pt 1. Which of the following best describes the process of finding the range for a set of data? ADD   the biggest and smallest numbers Place the number in order from least to greatest then find the middle. Find the difference between the Maximum and the Minimum. Find the average.

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