A graph is a mathematical representation of networks. The purpose of the graph is to show mathematical relations in visual form so that it can be easily understood. There are many types of graph:

- Bar graph
- Pie graph
- Line graph

**Bar Graph**

Bar graph, also known as bar chart which is a visual tool and is used to compare the data among categories. Bar graph may be horizontal or vertical.

Below is a bar graph which shows pollution level of each city.

Bar graph represents a discrete value on one axis and category on another axis and the motive is to show the relation between two axes.

We can easily make a comparison between the set of data.

### Pie Chart

A pie chart is a circular chart in which each data is represented in a portion of circle. As the chart is divided into wedge-like sectors the total value of pie chart is always 100%.

#### Steps for Creating a Pie Chart

**Step 1:** Decide the topic of your chart.

**Step 2: **Having all the information or data and divide it into a number of items, and the value of each item adding together should have a sum equal to 100%.

### Line Graph

A line graph is also known as a line chart. It is used to visualize the value of something over time. The line graph have horizontal x-axis and vertical y-axis. The point where axes intersect is called origin i.e. (0,0). Each axis having its own data type. For e.g. x-axis could have months, days, week and y-axis may have growth increase in shares and revenue.

Each data value is represented in points and later they connected by line from one to other i.e. in “dot-to-dot” fashion.

**Cartesian Planes and Coordinate Axes**

**Cartesian Planes**

A Cartesian plane is defined by the two perpendicular lines i.e. the x-axis(horizontal) and y-axis(vertical). With the help of these axes we can mark any point in the Cartesian plane.

The Cartesian plane is infinite however to shows this in book they put arrow in the end of the line. The Cartesian plane is divided into four quadrants.

**Coordinate Axes**

It is x and y-axis if we label x and y-axis as in above diagram this framework is known as coordinate axis and the point of intersection of coordinate axes is known as origin.

**Linear graphs**

A line graph having an unbroken line is called linear graph. To draw the unbroken line we need to locate some points on graph sheet.

**Location of point: **How can we describe the location of board?

Drawing a point on board namely A1, A2, A3, …., AN measuring these points from the left edge of board and it is found to be 45 cm. Now we can say that A1 is 45 cm from left edge and 150 cm from bottom edge.

Let’s make graph having x-coordinate and y-coordinate.

Suppose you go to an auditorium and search for your reserved seat. You need to know two numbers, the row number and the seat number. This is the basic method for fixing a point in a plane.

### Sample Problems on Linear Graphs

**Problem 1: Plot (3, 4) on the graph **

**Solution: **

As in graph points are denoted as in the form of (x, y)

So, on comparing the points we get x = 3 and y = 4. First, we draw x = 4 we will move in the forward direction till we reach x = 4.

We draw y = 3 we move in the upward direction we will reach at (4, 3).

y-axis | |||||||

3 | 4, 3 | To be made on Graph | |||||

2 | |||||||

1 | |||||||

0,0 | 1 | 2 | 3 | 4 | 5 | 6 | x-axis |

0, 0 is the origin.

**Problem 2: **Locate the given points on the graph.

- (1, 2)
- (2, 8)
- (4, 2)

**Solution:**

For x = 1 and y = 2. Starting from (0, 0) origin we move x = 1 direction forward and from there we move y = 2 direction upward then finally reaching state is our point.

Similarly, plot the rest of the two points on Graph.

**Problem 3: **Plot the following points and verify if they lie on line.

- (0, 1), (0, 2), (0, 4), (0, 3)
- (1, 1), (2, 2), (3, 3), (4, 4), (5, 5)

**Solution: **

1. Plotting (0, 1), (0, 2), (0, 4), (0, 3) on graph

We can draw every coordinate on graph sheet as follows .

Here it forms a line after joining all the points.

2. Plotting (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) on graph

Note:In each of the above cases, graph obtained by joining the plotted points is a line. Such graphs are called linear graphs.

**Application of Linear Graphs**

In everyday life, we observe variations in the value of different quantity, or we can say that the more we use facility the more we have to pay for it. for e.g. If the more electricity we consumed then we will have to pay the bill more and vice versa. So one quantity affects the other quantity. We can say that quantity of electricity is independent variable and the amount of bill is the dependent variable. These relations we can show with graphs.

**Problem:** Amit can ride a bike with a constant speed of 30 km/hour. Draw a time-distance graph for this situation and find it

- Time taken by Amit to ride 75km
- Distance covered by Amit in 3.5 hour.

**Solution: **

Hours of ride | Distance covered |

1hour 2 hours 3 hours 4 hours |
30 km 2*30=60km 3*30=90km 4*30=120km |

We get the table:

Time (in hours) | 1 | 2 | 3 | 4 |

Distance covered (in km) | 30 | 60 | 90 | 120 |

(i) Scale:

Horizontal: 2 units = 1 hour

Vertical: 1 unit = 10 km

(ii) Mark time on horizontal axis.

(iii) Mark distance on vertical axis

(iv) Plot points: (1, 30), (2, 60), (3, 90), (4, 120)

(v) On joining the points we get a linear graph.

1. Corresponding to 75 km on the vertical axis, we get the time to be 2.5 hours on the horizontal axis. Thus, 2.5 hours are needed to cover 75 km.

2. Corresponding to 3.5 hours on the horizontal axis, the distance covered is 105 km on the vertical axis.