## 2.5.1 Introduction to Graphs:

In May, when summer is at its peak. We may find some days are too hot and some days are not so hot. We may measure the maximum temperature on each day. We get data of the whole month. Perhaps we will tabulate it straight forward way in order to learn briefly on **introduction to graphs** chapter:

Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | …….. | 30 | 31 |

Maximum temperature | 33˚ | 35˚ | 32˚ | 37˚ | 35˚ | 31˚ | 30˚ | 36˚ | …….. | 36˚ | 36˚ |

Similarly, the rainfall for each month of a year can be measured in inches or centimetres and we obtain a data. Recording these data is very useful for future plans. For example, we can see the rain pattern over several years and agricultural crops accordingly. We may come for a better future. One easy way is to tabulate as earlier. However this is very cumbersome procedure. People look for efficient way of recording such data, which also help them to analyse it in a better way. **One of the effective methods is the use graphs.**

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So, what is a graph actually meant by, we can define it as **the visual representation of numerical data **collected during an experiment. By looking at the graph one can easily understand the data, the graphs also help us to analyse the data quickly.

**2.5.2 Bar graphs – Introduction to Graphs**

A diagram in which the numerical values of variables are represented by the height or length of lines or rectangles of equal width.

For example:

Note that each bar in the above picture represents the profit of the firm during that year which is given below the bar. You see that the firm has a good growth, but a set-back in last year.

It is not necessary that the vertical bars in a bar graph be separated. The adjacent graph shows the marks obtained by Priya and Sujatha in the examination. It is easy to compare their performance in individual subjects.

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**2.5.3 Pie charts – Introduction to graphs**

a type of graph in which a circle is divided into sectors that each represent a proportion of the whole.

Example:

**2.5.4 Coordinate system – Introduction to Graphs**

Another extremely useful and important method of representing data is Cartesian graphs. These are also called coordinate graphs as the basic principle depends upon the use of coordinate system.

A **Cartesian graph** is really just two number lines that cross at 0. These number lines are called the horizontal axis (also called the x-axis) and the vertical axis (also called the y-axis). The place where these two axes (plural of axis) cross is called the origin.

**Introduction to Graphs – Exercise 2.5.4**

**Fix up your own coordinate system on a graph paper and locate the following points on the sheet:**

**(i) P(-3, 5)**

*Solution:*

**(ii) Q(0, -8)**

**Solution :**

**iii)R(4,0)**

**Solution:**

**(iv)S(-4, -9)**

**Solution:**

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**2.Suppose you are given a coordinate system. Determine the quadrant in which the following points lie:**

**(i) A(4,5)**

**Solution:**

Both x and y values are positive so it lies in 1^{st} quadrant.

(ii) B(-4,-5)

**Solution:**

Both x and y are negative so it lies in third quadrant.

(iii) C(4, -5)

**Solution:**

Here x is positive and y is negative so it lies in fourth quadrant.

**Linear graphs – Introduction to Graphs**

A series of points, discrete or continuous, as in forming a curve or surface, each of which represents a value of a given function also called **linear graph**. A network of lines connecting points.

For example:

**Introduction to Graphs – Exercise 2.5.5**

**Draw the graphs of the following straight lines:**

**(i) y = 3-x**

**Solution:**

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Y | 3 | 2 | 1 | 0 | -1 | -2 | -3 | -4 | -5 |

**(ii) y = x-3**

**Solution:**

x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Y | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |

**(iii) y = 3x-2**

**Solution:**

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Y | -2 | 1 | 4 | 7 | 10 | 13 | 16 | 19 | 22 |

**(iv) y = 5 – 3x**

**Solution:**

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Y | 5 | 2 | -1 | -4 | -7 | -10 | -13 |

**(v) 4y = -x + 3**

**Solution:**

y = (-x+3)/4 = (-x/4) + 3/4

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Y | 0.75 | .5 | 0.25 | 0 | -0.25 | -0.5 | -0.75 |

**(vi) 3y = 4x+1**

**Solution:**

y = (4x + 1)/3 = 4x/3 +1/3 = 4x/3 + 0.333

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Y | 0.33 | 1.66 | 2.99 | 4.33 | 5.66 | 6.99 | 8.33 | 9.66 | 10.99 |

**(vii) x=4**

**Solution:**

The graph of x = 4, The slope-intercept form is where is the slope and is the **y-intercept**. Since there is no value of y, then the graph of x = 4 is a vertical line.

This vertical line indicates that, when x = 4, y can be any number. This vertical line is from plus infinity to minus infinity.

**(viii) 3y=1**

**Solution:**

3y = 1

y = 1/3

Here slope is zero, i.e., the values of y remains constant with respect x-axis. So the graph will be a line parallel to x-axis

### 2. Draw the graph of ^{y}/_{x} = ^{(y+1)}/_{(x+2)}

**Solution:**

y(x+2) = x(y+1)

xy + 2y = xy + 1

2y = 1

y = ½

Here slope is zero, i.e., the values of y remains constant with respect x-axis. So the graph will be a line parallel to x-axis

**Find the point of intersection of the straight-lines 3y+4x = 7 and 4y+3x = 7, by drawing their graphs and looking for the point where they meet.**

**Solution:**

3y+4x = 7

y = ^{(7-4x)}/_{3} = ^{7}/_{3} – ^{4x}/_{3} = 2.33 – ^{4x}/_{3}

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

y | 2.33 | 1 | -0.33 | -1.67 | -3.00 | -4.33 | -5.67 | -7.00 | -8.33 |

4y+3x = 7

y = ^{(7-3x)}/_{4} = ^{7}/_{4} – ^{3x}/_{4} = 1.75 – ^{3x}/_{4}

X | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

y | 1.75 | 1 | 0.25 | -0.5 | -1.25 | -2 | -2.75 | -3.5 | -4.25 |

So, the straight lines meet at the point (1,1).

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