Graphs

GRAPHS

A graph is a representation of a set of points or co-ordinates where they can be linked or joined using a line or curve. The interconnected points are called vertices. Typically, a graph is depicted in diagrammatical form as a set of dots/asterisks and the vertices are joined with lines or curves called edges.

We will majorly be making reference to linear (straight line) graphs as they are the type of graphs that result from linear equations or functions. Aside from representing information, graphs can be used for solving equations (whether linear, quadratic, etc).

Axes of reference:

Axes (axis for singular) simply refer to the axes that make up the graph. Normally, every graph has two axes; viz, the horizontal axis (x-axis) and the vertical axis (y-axis). These axes serve as a means for representing strategic points in any graph. Co-ordinates or points on a graph are usually represented in the form (x₁,y₁). The value of x₁ is located on the x-axis and that of y₁ is located on the y-axis as well. After this has been done, they are successively traced out until they reach an intersection point (where they meet). This point is referred to as a vertice (x₁,y₁). This same procedure is carried out for as many points or co-ordinates as made available. After this has been successfully done, all the successive points are joined with lines or curves.

Origin of axes:

The origin of either axis can be defined as the intersection point of the two axes. At this point, both axes have a value of x = 0 and y = 0.

Axis of co-ordinate:

In writing the values of the co-ordinates of any graph, all values for the x axis always come first before all values for the y-axis. That is, for instance, (x₁,y₁), (x₂,y₂), (x_n,y_n), etc. the two axes combine to form the Cartesian plane.

Plotting a graph:

In plotting a graph, one must establish all the necessary values of x and y respectively and represent them appropriately. In representing them appropriately, there must be a well-defined scale for anyone to read and accurately interpret the graph. In questions where one is asked to plot a graph, co-ordinates can be given outrightly. These co-ordinates are then plotted on the Cartesian plane and joined with a straight line or curve.

Example:

Plot the graph of X against Y, given the following values for x and y respectively.

x	Y
1	10
2	8
3	6
4	4
5	2

The graph of the table above is thus:

 The graph of y against x   

Scale:

let 1cm represent 1 unit on x-axis

let 1cm represent 2 units on the y-axis

In some cases, the co-ordinates for a graph may be directly provided. The same procedure is applicable.

Example:

Plot a graph of the following co-ordinates:

A(3,2); B(6,-7); C(0,11)

The co-ordinates can be rewritten as:

x	y
3	2
6	-7
0	11

Thus, the graph for the above given values of x and y is thus:

 

The graph of y against x

scale:

let 1cm represent 2 units on the y-axis

let 1cm represent 2 unit on the x-axis

Also, the values for plotting a graph could be given in form of an equation for y, where the corresponding values of x are given.

Example:

Plot the graph of 2x-y = 4 where x is defined by 0≤X≤4

First, we make y the subject of the formula

-y = 4-2x (moving 2x to the right side)

Y = 2x-4 (dividing through by -1)

Thus, we proceed to obtain values for y, when x = 0,1,2,3 and 4

x	0	1	2	3	4
2x	0	2	4	6	8
-4	-4	-4	-4	-4	-4
y	-4	-2	0	2	4

Bringing out the values of x and y, we have:

x	y
0	-4
1	-2
2	0
3	2
4	4

From the table above, the graph below can easily be obtained:

     The graph of y against x

Scale:

Let 1cm represent 1 unit on x-axis

Let 1cm represent 1 unit on y-axis

It is noteworthy that what we have been making reference to is the straight line/linear graph. Now, let us examine graphs in form of curves. Usually, quadratic equations or equations of higher orders (degrees) give rise to curves when plotted in graphical forms.

Example:

Plot the graph of y = 2x²-5x-3 given that the values of x is defined by -2≤x≤5

Solution

To plot this curve, we must tabulate our values as usual to avoid unnecessary hiccups or complications.

x	-2	-1	0	1	2	3	4	5
2x2	8	2	0	2	8	18	32	50
-5x	10	5	0	-5	-10	-15	-20	-25
-3	-3	-3	-3	-3	-3	-3	-3	-3
y	15	4	-3	-6	-5	0	9	22

We proceed to take out the relevant values, that is, x and y:

x	y
-2	15
-1	4
0	-3
1	-6
2	-5
3	0
4	9
5	22

From the values above, we can easily plot the graph of y against x thus:

              The graph of y against x

Scale: let 1cm represent 1 unit on x-axis

       Let 1cm represent 5 units on y-axis

Finally, attempt this: y = 6+x-2x²

^{For  more info on graphs, feel free to comments...................}