A point is a dimensionless quantity. But, to specify a point, we need a reference using certain dimensions. In single dimension, like a number line, a point specifies the magnitude and sign of a number by the distance and the direction with reference to a point called origin.
In two dimensions, a point is specified by the horizontal and vertical displacements from a set of axes. Let us first describe the system and the definitions of theses coordinates.
The set of axes consists of two axes: one in horizontal direction and the other at vertical directions. These two axes intersect at a point described as origin. The horizontal axis is called as x-axis and the vertical axis as y-axis. The horizontal distance of a point from origin is called its x-coordinate (x) and the vertical distance of a point from origin is called its y-coordinate (y).
The set of these two coordinates in a combined form is called as the ordered pair of the point, in the order of (x, y). The part of x-axis left to the origin and the part of y-axis below the origin represent negative signs of the measures.
Now, let us see how the ordered pair of a point is determined.
In the above diagram, a point P is shown at a location on a set of axes. Draw a vertical line from P and it intersects the positive x-axis at 3. It means, the point is 3 units away from the origin horizontally. A horizontal line from the same point intersects the positive y-axis at 2. It means, the point is 2 units away from the origin vertically. In other words, the x-coordinate of the point is 3 and the y-coordinate of the point is 2. Thus, the location of the point P is specified by an ordered pair (3, 2).
The very first shape in geometry is a point. All the subsequent shapes like lines and curves (graphs in general) actually contain infinite number of points connected by formulas and relations that describe the shape. That is, in all cases the shape is the locus of a point stipulated by such relations.
For example, the shape of a circle is the locus of a point that moves at a constant distance from a fixed point. The algebraic connection gives us the description of such locus by use of variables for a clear visualization. In the same example of a circle, the relation using the variables is x2
, assuming the fixed center to be at origin and the constant distance as "r".
As mentioned earlier, there is infinite number of points on a graph. All such points may not be of importance. Some points on a graph may be vital as follows:
- A point where a function is discontinuous at a particular for any reason is referred as point of discontinuity.
- The point of vertex of a parabola tells us where the maximum or minimum of the function occurs and also the corresponding values.
- The points of local maximum and minimum are called as the critical points of the function.
- The point of inflection of a function is that point, where the concavity of the graph of the function changes its direction.