The Rectangular Coordinate System

Keep tthe following in mind:

(i) When the coordinates of a point are written as a bracketed pair of numbers, we always write the horizontal coordinate first and the vertical coordinate second. This is why (a, b) is often called an ordered pair. For example, the point (5, 2) is at a quite different location than the point (2, 5):

(ii) Since the identity of a point comes from its measured position or location with respect to the scales on the coordinate axes, it is mandatory to show the scales on the axes explicitly , and to label the axes explicitly. If you omit the scale markings and labels, you end up with a meaningless graph.

(iii) The graph of a single point is just a dot at the appropriate location. No additional lines, etc. should be drawn unless they are requested.

(iv) Points on the horizontal axis have coordinates of the form (b, 0) – that is, their vertical coordinates are equal to zero. Points on the vertical axis have coordinates of the form (0, b) – that is, their horizontal coordinate is zero. The origin has coordinates (0, 0) – both of its coordinates are zero.

(v) The rectangular coordinate axes divide the plane into four regions, called quadrants . The quadrants are identified by number, with quadrant 1, or the first quadrant being the upper right one. They are arranged as shown:

Plotting Graphs of Formulas

To graph a formula by hand, the usual procedure is to

i) make a table of x and y values for a representative collection of values of x in the specified interval

ii) plot these pairs of values as points on a rectangular coordinate system

iii) join the points by a line or smooth curve

Example:

Plot the graph of y = 2x + 3 for x between -3 and +3 inclusive.

Solution:

(i) start by making a table of coordinates of representative points:

(ii) we need coordinate axes that have x going from -3 to +3 and y going from -3 to +9. Set these up and then plot the points as dots at the appropriate locations:

(iii) The points appear to lie on a straight line. Laying a straight edge on the graph confirms this, so in this case, just draw a straight line through the points to complete the graph, as shown above. Had we recognized that the formula, y = 2x + 3, is the type of formula that gives a straight line graph, we could have saved ourselves some work, since we would need to plot only two points to get the entire graph.

Example:

The area, A, of a square with sides of length s is given by the formula

A = s ²

Plot a graph of A vs s for s = 0 through s = 5.

Solution:

This example illustrates several issues:

(i) People often use the symbols x and y generically when speaking about graphs. However, we can create a rectangular coordinate system for any pair of symbols or variables we wish to use.

(ii) When asked to plot the graph of “A vs. s”, we are to make the vertical axis the A-axis, and the horizontal axis is the s-axis:

So, here, we start again by making a table of coordinates of representative points for the graph:

Now, plot the points on a rectangular coordinate system which has a horizontal (s) axis running from s = 0 to s = 5, at least, and a vertical (A) axis running at least from A = 0 to A = 25.

We see in this case that the points appear to follow a curved path, bending upwards as you move towards the right. The graph starts out with quite a shallow slope around s = 0, but appears to get steeper and steeper as we move towards higher values of s.