The Mathematics 11
Competency Test
When we write down a formula for some quantity, y, in terms of another quantity, x, we are expressing a relationship between the two quantities. For example, if we use the symbols y = the area of a square and x = the length of one side of the square, then the formula y = x 2 tells us the relationship between the length of a side of a square and the area of the square. It tells us that to calculate the area of the square, we must raise the length of one side of that square to the power 2 (which is one reason why raising a number to the power 2 is often referred to as “squaring” the number).
We can use this formula to calculate the value of y for any particular value of x. This formula relating y to x is itself informative, but often we can understand the nature of the relationship between y and x even better if we have a visual image of its characteristics as well. This is where graphing formulas is helpful.
A graph of a relationship is a way of drawing points and other geometric shapes at locations representing the values of x and y. This is most commonly done using a so-called rectangular coordinate system . When the formula expresses y in terms of x, the coordinate system is usually arranged as: y the “origin”
(x = 0, y = 0) vertical axis
or y - axis horizontal axis
or x- axis
5
4
3
2
1 positive values of y positive values of x
-5 -4 -3 -2 -1
0 negative values of x
-2
-3
-4
-5
1 2 3 4 5 negative values of y x
 the vertical axis and the horizontal axis (often called the y-axis and the x-axis, respectively, if the two variables are y and x) intersect at a central point called the origin , which corresponds to y = 0 and x = 0.

David W. Sabo (2003) The Rectangular Coordinate System Page 1 of 6

 a numerical scale is created on each axis. Values on the horizontal axis increase from 0 at the origin though positive values to the right, and from 0 at the origin through negative values: -1, -2, 3, …, etc., to the left. The scales along the axes should be uniform – the values of x should be spaced uniformly along the length available.
 values on the vertical scale increase from 0 at the origin through positive values as you go upwards, and from 0 at the origin through negative values: -1, -2, 3, …, etc., as you go down.
These two axes complete with the explicitly labelled numerical scales form what is called a rectangular coordinate system.
Then, the corresponding pair of values, x = a and y = b, written as a pair of numbers in this order in brackets, (a, b), corresponds to, or is plotted as a point at the location where the vertical line through x = a intersects the horizontal line through y = b: y
the point
(a, b)
x
The values x = a and y = b here are called the coordinates or rectangular coordinates of the point. You can also think of the coordinates, (a, b), of a point as indicating that to get to the point from the origin, you need to move ‘a’ units horizontally and ‘b’ units vertically. Positive movements are to the right horizontally, and upwards vertically. Negative movements are to the left horizontally and downwards vertically. y
Example: Plot the points
A = (4, 2)
B = (-3, 5)
C = (-4, -3) and
D = (5, -2) on the coordinate axes shown to the right. Be sure to x label the axes scales and label the points you plot.
Solution:
Recalling the meaning of this notation giving pairs of numbers in brackets, we know that the point
A is the point that occurs at x = 4 and y = 2 – the first number in the brackets gives the xcoordinate of the point, and the second number in brackets gives the y-coordinate of the point.

David W. Sabo (2003) The Rectangular Coordinate System Page 2 of 6

Now, the x-coordinates for these four points range from a minimum of -4 to a maximum of 5, so our horizontal scales must go at least to -4 on the left to at least +5 on the right. Also, we see that the y-coordinates must go at least to -3 on the bottom to at least 5 on the top. The result is: y
= (-3, 5)
5
4
3
2
1
= (4, 2)
-5 -4 -3 -2 -1
0
1 2 3 4 5 x
= (-4, -3)
-2
-3
-4
= (5, -2)
-5
The dotted lines show how the points line up with the appropriate scale positions on both axes.
Notice that the points are plotted as heavy dots – if you are just plotting points, there is no need to join them by lines or add any other features to the graph.
Remarks:
(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): y
5
4
3
2
1
(2, 5)
(5, 2)
0 x
1 2 3 4 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.

David W. Sabo (2003) The Rectangular Coordinate System Page 3 of 6

(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. y
5
4
3
2
1 (-4, 0)
-5 -4 -3 -2 -1
0
(0, 4)
(0, 0)
(5, 0)
1 2 3 4 5 x
-2
-3
-4
(0, -3)
-5
(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:

David W. Sabo (2003) The Rectangular Coordinate System Page 4 of 6

x
-3
-2
-1
0
1
2
3 y
-3 [ = 2 x (-3) + 3 = -3 ]
-1 [ = 2 x (-2) + 3 = -1 ]
1 [ = 2 x (-1) + 3 = 1 ]
3 etc….
5
7
9
(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:
(-1, 1)
3
2
1
6
5
4
9
8
7 y
(0, 3)
(1, 5)
(2, 7)
(3, 9)
-3
(-2, -1)
-1
0
1 2 3 x
-2
(-3, -3)
-3
(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:

David W. Sabo (2003) The Rectangular Coordinate System Page 5 of 6

A s
So, here, we start again by making a table of coordinates of representative points for the graph: s A = s 2
3
4
5
0
1
2
0
1
4
9
16
25
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.
A
25 (5, 25)
20
15
(4, 16)
10
(3, 9)
5
(2, 4)
(0, 0) (1, 1) s
0
1 2 3 4 5
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.

David W. Sabo (2003) The Rectangular Coordinate System Page 6 of 6