Graphs drawn on a rectangular coordinate system, known as the Cartesian coordinate system (invented by René Descartes), help provide information in a visual form. Understanding the rectangular coordinate system is crucial in order to be able to read and draw graphs, which is essential in many branches of mathematics.
The rectangular coordinate system uses a horizontal and a vertical number line, each known as an axis. These two perpendicular axes cross at the point (O), known as the origin.
The horizontal number line (moving to the left or the right) is called the X-axis and the vertical number line (moving up or down) is called the Y-axis, as illustrated in Exhibit 8.1-a.
The numbers to the right of the origin along the X-axis are positive ( + ) and those to the left are negative ( − ). The numbers above the origin along the Y-axis are positive ( + ) and those below are negative ( − ).
The purpose of the rectangular coordinate system and the sign convention is to locate a point relative to the X- and Y-axes and in reference to the origin 'O'.
A point in the Cartesian coordinate system is a location in the plane, represented as an ordered pair of numbers inside a set of brackets, called coordinates. The first number is called the x-coordinate, representing its horizontal position with respect to the origin, and the second number is called the y-coordinate, representing its vertical position with respect to the origin. The ordered pair of coordinates for a given point P is written as follows: P(x, y), or simply (x, y). For example, the origin (i.e., the point where the x-axis and y-axis intersect) is identified by the coordinates (0, 0) since both its x and y coordinates are 0.
As illustrated in Exhibit 8.1-b, the ordered pair (2, 3) refers to the coordinates of the point P which is 2 units to the right and 3 units above, in reference to the origin.
It is important to identify the coordinate numbers in their order. They are called ordered pairs because the order in which they appear determines their position on the graph. Changing the order of the coordinates will result in a different point.
For example, (2, 3) and (3, 2) are different points.
(2, 3) refers to a point 'P', which is 2 units to the right of the origin and 3 units above the origin.
(3, 2) refers to a point 'Q', which is 3 units to the right of the origin and 2 units above the origin.
It is called a rectangular coordinate system because the x- and y-coordinates form a rectangle with the X- and Y-axes, as seen in the exhibit above.
The X- and Y-axes divide the coordinate plane into four regions, called quadrants. Quadrants are numbered counter-clockwise from one (I) to four (IV), starting from the upper-right quadrant, as illustrated in Exhibit 8.1-c.
That is, the upper-right quadrant is Quadrant I, the upper-left quadrant is Quadrant II, the lowerleft quadrant is Quadrant III, and the lower-right quadrant is Quadrant IV. Table 8.1 shows the sign convention of coordinates in each of the quadrants with examples that are plotted on the graph in Exhibit 8.1-d.
Quadrant, Axis, Origin |
Sign of x-coordinate |
Sign of y-coordinate |
Example (plotted in Exhibit 8.1-d) |
---|---|---|---|
Quadrant I | Positive (+) | Positive (+) | A (3, 2) |
Quadrant II | Negative (−) | Positive (+) | B (−3, 4) |
Quadrant III | Negative (−) | Negative (−) | C (−5, −2) |
Quadrant IV | Positive (+) | Negative (−) | D (5, −3) |
X−Axis | Positive (+) or Negative (−) |
Zero (0) | E (4, 0), F (−2, 0) |
Y−Axis | Zero (0) | Positive (+) or Negative (−) |
G (0, 3), H (0, −4) |
Origin | Zero (0) | Zero (0) | 0 (0, 0) |
Determine the x- and y-coordinates of the points A, B, C, D, E, F, G, and H labelled in the graph.
A: (6, 3) B: (−2, 5) C: (−7, −3) D: (4, −6)
E: (0, 4) F: (−5, 0) G: (0, −6) H: (3, 0)
Identify the quadrant or the axis in which the following points are located:
A (−15, 20)
B (20, 5)
C (9, 0)
D (0, 20)
E (12, −18)
F (0, −6)
G (−30, −15)
H (−1, 0)
A (−15, 20) [latex] \longrightarrow [/latex] (−, +) = 2^{nd} Quadrant
B (20, 5) [latex] \longrightarrow [/latex] (+, +) = 1^{st} Quadrant
C (9, 0) [latex] \longrightarrow [/latex] (+, 0) = X-Axis (Right)
D (0, 20) [latex] \longrightarrow [/latex] (0, +) = Y-Axis (Up)
E (12, −18) [latex] \longrightarrow [/latex] (+, −) = 4^{th} Quadrant
F (0, −6) [latex] \longrightarrow [/latex] (0, −) = Y-Axis (Down)
G (−30, −15) [latex] \longrightarrow [/latex] (−, −) = 3^{rd} Quadrant
H (−1, 0) [latex] \longrightarrow [/latex] (−, 0) = X-Axis (Left)
Three vertices of a rectangle ABCD have points A (–3, 3), B (4, 3), and C (4, –2). Find the coordinates of the 4^{th} vertex D.
Plotting points A, B, and C:
A (–3, 3): 3 units to the left of the origin and 3 units above the origin
B (4, 3): 4 units to the right of the origin and 3 units above the origin
C (4, –2): 4 units to the right of the origin and 2 units below the origin
Connecting point A to point B results in a horizontal line (since they share the same y-coordinate), and connecting point B to point C results in a vertical line (since they share the same x-coordinate).
The 4^{th} vertex of the rectangle, D, will have the same x-coordinate as point A and the same y-coordinate as point C.
Therefore, the coordinates for the 4^{th} vertex are D (–3, –2).
A vertical line has a length of 3 units and the coordinates at one end of the line are P (–2, 1). Find the possible coordinates of the other end of the line, Q.
Plotting point P:
P (–2, 1): 2 units to the left of the origin and 1 unit above the origin
Since we are drawing a vertical line, point Q will have the same x-coordinate as point P.
One possible set of coordinates for the other end of the line is 3 units above point P, i.e., Q (–2, 4).
The other possible set of coordinates for the other end of the line is 3 units below point P, i.e., Q (–2, –2).
For Problems 1 to 4, plot the points on a graph.
a. A (−3, 5)
b. B (5, −3)
c. C (0, −4)
a. A (−6, 0)
b. B (4, −2)
c. C (0, −7)
a. D (6, 0)
b. E (−2, 4)
c. F (5, 2)
a. D (8, 0)
b. E (−3, −5)
c. F (5, 5)
For Problems 5 to 8, determine the quadrant or axis in which the points lie.
a. A (−1, 2)
b. B (5, −1)
c. C (3, 5)
a. A (1, 6)
b. B (4, −3)
c. C (−7, 3)
a. D (−4, 0)
b. E (−2, −7)
c. F (0, 5)
a. D (6, 0)
b. E (−1, −13)
c. F (0, −7)
For Problems 9 to 12, plot the pairs of points on a graph and calculate the length of each horizontal line joining the pair of points.
a. (3, 4) and (5, 4)
b. (−7, 1) and (2, 1)
a. (2, –6) and (7, –6)
b. (–5, –4) and (0, –4)
a. (−5, 3) and (0, 3)
b. (−2, −2) and (6, −2)
a. (–6, 8) and (–1, 8)
b. (7, –5) and (2, –5)
For Problems 13 to 16, plot the pairs of points on a graph and calculate the length of each vertical line joining the pair of points.
a. (3, 6) and (3, 1)
b. (5, 2) and (5, –5)
a. (–3, –5) and (–3, –9)
b. (–3, 0) and (–3, 6)
a. (5, 6) and (5, 2)
b. (7, 2) and (7, −4)
a. (−3, 5) and (−3, −4)
b. (−3, 5) and (−3, 0)