Coordinates in Two Dimensions Help

Example 1

Plot the points P = (3, −2), Q = (−4, 6), R = (2, 5), S = (−5, −3).

Solution 1

The first coordinate 3 of the point P tells us that the point is located 3 units to the right of the y -axis (because 3 is positive ). The second coordinate −2 of the point P tells us that the point is located 2 units below the x -axis (because −2 is negative). See Fig. 1.7 .

The first coordinate −4 of the point Q tells us that the point is located 4 units to the left of the y -axis (because −4 is negative ). The second coordinate 6 of the point Q tells us that the point is located 6 units above the x -axis (because 6 is positive). See Fig. 1.7 .

The first coordinate 2 of the point R tells us that the point is located 2 units to the right of the y -axis (because 2 is positive ). The second coordinate 5 of the point R tells us that the point is located 5 units above the x -axis (because 5 is positive). See Fig. 1.7 .

The first coordinate −5 of the point S tells us that the point is located 5 units to the left of the y -axis (because −5 is negative ). The second coordinate −3 of the point S tells us that the point is located 3 units below the x -axis (because −3 is negative). See Fig. 1.7 .

Fig. 1.7

Example 2

Give the coordinates of the points X, Y, Z, W exhibited in Fig. 1.8 .

Fig. 1.8

Solution 2

The point X is 1 unit to the right of the y -axis and 3 units below the x -axis. Therefore its coordinates are (1, −3).

The point Y is 2 units to the left of the y -axis and 1 unit above the x -axis. Therefore its coordinates are (−2, 1).

The point Z is 5 units to the right of the y -axis and 4 units above the x -axis. Therefore its coordinates are (5, 4).

The point W is 6 units to the left of the y -axis and 5 units below the x -axis. Therefore its coordinates are (−6, −5).

Example 3

Sketch the set of points ℓ = {( x , y ): y = 3}. Sketch the set of points k = {( x , y ): x = −4}.

Solution 3

The set ℓ consists of all points with y -coordinate equal to 3. This is the set of all points that lie 3 units above the x -axis. We exhibit ℓ in Fig. 1.9 . It is a horizontal line.

Fig. 1.9

The set k consists of all points with x -coordinate equal to −4. This is the set of all points that lie 4 units to the left of the y -axis. We exhibit k in Fig. 1.10 . It is a vertical line.

Fig. 1.10

Example 4

Sketch the set of points S = {( x , y ): x > 2} on a pair of coordinate axes.

Solution 4

Notice that the set S contains all points with x -coordinate greater than 2. These will be all points to the right of the vertical line x = 2. That set is exhibited in Fig. 1.11 .

Fig. 1.11