Is X Up Or Down
The Coordinate Plane
Learning Objective(s)
· Plot ordered pairs on a coordinate plane.
· Given an ordered pair, decide its quadrant.
Introduction
The coordinate plane was adult centuries ago and refined by the French mathematician René Descartes. In his honor, the system is sometimes chosen the Cartesian coordinate arrangement. The coordinate plane can be used to plot points and graph lines. This organisation allows us to describe algebraic relationships in a visual sense, and also helps u.s. create and interpret algebraic concepts.
Getting to Know the Coordinate Plane
You have probable used a coordinate aeroplane earlier. For instance, have you lot ever used a gridded overlay to map the position of an object? (This is often done with road maps, too.)
This "map" uses a horizontal and vertical grid to convey information about an object's location. Notice that the letters A-F are listed along the top, and the numbers one-vi are listed forth the left edge. The general location of whatever item on this map can be constitute by using the alphabetic character and number of its grid foursquare. For example, y'all can notice the item that exists at square "4F" by moving your finger forth the horizontal to letter F and then straight downward so you are in line with the 4. Yous'll find a blue disc is at this location on the map.
The coordinate plane has like elements to the grid shown in a higher place. It consists of a horizontal axis and a vertical centrality, number lines that intersect at right angles. (They are perpendicular to each other.)
The horizontal axis in the coordinate plane is chosen the x-axis. The vertical axis is called the y-centrality. The point at which the two axes intersect is called the origin. The origin is at 0 on the 10-centrality and 0 on the y-axis.
The intersecting x- and y-axes separate the coordinate plane into iv sections. These four sections are called quadrants. Quadrants are named using the Roman numerals I, II, III, and Four beginning with the top right quadrant and moving counter clockwise.
Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a betoken by relating the point's location along the x-axis (the commencement value of the ordered pair) and along the y-centrality (the 2nd value of the ordered pair).
In an ordered pair, such as (x, y), the starting time value is called the x-coordinate and the 2d value is the y-coordinate. Note that the 10-coordinate is listed earlier the y-coordinate. Since the origin has an x-coordinate of 0 and a y-coordinate of 0, its ordered pair is written (0, 0).
Consider the point beneath.
To identify the location of this point, start at the origin (0, 0) and move right along the 10-axis until you are under the indicate. Look at the characterization on the x-axis. The iv indicates that, from the origin, you have traveled four units to the correct along the 10-centrality. This is the ten-coordinate, the first number in the ordered pair.
From iv on the x-axis move upwardly to the point and notice the number with which it aligns on the y-axis. The 3 indicates that, after leaving the ten-axis, you traveled three units upward in the vertical direction, the direction of the y-axis. This number is the y-coordinate, the 2d number in the ordered pair. With an ten-coordinate of 4 and a y-coordinate of iii, you have the ordered pair (4, 3).
Let'south expect at another case.
| Example | ||
| Problem | Describe the point shown as an ordered pair. | |
| | ||
| (v, y) | Begin at the origin and movement along the x-centrality. This is the x-coordinate and is written first in the ordered pair. | |
| (5, two) | Move from five up to the ordered pair and read the number on the y-axis. This is the y-coordinate and is written second in the ordered pair. | |
| Answer | The point shown equally an ordered pair is (5, 2). | |
Plotting Points in the Coordinate Plane
Now that you lot know how to use the x- and y-axes, you can plot an ordered pair too. Just remember, both processes commencement at the origin—the beginning! The example that follows shows how to graph the ordered pair (ane, iii).
| Case | ||
| Problem | Plot the bespeak (1, 3). | |
| | ||
| The x-coordinate is ane because it comes first in the ordered pair. Start at the origin and motion a distance of 1 unit in a positive direction (to the right) from the origin along the 10-centrality. | The y-coordinate is 3 because it comes 2d in the ordered pair. From here motility directly three units in a positive direction (up). If yous await over to the y-axis, you lot should be lined up with three on that centrality. | |
| Respond | Draw a point at this location and label the point (1, 3). | |
In the previous example, both the ten- and y-coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you lot will need to movement in the negative management along ane or both axes. Consider the example below in which both coordinates are negative.
| Example | ||
| Problem | Plot the betoken (−iv, −2). | |
| | ||
| The x-coordinate is − 4 because it comes first in the ordered pair. Showtime at the origin and move iv units in a negative direction (left) along the x-axis. | The y-coordinate is − 2 because information technology comes second in the ordered pair. Now move 2 units in a negative direction (downwards). If y'all look over to the y-centrality, yous should be lined upwards with − 2 on that axis. | |
| Reply | Depict a bespeak at this location and label the indicate (−iv, −2). | |
The steps for plotting a signal are summarized below.
Steps for Plotting an Ordered Pair (x, y) in the Coordinate Plane
o Determine the ten-coordinate. Start at the origin, move horizontally, the management of the ten-axis, the distance given by the x-coordinate. If the x-coordinate is positive, move to the correct; if the x-coordinate is negative, move to the left.
o Determine the y-coordinate. Start at the x-coordinate, motility vertically, the direction of the y-axis, the altitude given by the y-coordinate. If the y-coordinate is positive, movement up; if the y-coordinate is negative, move down.
o Draw a bespeak at the ending location. Label the point with the ordered pair.
Which point represents the ordered pair ( − 2, −iii)?
Show/Hide Answer
Point B is right. Begin at the origin and move 2 units in the negative direction (left) along the x-axis. Then, move 3 units in a positive direction along the y-centrality (3 units up). Plot a betoken there.
The Iv Quadrants
Ordered pairs within any detail quadrant share certain characteristics. Look at each quadrant in the graph beneath. What do you observe near the signs of the ten- and y-coordinates of the points within each quadrant?
Within each quadrant, the signs of the x-coordinates and y-coordinates of each ordered pair are the aforementioned. They also follow a pattern, which is outlined in the table below.
| Quadrant | General Form of Point in this Quadrant | Example | Description |
| I | (+, +) | (5, 4) | Starting from the origin, go along the x-centrality in a positive direction (right) and along the y-axis in a positive direction (up). |
| Two | ( − , +) | ( − v, 4) | Starting from the origin, go along the x-axis in a negative direction (left) and forth the y-axis in a positive direction (up). |
| III | ( − , − ) | ( − 5, − 4) | Starting from the origin, go along the x-axis in a negative direction (left) and forth the y-centrality in a negative direction (down). |
| Four | (+, − ) | (5, − 4) | Starting from the origin, go along the 10-axis in a positive management (right) and along the y-axis in a negative direction (downwardly). |
In one case y'all know nigh the quadrants in the coordinate aeroplane, yous tin can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here's another style to think about it.
The example below details how to decide the quadrant location of a point only by thinking about the signs of its coordinates. Thinking nigh the quadrant location earlier plotting a point tin help you preclude a mistake. It is also useful noesis for checking that you take plotted a point correctly.
| Instance | ||
| Problem | In which quadrant is the point ( − 7, x) located? | |
| ( − seven, ten) | Look at the signs of the x- and y-coordinates. For this ordered pair, the signs are ( − , +). | |
| Points with the pattern | Using the tabular array or filigree higher up, locate the pattern ( − , +). | |
| Answer | The indicate ( − vii, 10) is in Quadrant Ii. | |
| Example | ||
| Problem | In which quadrant is the betoken ( − 10, − 5) lo cated? | |
| ( − 10, − 5) | Look at the signs of the x- and y-coordinates. For this ordered pair, the signs are ( − , − ). | |
| Points with the design | Using the table or grid in a higher place, locate the design ( − , − ). | |
| Reply | The point ( − 10, − 5) is in Quadrant III. | |
What happens if an ordered pair has an x- or y-coordinate of zero? The example beneath shows the graph of the ordered pair (0, four).
A point located on one of the axes is not considered to be in a quadrant. Information technology is but on one of the axes. Whenever the x-coordinate is 0, the indicate is located on the y-axis. Similarly, any point that has a y-coordinate of 0 will be located on the 10-centrality.
Which of the descriptions below best describes the location of the point (8, 0)?
A) Quadrant I
B) It is on the ten-axis
C) It is on the y-axis
D) The coordinate plane
Show/Hibernate Answer
A) Quadrant I
Wrong. The axes are not considered part of the quadrant. Rather, they are the boundaries that define the quadrant. The correct answer is the x-axis.
B) It is on the x-axis
Right. To plot this point, you would begin at the origin and move viii units in the positive direction along the x-axis. Since the y-coordinate is 0, you do non motility off the location on the ten-axis.
C) Information technology is on the y-axis
Incorrect. Since the y-coordinate is 0, this point is located on the x-axis. To plot this bespeak, you would begin at the origin and motion 8 units in the positive direction along the x-axis. Since the y-coordinate is 0, you exercise non move off the location on the x-centrality. The correct respond is the 10-centrality.
D) The coordinate plane
Incorrect. Even though this point in located in the coordinate plane, yous can provide a meliorate description of its location. The correct reply is the ten-axis.
The coordinate plane is a system for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (10-) axis and a vertical (y-) axis. The intersection of these lines creates the origin, which is the point (0, 0). The coordinate aeroplane is split up into 4 quadrants. Together, these features of the coordinate arrangement allow for the graphical representation and communication nigh points, lines, and other algebraic concepts.
Is X Up Or Down,
Source: http://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U13_L1_T1_text_final.html
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