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Here are a few recommended readings before getting started with this lesson.
A coordinate system is a reference framework used to describe the positions of objects like points, lines, and surfaces in a space. A fixed point called the origin is used as a reference in the coordinate system. The most common types of coordinate systems are one-, two-, or three-dimensional.
A coordinate plane is a two-dimensional coordinate system. It is a grid that results from intersecting a vertical number line with a horizontal number line at their zero points. The horizontal number line is usually named the x-axis and the vertical number line is usually the y-axis.
The positive numbers on the y-axis are above zero and the negative ones are below zero. The origin is where the lines intersect, which is the point (0,0).A coordinate is the position of an object in a coordinate system relative to the corresponding axis. Coordinates are often seen together with other coordinates, which can describe the position of an object in a coordinate system with one, two, three, or even more dimensions.
In a two-dimensional coordinate system, points are usually expressed as a coordinate pair — also called an ordered pair — denoted by (x,y). The first coordinate states the position along the x-axis and the second coordinate states the position along the y-axis.
Identify the coordinates of the given point. Write the coordinates as an ordered pair (x,y), where x represents the x-coordinate and y the y-coordinate.
Dominika plays basketball with her friends every afternoon. She warms up by running and picks up two friends, Jordan and Emily, on the way to the court. Dominika uses ordered pairs to represent the positions of her and her friends' houses.
Position | |
---|---|
Dominika's House | D=(2,1) |
Jordan's House | J=(2,4) |
Emily's House | E=(-1,4) |
Moving one unit on the x-axis or y-axis on the coordinate plane is counted as moving one block. Since Dominika can only stay on the straight roads, her position can only move up, down, left, and right on the graph.
Moving one unit on the x-axis or y-axis on the coordinate plane is the same as Dominika moving one block. Dominika only runs on the straight roads, so her progress on the graph can only move in four directions — up, down, left, and right.
Keeping this in mind, one route she can take is moving five blocks directly to the right. Move 5 units to the right starting her house to show this possible destination. Remember that Dominika's house is located at (2,1), not the origin!
Another spot Dominika might go is one block up and four blocks to the right of her starting point. It does not matter if she moves up on her first, second, third, fourth, or final move. Her destination will always be the same if she moves one block up out of the five total moves.
Follow the same reasoning to find the other places that Dominika can go. Plot each as ordered pairs on the coordinate plane.
Now that all the points are plotted on the graph, connect them with lines to see the overall shape.
The shape is a rotated square.
The places Dominika can go can be determined by identifying all the possible routes she can take. Remember, she can only go five blocks away from her home. Moving left or down indicates a negative direction on the corresponding axis, represented by negative numbers.
Route from Home | Calculations | Destination |
---|---|---|
5 blocks to the right | (2+5,1) | (7,1) |
4 blocks to the right and 1 block up | (2+4,1+1) | (6,2) |
3 blocks to the right and 2 blocks up | (2+3,1+2) | (5,3) |
2 blocks to the right and 3 blocks up | (2+2,1+3) | (4,4) |
1 block to the right and 4 blocks up | (2+1,1+4) | (3,5) |
5 blocks up | (2,1+5) | (2,6) |
1 block to the left and 4 blocks up | (2−1,1+4) | (1,5) |
2 blocks to the left and 3 blocks up | (2−2,1+3) | (0,4) |
3 blocks to the left and 2 blocks up | (2−3,1+2) | (-1,3) |
4 blocks to the left and 1 block up | (2−4,1+1) | (-2,2) |
5 blocks to the left | (2−5,1) | (-3,1) |
4 blocks to the left and 1 block down | (2−4,1−1) | (-2,0) |
3 blocks to the left and 2 blocks down | (2−3,1−2) | (-1,-1) |
2 blocks to the left and 3 blocks down | (2−2,1−3) | (0,-2) |
1 block to the left and 4 blocks down | (2−1,1−4) | (1,-3) |
5 blocks down | (2,1−5) | (2,-4) |
1 block to the right and 4 blocks down | (2+1,1−4) | (3,-3) |
2 blocks to the right and 3 blocks down | (2+2,1−3) | (4,-2) |
3 blocks to the right and 2 blocks down | (2+3,1−2) | (5,-1) |
4 blocks to the right and 1 block down | (2+4,1−1) | (6,0) |
In a coordinate plane, the intersection of the x-axis and the y-axis produces four regions called quadrants. The quadrants are numbered counterclockwise from the top right quadrant as Quadrant I to Quadrant IV in the bottom right.
The signs of the coordinates of a point can be determined based on which quadrant the point lies in.
Conversely, the quadrant of a point can be found by looking at the signs of its coordinates.A coordinate system has four quadrants that provide information about the signs of the x- and y-coordinates of a point. In the following applet, identify in which quadrant the given point lies.
Look at the signs of the coordinates of each point to identify their quadrants.
Point | Signs of Coordinates | Quadrant |
---|---|---|
(-2,-8) | (−,−) | Quadrant III |
(4,-2) | (+,−) | Quadrant IV |
(-11,3) | (−,+) | Quadrant II |
(4,3) | (+,+) | Quadrant I |
Reflecting a point means making a mirror image of that point by flipping it across a certain line or axis. The following rules show how to find reflected points across a specific axis.
Point | Axis of Reflection | Procedure | Reflected Point |
---|---|---|---|
(x1,y1) | x | Change the y-coordinate to its opposite. | (x1,-y1) |
(x1,y1) | y | Change the x-coordinate to its opposite. | (-x1,y1) |
(x1,y1) | x and y | Change the x- and y-coordinates to their opposites. | (-x1,-y1) |
Next, the process for reflecting points across the x-axis, the y-axis, or both, on a coordinate plane will be shown. Consider the points A, B, and C.
Point | Axis of Reflection | Reflected Point |
---|---|---|
A=(5,8) | x | ? |
B=(10,-5) | y | ? |
C=(-4,-7) | x and y | ? |
Once each initial point and its reflection across a given axis are found, plot both points on the coordinate plane.
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