0580 · 3.1Coordinates

Coordinate axes and ordered pairs

0:00A point on a grid needs a precise address. Saying that it is near the top leaves many possible positions. Coordinates remove that uncertainty by describing its horizontal and vertical position separately. By the end of this video, you will understand how to read and plot that address, including negative values and different scales. You need one familiar idea before we begin. On a number line, positive numbers lie to the right of zero, while negative numbers lie to the left.

0:24A vertical number line uses the same ordering, with positive values above zero and negative values below. Coordinates combine these two directions into one diagram. A Cartesian coordinate system uses two perpendicular number lines called axes. The horizontal one is the x axis, and the vertical one is the y axis. They meet at the origin, where both coordinates are zero. Together, they let us describe positions in two dimensions.

0:50Each point has an ordered pair of coordinates. The first value tells us its x position, and the second gives its y position. The word ordered matters because these values have different jobs. Swapping them usually changes the position, just as swapping a house number and a street name changes an address. Consider the point with x equal to three and y equal to two. Its horizontal position is three units to the right of the origin.

1:14Its vertical position is two units above the horizontal axis. The point lies where those two positions meet, so both coordinates describe it together. Moving across and then vertically is a useful way to locate a point. However, the coordinates describe its position, not a journey it must follow. You could reach the same location by moving vertically first. You would still write the horizontal coordinate first because that is the agreed order.

Plotting negative coordinates

1:37For our first worked example, plot the point P with x equal to negative three and y equal to two. Start at the origin and move three units left because the horizontal coordinate is negative. From that position, move two units upwards because the vertical coordinate is positive. Mark the intersection with a small cross and label it P. Negative coordinates do not mean that a point is impossible or that a distance has become negative.

2:00They identify which side of an axis contains the point. A negative x coordinate places it left of the vertical axis, and a negative y coordinate puts it below the horizontal one. The axes divide the plane into four regions called quadrants. Both coordinates are positive in the upper right region. In the upper left, x is negative while y is positive. Both are negative in the lower left, while only y is negative in the lower right.

Reading coordinates and scales

2:26Reading coordinates reverses the plotting process. Locate the point horizontally and read its x value from the horizontal axis. Then use its vertical position to find the y value. Before counting grid spaces, inspect the scale because one square does not always represent one unit. For our second example, write the coordinates of Q on this scaled grid. Each horizontal square represents zero point five, so three squares left represent one point five units.

2:51That gives an x coordinate of negative one point five. Each vertical square represents two units, so two squares upwards give y equal to four. The answer is the point negative one point five comma four. Notice that the two axes use different scales. Counting squares without reading their labels would give the wrong coordinates. Coordinates can also contain fractions or decimals because points are not restricted to whole number positions.

3:16For our third example, write the coordinates of the two points on the axes. Point R lies four units left of the origin, so its x coordinate is negative four. It has no vertical displacement, giving y equal to zero. Therefore, R is the point negative four comma zero. Point S lies directly below the origin, so its x coordinate is zero. Its vertical position is three units down, giving y equal to negative three.

3:41So S is zero comma negative three. Every point on the vertical axis has x equal to zero, while points on the horizontal axis have y equal to zero. Common mistakes include reversing the coordinates, overlooking negative signs, and assuming every square represents one unit. Keep the horizontal value first because it controls the left or right position. Check the signs against the point’s region before finishing. When a point lies on an axis, include its zero coordinate rather than leaving that value out.

4:09An examination may ask you to plot points or write their coordinates from a diagram. These methods must work without a calculator on Paper two, and remain useful on Paper four. Use a small, precise mark when plotting and check both axis scales. Keep exact coordinate values rather than rounding them unnecessarily. Coordinates give each point a precise position using two values. The x coordinate describes the horizontal position, and the y coordinate describes the vertical position.

4:35Their signs identify direction from the origin, while the scales determine the value of each grid interval. Understanding those roles makes reading and plotting coordinates the same idea used in opposite directions.