C4.1; E4.1 - Cartesian coordinates

C4.1; E4.1 - Cartesian coordinates

Does (−3,2)(-3,2) name the same place as (2,−3)(2,-3)? Coordinates give a precise address on a flat grid, but the order and scale both matter. C4.1 and E4.1 develop reading and placing points in all four quadrants, including grids with fractional and decimal scales.

Two axes

Two number lines cross at right angles to locate a point. The horizontal line is the xx-axis, the vertical line is the yy-axis, and their crossing is the origin, (0,0)(0,0). Values increase to the right and upwards.

Cartesian coordinates

The ordered pair (x,y)(x,y) gives a point's horizontal position first and vertical position second. From the origin, positive xx means right and negative xx means left; positive yy means up and negative yy means down.

Start with the horizontal coordinate even when it is negative. You can imagine walking along the xx-axis, then moving vertically to the point. These movements locate the point; they do not prescribe a journey that an object must take.

[DIAGRAM: asset_name: Ordered pairs in four quadrants; asset_slug: coordinates_four_quadrants; caption: Changing the order changes the point: A is at minus 3, 2 and B is at 2, minus 3.; description: An accurately scaled Cartesian grid from minus four to four on both axes shows A at minus three comma two, B at two comma minus three, C at three comma two and D at minus three comma minus two. The four quadrants are labelled I positive positive, II negative positive, III negative negative and IV positive negative. Dashed projection lines from A and B show their two coordinate readings.]
Changing the order changes the point: A is at minus 3, 2 and B is at 2, minus 3.

Worked example 1: interpreting an ordered pair

Question. Describe the position of A(−3,2)A(-3,2) relative to the origin.

Plan. Read the first coordinate horizontally, then the second vertically.

x=−3⟶3 units lefty=2⟶2 units up\begin{aligned} x=-3&\quad\longrightarrow\quad 3\text{ units left}\\ y=2&\quad\longrightarrow\quad 2\text{ units up} \end{aligned}

Answer. AA is 3 units left and 2 units above the origin.

Check. A point left of the yy-axis has negative xx; a point above the xx-axis has positive yy. Both signs match (−3,2)(-3,2).

Now swap the same two numbers. The horizontal instruction changes as well as the vertical one.

Worked example 2: reversing the order

Question. Describe the position of B(2,−3)B(2,-3) relative to the origin.

Plan. Apply the same horizontal-first convention to the new order.

x=2⟶2 units righty=−3⟶3 units down\begin{aligned} x=2&\quad\longrightarrow\quad 2\text{ units right}\\ y=-3&\quad\longrightarrow\quad 3\text{ units down} \end{aligned}

Answer. BB is 2 units right and 3 units below the origin, so it is a different point from AA.

Check. Its right-and-down position has signs (+,−)(+,-), matching (2,−3)(2,-3).

The axes divide the plane into four quadrants, numbered anticlockwise from the upper-right region: I has signs (+,+)(+,+), II (−,+)(-,+), III (−,−)(-,-), and IV (+,−)(+,-). A point on an axis belongs to no quadrant. For (0,5)(0,5), zero horizontal movement puts it on the yy-axis; for (−4,0)(-4,0), zero vertical movement puts it on the xx-axis.

Watch out

Coordinates are an ordered pair, not an unordered list. Write (−3,2)(-3,2), with brackets and a comma; reversing the entries usually gives another point. Zero is a coordinate value, not an instruction to omit an entry.

Use a zero coordinate to check the axis rule.

Plotting points

Read the scale on each axis before counting squares. If two adjacent labelled values are 0 and 2 with four equal small squares between them, each small square represents 2÷4=0.52\div4=0.5 units. The two axes can have different scales.

Plotting a point

To plot (x,y)(x,y), find the required xx-position and yy-position using the axes' scales, then mark their intersection with a small clear cross. Put the point's letter nearby, without covering the cross.

Choose axes that include all the required coordinates, label them xx and yy, and keep each axis's scale uniform. A plot is accurate to scale; a sketch merely communicates important features. Aim for the exact intersection; graph readings and plots are expected within half the smallest square. Separate points stay separate unless a question asks you to join them. When a straight line is required, draw it with a ruler.

[DIAGRAM: asset_name: One point on two scales; asset_slug: plotting_point_scales; caption: The point is three units right and two units down in both grids, but the number of squares changes.; description: Two accurate grids show P at three comma minus two. On the left one small square is one unit on each axis, so P is three squares right and two down. On the right one small square is half a unit horizontally and one unit vertically, so P is six squares right and two down. Both axes have clearly labelled numeric scales and the point cross and letter are separated.]
The point is three units right and two units down in both grids, but the number of squares changes.

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