Coordinates and the coordinate plane
Introduced| English | Meaning |
|---|---|
| origin/ˈɒrɪdʒɪn/ | The point (0, 0) where the axes cross. |
| coordinate plane/kəʊˈɔːdɪnət pleɪn/ | A flat system with a horizontal x-axis and a vertical y-axis, used to locate points with ordered pairs. |
| quadrants/ˈkwɒdrənts/ | The four regions of a coordinate plane, divided by the axes. |
| coordinates/kəʊˈɔːdɪnəts/ | Ordered values specifying the position of a point relative to a chosen system. |
A city, a battleship, a treasure map
- A friend says: "Walk 3 blocks east, then 2 blocks south." You can find the spot exactly — because you have two directions.
- A distance alone does not tell you the direction. An ordered pair gives the horizontal and vertical positions from a fixed origin 原点.
- That is the whole idea behind the coordinate plane 坐标平面: every point gets a unique address.
The coordinate plane
- Two perpendicular number lines — the $x$-axis (horizontal) and the $y$-axis (vertical) — meet at the origin $O = (0, 0)$.
- The axes split the plane into four quadrants 象限 (I, II, III, IV).
- Any point is described by its coordinates 坐标 $(x, y)$ — the $x$-value first, then the $y$-value.
The axes meet at the origin; $(3, -2)$ means 3 right, then 2 down — fourth quadrant.
The coordinate plane
y = mx + c
Every point has an (x, y) coordinate. A straight line is the set of points where y depends on x in a fixed way.
The point where the $x$-axis and $y$-axis cross is called the ______.
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The origin is the point $(0, 0)$ where the two axes meet.
Match each point to its quadrant.
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Quadrant I: both positive. II: $x$ negative, $y$ positive. III: both negative. IV: $x$ positive, $y$ negative.
Reading a point
- Start at the origin. Read the $x$-value by going across (right is positive, left is negative).
- Then read the $y$-value by going up (positive) or down (negative).
- The point $(3, -2)$: go $3$ right, then $2$ down. It sits in quadrant IV.
The order matters. $(3, -2)$ and $(-2, 3)$ are completely different points. Always read across first, then up/down — like walking along a street before climbing stairs.
A city street grid: every place is fixed by its coordinates
The point $(-4, 3)$ is in which quadrant?
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Negative $x$ (left) and positive $y$ (up) → quadrant II (top-left).
A point starts at $(2, 5)$. It moves $5$ right and $2$ down. What is its new $y$-coordinate?
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New position: $(2+5,\; 5-2) = (7, 3)$. The $y$-coordinate is $3$.
The points $(3, -2)$ and $(-2, 3)$ are the same point.
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Coordinates are ordered: $(x, y)$. $(3, -2)$ is in quadrant IV; $(-2, 3)$ is in quadrant II — completely different locations.
Core and Extended: on labelled axes, plot A(−3,2), B(2,−3), C(0,−2) and D(−2,0). State each quadrant, or the axis on which the point lies. Show why A and B are different points.
Writing the quadrant names alone does not replace plotting the points. Ask a teacher to check your drawing.
Drawing a straight-line graph
- For Core and Extended, draw a non-vertical line given as $y=mx+c$ using either method:
- Method 1 — mark the intercept $c$ on the $y$-axis, then step using the gradient $m$.
- Method 2 — make a table of values: pick a few $x$-values, calculate $y$, plot the points, and join them.
The axes meet at the origin $O$ and split the plane into four quadrants; $(3,-2)$ means 3 right and 2 down
Worked example — table of values
- Draw $y = 2x + 1$.
| $x$ | $y = 2x + 1$ |
|---|---|
| $0$ | $1$ |
| $1$ | $3$ |
| $2$ | $5$ |
- Plot $(0, 1)$, $(1, 3)$, $(2, 5)$ and join with a straight line.
Pick $x$-values, find $y$, and plot carefully. A third point checks the line through the first two.
Special lines. $x = k$ is a vertical line (fixed $x$ for every $y$); $y = k$ is a horizontal line (fixed $y$ for every $x$). A horizontal line has $m=0$ and fits $y=mx+c$; a vertical line does not.
Which equation gives a horizontal line?
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$y = k$ fixes the $y$-value for every $x$, giving a horizontal line. $x = k$ is vertical.
Check the scale and the whole line
- Read the numbered axes before plotting: one grid square does not always mean one unit. Negative coordinates go left or down; a point on an axis belongs to no quadrant.
- Use a ruler and extend the straight line across the requested interval, rather than drawing only the short segment between two points. A third calculated point checks the graph. Extended: you may also be given a form such as $3x+2y=6$; rearrange to $y=-\tfrac32x+3$ or calculate a table first.
Core and Extended: draw y=−2x+3 for −2≤x≤3. Use a table with x=−2,0,1,3, label the axes and join the points with a ruler. Then draw x=−1 on the same axes. Check a third point on your sloping line.
A table is preparation for the actual graph; it is not the final drawing.
Extended: draw 3x+2y=6 for −2≤x≤4. Show a table using x=−2,0,2,4. State where the line crosses each axis.
This equation form is an Extended drawing demand; Core drawing equations are normally supplied as y=mx+c unless a table is given.
You've got it
- a point is $(x, y)$ — across first, then up/down; the axes meet at the origin $(0, 0)$
- the four quadrants are numbered I (top-right) → IV (bottom-right) anti-clockwise
- $(3, -2)$ → 3 right, 2 down (quadrant IV); $(-2, 3)$ is a different point entirely
- draw $y = mx + c$ by marking $c$ on the $y$-axis, then stepping with the gradient — or use a table of values
Plot and draw independently
- A point starts at (1, 2).
- It moves 4 right and 3 up.
A point starts at (1, 2) and moves 4 right and 3 up. What is the x-coordinate of its new position?
1 plus 4 is 5.
A point starts at (1,2) and moves 4 right and 3 up. What is its new x-coordinate?
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Only the move to the right changes x.