Coordinates and the coordinate plane
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 single number ("go 3 blocks") leaves you on a whole circle. Two numbers pin you to a single point.
- That is the whole idea behind the coordinate plane: 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 (I, II, III, IV).
- Any point is described by its coordinates $(x, y)$ — 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 point where the $x$-axis and $y$-axis cross is called the ______.
The origin is the point $(0, 0)$ where the two axes meet.
Match each point to its quadrant.
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.
The point $(-4, 3)$ is in which quadrant?
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?
New position: $(2+5,\; 5-2) = (7, 3)$. The $y$-coordinate is $3$.
The points $(3, -2)$ and $(-2, 3)$ are the same point.
Coordinates are ordered: $(x, y)$. $(3, -2)$ is in quadrant IV; $(-2, 3)$ is in quadrant II — completely different locations.
Drawing a straight-line graph
- Most lines are $y = mx + c$. The quickest way to draw one:
- 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.
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 $y$, plot the points, and join — the table guarantees accuracy.
Special lines. $x = k$ is a vertical line (fixed $x$ for every $y$); $y = k$ is a horizontal line (fixed $y$ for every $x$). These are the only lines that don't fit $y = mx + c$.
Which equation gives a horizontal line?
$y = k$ fixes the $y$-value for every $x$, giving a horizontal line. $x = k$ is vertical.
Fun fact
- The coordinate plane is named after René Descartes (1596–1650), who — legend has it — invented it while lying in bed watching a fly on the ceiling and wondering how to describe its exact position.
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