Coordinates and the coordinate plane · Coordenadas y el plano cartesiano
| English | Español |
|---|---|
| coordinate plane/kəʊˈɔːdɪnət pleɪn/ | plano cartesiano |
| origin/ˈɒrɪdʒɪn/ | origen |
| quadrants/ˈkwɒdrənts/ | cuadrantes |
| coordinates/kəʊˈɔːdɪnəts/ | coordenadas |
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 坐标平面: every point gets a unique address.
The coordinate plane · El plano cartesiano
- Two perpendicular number lines — the $x$-axis (horizontal) and the $y$-axis (vertical) — meet at the origin · origen 原点 $O = (0, 0)$.
- 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 · El plano cartesiano
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. · Cada punto tiene una coordenada (x, y). Una línea recta es el conjunto de puntos donde y depende de x de una manera fija.
The point where the $x$-axis and $y$-axis cross is called the ______. · El punto donde los ejes $x$ e $y$ se cruzan se llama ______.
The origin is the point $(0, 0)$ where the two axes meet. · El origen es el punto $(0, 0)$ donde se encuentran los dos ejes.
Match each point to its quadrant. · Empareja cada punto con su cuadrante.
Quadrant I: both positive. II: $x$ negative, $y$ positive. III: both negative. IV: $x$ positive, $y$ negative. · Cuarto I: ambos positivos. II: $x$ negativo, $y$ positivo. III: ambos negativos. IV: $x$ positivo, $y$ negativo.
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 · arriba (positive) or down · hacia abajo (negative).
- The point $(3, -2)$: go $3$ right, then $2$ down. It sits in quadrant IV.
The order matters. $(3, -2)$ and · y $(-2, 3)$ 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? · ¿En qué cuadrante se encuentra el punto $(-4, 3)$?
Negative · Negativo $x$ (left) and positive $y$ (up) → quadrant II (top-left). · $x$ negativo (izquierda) e $y$ positivo (arriba) → cuadrante II (arriba-izquierda).
A point starts at $(2, 5)$. It moves $5$ right and $2$ down. What is its new $y$-coordinate? · Un punto comienza en $(2, 5)$. Se mueve $5$ a la derecha y $2$ hacia abajo. ¿Cuál es su nueva coordenada $y$?
New position: $(2+5,\; 5-2) = (7, 3)$. The $y$-coordinate is $3$. · Nueva posición: $(2+5,\; 5-2) = (7, 3)$. La coordenada $y$ es $3$.
The points $(3, -2)$ and · y $(-2, 3)$ are the same point. · Los puntos $(3, -2)$ y $(-2, 3)$ son el mismo punto.
Coordinates are ordered: $(x, y)$. $(3, -2)$ is in quadrant IV; $(-2, 3)$ is in quadrant II — completely different locations. · Las coordenadas están ordenadas: $(x, y)$. $(3, -2)$ está en el cuadrante IV; $(-2, 3)$ está en el cuadrante II — ubicaciones completamente diferentes.
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.
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 $y$, 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 $x$). A horizontal line has $m=0$ and fits $y=mx+c$; a vertical line does not.
Which equation gives a horizontal line? · ¿Qué ecuación da como resultado una línea horizontal?
$y = k$ fixes the $y$-value for every $x$, giving a horizontal line. $x = k$ is vertical. · $y = k$ fija el valor de $y$ para cada $x$, lo que da una línea horizontal. $x = k$ es 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 · origen $(0, 0)$
- 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