The Coordinate Plane

The coordinate plane is made from two number lines that cross at right angles. The horizontal number line is the $x$-axis. The vertical number line is the $y$-axis. The point where they cross is the origin, written $(0,0)$.

Every location on the plane is named by an ordered pair $(x,y)$. The first number tells you how far to move left or right along the $x$-axis. The second number tells you how far to move up or down parallel to the $y$-axis. Because the order matters, we call it an ordered pair: $(3,5)$ is a different point than $(5,3)$.

The two axes split the plane into four regions called quadrants. We number them with Roman numerals, starting at the top-right and going counter-clockwise:

• Quadrant I — top right — $x$ positive, $y$ positive: $(+,+)$
• Quadrant II — top left — $x$ negative, $y$ positive: $(-,+)$
• Quadrant III — bottom left — $x$ negative, $y$ negative: $(-,-)$
• Quadrant IV — bottom right — $x$ positive, $y$ negative: $(+,-)$

If a point sits on an axis, it is not in any quadrant.

• A point on the $x$-axis has the form $(x,0)$ — its $y$-coordinate is $0$. Example: $(4,0)$.
• A point on the $y$-axis has the form $(0,y)$ — its $x$-coordinate is $0$. Example: $(0,-3)$.
• The origin $(0,0)$ sits on both axes.

To plot the point $(x,y)$:

• Start at the origin $(0,0)$.
• Move along the $x$-axis: right if $x>0$, left if $x<0$.
• From there move parallel to the $y$-axis: up if $y>0$, down if $y<0$.
• Mark a dot where you stop.

To name the coordinates of a point you already see, do the reverse: count how far it is left/right, then up/down, from the origin.

A table of values is a neat list of ordered pairs. We usually pick a few $x$-values, apply a rule, and record the matching $y$-values. Each row of the table $(x,y)$ becomes one point on the plane.

A rule like $y = x+2$ or $y = 2x$ describes how $y$ depends on $x$. When you plot all the ordered pairs from its table, the points fall in a perfectly straight line — this is a linear relationship. Connecting the dots gives the graph of the rule.

Two points lie on the same horizontal line when they share the same $y$-coordinate. Two points lie on the same vertical line when they share the same $x$-coordinate. In either case the distance between them is easy to find:

• Horizontal: subtract the $x$-coordinates and take the positive difference.
• Vertical: subtract the $y$-coordinates and take the positive difference.

Coordinates appear everywhere: on maps, game boards, and city grids. A treasure map might place a chest at $(5,3)$, meaning $5$ blocks east and $3$ blocks north of a starting corner. Counting blocks between two spots on a grid is exactly the horizontal or vertical distance from the last section.

For two points not on the same horizontal or vertical line, the straight-line distance still comes from a right triangle. Between $(x_1,y_1)$ and $(x_2,y_2)$, the horizontal leg has length $|x_2-x_1|$ and the vertical leg has length $|y_2-y_1|$. The distance is the hypotenuse, so by the Pythagorean theorem

Because each difference is squared, the sign of the subtraction never matters — squaring erases it.

The midpoint of the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is the point exactly halfway between them. You simply average the $x$-coordinates and average the $y$-coordinates:

The average of two numbers always lands in the middle, so this works even when the coordinates are negative.

The slope of a line measures how steeply it rises. It is the ratio of vertical change to horizontal change between any two points on the line:

A positive slope climbs left-to-right; a negative slope falls. A larger $|m|$ means a steeper line. In a real setting, slope is a rate of change — for the plant rule $y=3x+1$, the slope $3$ means “$3$ cm of growth per day.”

Every non-vertical line can be written as

where $m$ is the slope and $b$ is the $y$-intercept — the $y$-value where the line crosses the $y$-axis (that is, where $x=0$). For $y=2x+1$: the slope is $m=2$, and the line crosses the $y$-axis at $(0,1)$. Each step right of $1$ raises $y$ by $2$. This is the same “starting amount plus steady change” idea from the tables, written in one tidy formula.

Moving or flipping a shape changes each point by a simple rule:

• Translation (slide) by $a$ right and $b$ up: $(x,y)\rightarrow(x+a,\ y+b)$.
• Reflection over the $x$-axis: $(x,y)\rightarrow(x,\ -y)$.
• Reflection over the $y$-axis: $(x,y)\rightarrow(-x,\ y)$.
• Reflection over the origin: $(x,y)\rightarrow(-x,\ -y)$.

Apply the rule to every vertex, then reconnect the new points to get the image.

You can find the area of a triangle straight from its corner coordinates $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ — no height needed:

The absolute-value bars keep the area positive no matter the order of the vertices. This trick is called the shoelace formula because of the criss-cross pattern of multiplications.

A lattice point is a point whose coordinates are both integers, like $(3,-2)$ — exactly where grid lines cross. On a city grid you cannot cut diagonally through buildings, so the travel distance is the taxicab distance: add the horizontal and vertical blocks,

From $(1,2)$ to $(5,5)$ this is $|5-1|+|5-2|=4+3=7$ blocks, even though the straight-line distance is only $5$. The taxicab distance is never shorter than the straight-line distance.