The Coordinate Plane & Slope

Giving numbers a place to live

“Meet me at 5th Avenue and 3rd Street.” Two numbers, one exact corner — a city grid turns any location into a pair of numbers, and that trick is the whole coordinate plane. Take the number line you’ve used since F1, lay a second one across it vertically, and let them cross at zero: the horizontal x-axis, the vertical y-axis, meeting at the origin $(0, 0)$. Now every point on the page has an address, an ordered pair $(x, y)$: x first (how far over, left or right), then y (how far up or down). Say it as “over, then up.”

The order is pure convention — nothing about $3$ makes it more “horizontal” than $5$ — and that’s exactly why $(3, 5)$ and $(5, 3)$ are so easy to swap: the two numbers carry no visible labels. The convention is alphabetical ($x$ before $y$) and universal, so drill the habit now, while it’s cheap. The axes also slice the plane into four quadrants, numbered counter-clockwise from the top right.

Plot $(3, 5)$, then $(5, 3)$ — genuinely different places. Then try $(0, -4)$: a point sitting on an axis belongs to no quadrant at all.

Slope: steepness as a number

Building codes say a wheelchair ramp may rise at most $1$ inch for every $12$ inches it runs forward. That ”$1$ per $12$” is a rate — the unit-rate thinking from F6 — and drawing it on the grid gives the rate a shape: a line whose steepness is the number. Slope (written $m$) is exactly that:

$m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$

Concretely: a staircase line passes through $(2, 1)$ and $(6, 9)$. Going from the first point to the second, you travel $9 - 1 = 8$ upward (the rise) while covering $6 - 2 = 4$ across (the run): the slope is $\frac{8}{4} = 2$ — two units of climb per unit forward, everywhere on the line.

One discipline keeps the formula honest: subtract in the same order top and bottom. Rise and run are signed journeys — up or down, forward or backward — made between the same two points. Mix the order ($\frac{y_2 - y_1}{x_1 - x_2}$) and you’ve measured the climb on one trip but the distance on the return trip: for our staircase that computes $\frac{8}{-4} = -2$, calling an uphill line downhill.

The sage leg is the run (across); the terracotta leg is the rise. Drag $B$ upward and predict the slope before the readout settles; then drag $B$ below $A$ and watch the sign go negative while the triangle flips.

The four kinds of slope

• Positive ↗ — uphill left to right (rise and run share a sign).
• Negative ↘ — downhill (rise and run have opposite signs).
• Zero — — a flat, horizontal line: the rise is $0$, so $m = 0$.
• Undefined | — a vertical line: the run is $0$, and dividing by zero has no answer.

The classic mix-up is the last two, and everyday language is the culprit: a flat road and a vertical wall can both be described as having “no slope.” But they’re opposites. A flat road has a perfectly good steepness — zero, a real number you could walk all day. A vertical wall breaks the question itself: with no run at all, “rise per run” divides by zero, and the slope is undefined. Flat is a zero; vertical is a shrug. Tap the preset chips in the tool above (positive ↗, negative ↘, zero —, undefined |) and check each triangle.

Intercepts — where the line meets the axes

Two points on any line matter more than the rest: where it crosses the y-axis (the y-intercept, a point $(0, b)$) and where it crosses the x-axis (the x-intercept, $(a, 0)$). The address system tells you how to find them: every point on the y-axis has $x = 0$, so set the other variable to zero. And intercepts mean things. If a line charts your phone bill against data used — climbing $$3$ per gigabyte — its y-intercept is the bill at zero usage: the $$20$ fixed fee, sitting on the axis before the line even starts to climb. The Slope Calculator reads both intercepts off any two points for you.

The one thing to remember

A point is an address — over, then up. A line’s slope is a rate with a shape: rise over run, subtracted in the same order, positive uphill, negative downhill, zero flat, undefined vertical. And the intercepts are where the line tells its story to the axes: set the other variable to zero to hear it.

The coordinate plane

Two number lines cross at the origin $(0, 0)$: the horizontal x-axis and the vertical y-axis. Any point is named by an ordered pair $(x, y)$ — x first (left/right), y second (up/down). “Over, then up.”

The axes split the plane into four quadrants, numbered counter-clockwise from the top-right:

Slope — the steepness of a line

Slope $m$ is the rise over the run: how much the line goes up (rise, the change in $y$) for how much it goes across (run, the change in $x$).

$m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$

Label your two points $(x_1, y_1)$ and $(x_2, y_2)$, then subtract in the same order top and bottom.

Worked example — through $(1, 2)$ and $(4, 8)$

The four kinds of slope

Intercepts

Where a line crosses an axis:

• y-intercept — where it crosses the y-axis. Here $x = 0$. Written as the point $(0, b)$.
• x-intercept — where it crosses the x-axis. Here $y = 0$. Written as the point $(a, 0)$.

To find an intercept, set the other variable to $0$. These show up constantly in the next units on lines and functions.