A visual guide · High school algebra
The coordinate plane, from first point to full function.
Every graph you'll ever draw starts with two numbers. Plane & Point walks you through the coordinate plane, plotting practice, function families, a full graphing calculator, and a game of Coordinate Battleship — all in one place.
Before the plane: the number line
Every graph starts here. One line, one number per position. Get comfortable with the number line and the coordinate plane feels obvious — it's just two of them, stacked at a right angle.
One dimension, one number
A number line is a straight line where every point corresponds to exactly one number. Zero sits in the middle. Positive numbers go right, negative numbers go left. The spacing between marks is constant — that's what makes it a number line and not just a row of tick marks.
Reading a position
Slide the marker below. Its value is its distance from zero, positive if it's to the right and negative if it's to the left. Distance is always positive — that's the number's absolute value, written |x|.
A number line for time
Rotate the idea a little and you get a timeline. Instead of “distance from zero,” each position is “distance from a reference year.” Everything else works exactly the same: earlier dates are to the left, later dates are to the right, and the space between two events is the number of years between them.
BCE, CE, and the missing year zero
The one quirk: the traditional Gregorian calendar has no year 0. The year 1 BCE is followed directly by 1 CE. Historians usually treat the boundary as the number zero for arithmetic — an event in 500 BCE was 2,525 years before 2025.
The coordinate plane
Two number lines crossed at a right angle. That's it — that's the whole idea. But those two lines let you name every location on a flat surface with just two numbers.
Axes and the origin
The horizontal line is the x-axis. The vertical line is the y-axis. They cross at the origin, written (0, 0). Every point is described by an ordered pair (x, y) — how far right or left, then how far up or down.
Quadrants
The two axes divide the plane into four quadrants, numbered counterclockwise from the top-right:
- I x positive, y positive
- II x negative, y positive
- III x negative, y negative
- IV x positive, y negative
Plotting points
Reading a coordinate is one thing. Plotting it is another. Given a target ordered pair, click where you think it goes. You'll know instantly if you're right.
How to plot a point
To plot the ordered pair (x, y):
- Start at the origin,
(0, 0). - Move right if x is positive, left if x is negative.
- Then move up if y is positive, down if y is negative.
- Mark the spot. That's your point.
Watch the sign
The sign of a coordinate tells you which quadrant it's in. A negative x always means left of the y-axis. A negative y always means below the x-axis. If either coordinate is zero, the point sits on an axis, not inside any quadrant.
(3, -2) is a completely different point from (-2, 3).
Lines & slope
A line is what you get when a point moves at a steady rate. Slope is that rate: how much y changes for every unit x changes.
Slope-intercept form
y = m·x + b
m is the slope — the tilt of the line. Positive slopes climb to the right; negative slopes fall. A slope of 0 is flat.
b is the y-intercept — the y-value where the line crosses the y-axis. Drag the sliders to see how each one moves the line independently.
Rise over run
Pick any two points on a line. Slope is (y₂ − y₁) / (x₂ − x₁). The rise is how far you climb; the run is how far you go across. A slope of 2 means: every 1 step right, 2 steps up.
Functions & curves
Once you leave lines behind, the plane fills with curves. Each family — parabolas, cubics, exponentials, trig, absolute value — has its own shape and its own knobs to turn.
Function families
A function is a rule that takes an input x and returns exactly one output y. Change the rule and you change the shape of the curve. Pick a family below to explore:
y = a(x − h)² + k
A parabola. a controls the width and direction (positive opens up, negative opens down). (h, k) is the vertex — the lowest or highest point on the curve, marked with a star. Parabolas are symmetric across the vertical line through the vertex.
Systems & inequalities
A single equation draws a curve. A system of two draws several — and the answer lives wherever they meet. Swap the equals sign for an inequality and instead of a line you shade a whole region.
Where lines meet
A system of two linear equations has three possible outcomes:
- One solution — the lines cross at exactly one point.
- No solutions — the lines are parallel and never meet.
- Infinite solutions — the lines are the same line, overlapping everywhere.
Inequalities shade regions
Replace = with <, ≤, >, or ≥, and instead of a line you get a half-plane — the whole region on one side of the line. When two inequalities are combined, the solution is where their shaded regions overlap.
The graphing calculator
Type any function of x. The calculator turns your equation into a picture. Add as many as you want — each gets its own color. Drag to pan, scroll to zoom, hover to inspect exact values.
Open the full-screen calculator → — with parameter sliders, tables of values, zeros, extrema, and intersections.
Coordinate Battleship
The best way to remember ordered pairs is to hunt with them. Each fleet has five single-cell ships hidden on the grid. Type coordinates like 3, -2 to fire at the enemy grid on the right. Sink all five before the enemy sinks yours.
Your fleet
Enemy waters
More on the way
Plane & Point is going to grow. Here's what's planned for the next round.
Printable worksheets
Auto-generated PDFs for plotting points, identifying quadrants, graphing lines, reflections, and translations.
Adaptive practice tests
10-question, 20-question, and SAT-style quizzes that adjust difficulty as you go.
Calculator tutorials
Step-by-step lessons on graphing quadratics, circles, inequalities, systems, piecewise functions, and trig.
More games
Alien Coordinate Hunt, Treasure Map, Race to the Origin, and Plot the Hidden Picture.
Mini calculators
Standalone tools for slope, midpoint, distance, x/y-intercept, domain & range, and quadrant lookup.
Reference library
Parent functions, transformations, and common curve gallery — a searchable visual index.
You now speak the language of graphs.
The coordinate plane is a shared canvas that every part of math — geometry, algebra, calculus, statistics — writes on. Every equation is a shape. Every intersection is an answer. Come back to the calculator any time.
Open the calculator