Coordinate Plane Grapher

The coordinate plane is built from two perpendicular number lines: the horizontal $x$-axis and the vertical $y$-axis, crossing at the origin $(0,0)$.

Every point is named by an ordered pair $(x, y)$, and the order matters absolutely: $x$ is the horizontal displacement, $y$ the vertical one. The point $(3, 5)$ is nowhere near $(5, 3)$.

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

A point with a zero coordinate sits on an axis, not in any quadrant: $(0, -6)$ is on the $y$-axis, $(7, 0)$ on the $x$-axis. Both axes must use a consistent, evenly spaced scale, though the $x$ and $y$ scales need not match each other.

Plotting a single point

1. Start at the origin.
2. Move horizontally by $x$ — right if positive, left if negative.
3. From there move vertically by $y$ — up if positive, down if negative.
4. Mark the point and label it.

The order is always across first, then up or down.

Graphing an equation by plotting points

Build a small table of values. Choose three or four convenient $x$ values, substitute each into the equation, and record the resulting $y$. Plot the pairs and join them.

Three points are the practical minimum: two define a line, and the third catches an arithmetic slip — if one point sits off the line, recheck that row.

Special lines

• $x = 4$ is a vertical line through every point with $x$-coordinate $4$. It has undefined slope and is not a function.
• $y = -2$ is a horizontal line, slope $0$.

Graphing a shape

Plot the vertices, then connect them in the order given. Connecting them out of order produces a self-crossing figure instead of the intended polygon. Once plotted, side lengths come from counting grid units for horizontal or vertical sides, and from the distance formula for slanted ones.

• Reversing the coordinates. Plotting $(2, 7)$ as "up 2, across 7" is the single most common error. The $x$-value always comes first.
• Counting from the wrong place. Every point is measured from the origin, not from the previously plotted point.
• Uneven or inconsistent scales. If one grid square is 1 unit near the origin it must be 1 unit everywhere on that axis, or the picture lies about slope and distance.
• Putting axis points in a quadrant. A zero in either coordinate means the point lies on an axis and belongs to no quadrant.
• Sign errors in quadrants II and III. Both have negative $x$; only III has negative $y$ as well.
• Joining scattered points with straight segments by default. Only connect points when the relationship is genuinely linear — otherwise plot more values first and look at the shape they trace.