12.5 The Coordinate Plane: Plotting Points, Quadrants, & Polygons on the Grid

The Coordinate Plane: Plotting Points, Quadrants, & Polygons on the Grid

Section 12.2 uses the distance and midpoint formulas. Before those come the Level M and Level D standards that build the plane itself: plotting ordered pairs, interpreting them in context, extending to negative coordinates in all four quadrants, and drawing polygons whose side lengths you can read off the grid.

The Ordered Pair

Two perpendicular number lines meet at the origin, $(0, 0)$. The horizontal line is the x-axis, the vertical line is the y-axis.

An ordered pair $(x, y)$ gives directions from the origin:

1. The first number is the horizontal move: right if positive, left if negative.
2. The second number is the vertical move: up if positive, down if negative.

Order matters absolutely. $(3, 5)$ means right 3, up 5. $(5, 3)$ means right 5, up 3. They are different points, and reversing them is the most common plotting error. The mnemonic that survives test pressure: "in the door, then up the stairs."

The numbers are called coordinates: $x$ is the x-coordinate or abscissa, $y$ is the y-coordinate or ordinate.

The Four Quadrants

The axes divide the plane into four quadrants, numbered counterclockwise starting from the upper right.

            y
            │
   II       │      I
 (−, +)     │    (+, +)
            │
 ───────────┼───────────  x
            │
   III      │      IV
 (−, −)     │    (+, −)
            │

Points on an axis belong to no quadrant. $(0, 5)$ sits on the y-axis; $(-3, 0)$ sits on the x-axis; $(0, 0)$ is the origin.

Reflections and signs. Reflecting a point across an axis changes exactly one sign:

• Across the x-axis: $(x, y) \rightarrow (x, -y)$ — so $(4, 7)$ becomes $(4, -7)$.
• Across the y-axis: $(x, y) \rightarrow (-x, y)$ — so $(4, 7)$ becomes $(-4, 7)$.
• Across both: $(x, y) \rightarrow (-x, -y)$.

Two points differing only in the sign of one coordinate are reflections of each other across the axis named by the unchanged coordinate.

Coordinates That Are Not Whole Numbers

Coordinates may be fractions or decimals: $\left(-2\frac{1}{2}, 3.75\right)$ is a perfectly ordinary point. Locate it by counting whole units first, then estimating the fractional part between gridlines.

Distance Along a Gridline

When two points share a coordinate, the distance between them is the absolute difference of the coordinates that differ — no square roots required.

Crossing an axis is where errors happen. From $-6$ to $5$ the distance is 11, not 1. Subtracting a negative adds. Sketching a quick number line prevents this every time.

For points sharing neither coordinate, use the Pythagorean theorem — see Section 12.3.

Polygons on the Grid

Given the vertices, you can find side lengths, perimeter, and area without any formula beyond the ones you already have.

Problem. A rectangle has vertices at $A(-3, 2)$, $B(5, 2)$, $C(5, -4)$, and $D(-3, -4)$. Find its perimeter and area.

Problem. A triangle has vertices at $(1, 1)$, $(7, 1)$, and $(7, 9)$. Find its area.

The side from $(1, 1)$ to $(7, 1)$ is horizontal with length 6; the side from $(7, 1)$ to $(7, 9)$ is vertical with length 8. They meet at a right angle at $(7, 1)$, so they are the base and height: $A = \tfrac{1}{2}(6)(8) = \mathbf{24} \text{ square units}$

Finding a missing vertex. Three vertices of a rectangle are $(2, 1)$, $(2, 7)$, and $(9, 7)$. The fourth must share an x-coordinate with $(9, 7)$ and a y-coordinate with $(2, 1)$, giving $\mathbf{(9, 1)}$.

Coordinates in Context

Level M places the plane in real settings, usually in the first quadrant only:

• A city grid where $(4, 7)$ means 4 blocks east and 7 blocks north of a landmark. Walking distance from $(4, 7)$ to $(4, 12)$ is 5 blocks.
• A seating chart where the pair identifies row and seat.
• A graph of time and quantity, where the horizontal axis is always the independent variable — see Section 5.4.

Read the axis labels before interpreting any point. In context, $(4, 7)$ means "4 of whatever the x-axis measures, 7 of whatever the y-axis measures," and answering with the numbers alone is an incomplete response.