Plotting Points
Plotting points means locating and marking ordered pairs on the coordinate plane using the x- and y-coordinates.
Definition
Plotting a point means finding its location on the coordinate plane using its ordered pair $(x, y)$: start at the origin, move left or right by the x-coordinate, then up or down by the y-coordinate, and mark the spot. Multiple points can be plotted to visualize patterns, lines, curves, or geometric shapes, since plotting is the bridge between algebra (equations) and geometry (shapes); in analysis, the graph of a function $f: \mathbb{R} \to \mathbb{R}$ is formally the set $\{(x, f(x)) : x \in \text{dom}(f)\}$ as a subset of $\mathbb{R}^2$. In computational settings, plotting is achieved by evaluating a function at a finite set of sample points and rendering them, with the density of sample points affecting perceived continuity.
Example
To plot $(4, -3)$: start at $(0,0)$, move $4$ spaces right, then $3$ spaces down, and mark the point. Plotting $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$, $(-1,1)$, $(-2,4)$ reveals that these points all satisfy $y = x^2$, together showing the parabolic shape of the function. Lissajous figures are parametric curves $(A\sin(at + \delta), B\sin(bt))$ traced by plotting many ordered pairs as $t$ varies; the ratio $a:b$ determines the shape, and irrational ratios produce dense space-filling curves.
Key Insight
Think of the x-coordinate as how far along the street you walk (east-west) and the y-coordinate as how many floors up or down you go; every point has its own unique address. The connection between discrete point sets and continuous curves is studied in approximation theory and fractal geometry: Cantor sets and fractal curves have graphs that are neither finite point sets nor smooth curves, revealing the richness of subsets of $\mathbb{R}^2$.