Table of Values
A table of values is an organized chart of input (x) and output (y) pairs generated from an equation or rule, used to analyze relationships and create graphs.
Definition
A table of values lists pairs of numbers that satisfy a rule or equation: you choose $x$-values, plug them into the equation, and record the resulting $y$-values, turning an abstract equation into a list of specific points you can plot on a graph. Patterns in a table of values reveal properties of the underlying relationship: constant differences suggest a linear rule, constant second differences suggest quadratic, and constant ratios suggest exponential. Formally, a table of values is a finite sampling of the graph of a function $f: D \to \mathbb{R}$, listing pairs $(x_i, f(x_i))$ for chosen $x_i \in D$; numerical methods use such tables to approximate functions and integrals (for example, Riemann sums), and finite difference tables underpin interpolation via Newton's and Lagrange's methods.
Example
For $y = 2x + 1$: when $x = 0$, $y = 1$; when $x = 1$, $y = 3$; when $x = 2$, $y = 5$. For $y = x^2 - 2$: $x = -2$ gives $y = 2$; $x = -1$ gives $y = -1$; $x = 0$ gives $y = -2$; the symmetry in the table reflects the parabola's axis of symmetry. Newton's forward difference formula uses a table of equally spaced values to construct an interpolating polynomial: if $f(0) = 1$, $f(1) = 4$, $f(2) = 9$, $f(3) = 16$, the differences reveal the underlying polynomial $f(x) = (x+1)^2$.
Key Insight
A table of values turns an abstract equation into a list of specific points you can plot on a graph. Finite difference tables are the discrete analog of derivatives: the $n$-th order finite difference corresponds to the $n$-th derivative in the continuous setting, connecting discrete numerical methods to calculus.