Table of Values
Pre-AlgebraA 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.
Example
For $y = 2x + 1$: when $x = 0$, $y = 1$; when $x = 1$, $y = 3$; when $x = 2$, $y = 5$. Record these as a table with an $x$ column and a $y$ column.
Key Insight
A table of values turns an abstract equation into a list of specific points you can plot on a graph.
Definition
A table of values is a systematic list of input-output pairs satisfying an equation. Choosing a range of x-values and evaluating y = f(x) for each produces points that can be graphed. The table reveals patterns in the relationship.
Example
For $y = x^2 - 2$: $x = -2$ gives $y = 2$; $x = -1$ gives $y = -1$; $x = 0$ gives $y = -2$; $x = 1$ gives $y = -1$; $x = 2$ gives $y = 2$. The symmetry in the table reflects the parabola's axis of symmetry.
Key Insight
Patterns in a table of values reveal properties of the function: constant differences suggest linear, constant second differences suggest quadratic, and constant ratios suggest exponential.
Definition
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 tables of values to approximate functions and integrals (e.g., Riemann sums, Newton's forward difference formula). Finite difference tables underpin interpolation via Newton's and Lagrange's methods.
Example
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
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.