Converting Between Table, Graph, Equation and Words

A function is just a rule pairing each input with one output. That rule can be displayed as a table (rows of x and y), a graph (a curve or line in the plane), an equation (y in terms of x), or words (a real-world description like a pricing plan). None of these is more "true" than the others — they're just different costumes on the same rule. Being fluent means you can start in any one and produce the other three.

To go from a table to an equation, check whether y changes by a constant amount each time x increases by 1 — that constant is the slope, and plugging one point in gets you the intercept. Words convert the same way: 'charges 30 dollars plus 8 dollars an hour' means a flat 30 (the y-intercept) plus 8 times the number of hours (the slope), so cost = 30 + 8h. To sketch that, start at 30 on the vertical axis and rise 8 for every 1 you move right.

Here's the trap: a table only tells you the function's behavior at the specific points listed. Four points that look linear could also sit on a curve that happens to pass through exactly those four spots and bends somewhere else. A table never proves linearity — it just fails to disprove it. This matters right alongside domain, range, and asymptotes, because once you commit to an equation from a table, you're also committing to a shape everywhere else, including places the table never showed you.