Graphs of Functions

A rigorous guide to reading, sketching, and verifying function graphs through features and transformations.

A function connects an allowed input set to a resulting output set. A curve appears with a highlighted horizontal domain span and vertical range span. The two labelled spans distinguish allowed inputs from resulting outputs.

Precise definition

The graph of $f$ is the set $\{(x,f(x)):x\in\operatorname{dom}f\}$. Key features include intercepts, domain, range, intervals of increase/decrease, extrema, symmetry, continuity, asymptotes, concavity, and end behaviour. A graph is a representation of a rule, not the function's full definition by itself.

Notation and mathematical language

Transformations of $y=f(x)$ include $f(x-h)+k$ (right $h$, up $k$), $af(x)$ (vertical scaling/reflection), and $f(bx)$ (horizontal scale by $1/|b|$ and reflection if $b<0$). Inside transformations act inversely on input coordinates.

Conceptual picture

Parent-function features move predictably, while algebra identifies exact intercepts and restrictions. Tables sample points but do not prove behaviour between them. A graphing window can conceal asymptotes or create apparent intersections.

Fully worked example

Interpretation and application

Graphs model costs, trajectories, populations, and data trends. An exact formula defines a graph; a fitted graph summarizes data approximately. Neither graphical association nor visual overlap alone proves causation.