Graphs of Functions

For $f:\mathbb{R}\to\mathbb{R}$, the graph of $f$ is the set of ordered pairs $\{(x,f(x)):x\text{ in the domain of }f\}$ plotted in the $xy$-plane. Not every curve in the plane is the graph of a function: a curve represents a function exactly when it passes the vertical line test — every vertical line meets the curve at most once (a single input $x$ cannot legitimately produce two different outputs). Since trigonometric functions describe naturally periodic phenomena (tides, day/night cycles, oscillations), the standard convention is to let $x$ measure an angle in radians and $y$ the function's value, writing e.g. $y=\sin x$ for the sine function. Studying a function through its graph — its rises, falls, turning points, and repeating pattern — is one of the m …