Reading a function's graph

A graph is a function's portrait: every input x paired with its output f(x), plotted as the point (x, f(x)). Once you can read one, you can describe a function at a glance.

The features worth naming

Four things tell you almost everything about a graph's behavior:

• y-intercept — where the curve crosses the y-axis. It's the output at x = 0, so it's just f(0). A function has at most one y-intercept.
• x-intercepts (roots) — where the curve crosses the x-axis. These are the inputs where f(x) = 0. A function can have many.
• Increasing / decreasing — reading left to right, the curve rises (going up) or falls (going down).
• Turning points — a local maximum is a peak; a local minimum is a valley. The curve changes direction there.

The vertical-line test

Not every curve is a function. The rule "exactly one output per input" has a visual fingerprint:

If any vertical line crosses the curve more than once, it is not a function.

Two crossings on one vertical line means one input x has two outputs — exactly what a function isn't allowed to do. A sideways parabola fails the test; a circle fails it. A standard upward parabola passes.

Why this matters

Reading a graph is faster than crunching a formula. "Where is profit highest?" is a local maximum. "When does the tank hit empty?" is an x-intercept. "Is it still growing?" is increasing vs. decreasing. The same four features answer questions in every field that uses math.