Features of a graph
Two people look at the same graph. What words let them point at the same places on it, and agree on exactly what happens there?
Where you are 10 inside
What this area is about
Every function graph has landmarks: where it meets the axes, where it rises or falls, its peaks and valleys, where it breaks, and where its ends are heading. This area gives each landmark a name and a precise statement, as inputs, intervals or outputs, so a description can be checked, not guessed.
How these fit together
Where it sits and moves. Intercepts and zeros are where the graph meets the axes. Increasing, decreasing, positive and negative intervals name the stretches where it rises, falls, or sits above or below the axis. Turning points, maximums and minimums locate peaks and valleys: the largest or smallest outputs nearby or overall.
Shape. Symmetry is a mirror fold along the y-axis or a half turn about the origin, and even and odd functions test for it from the rule alone. Continuous graphs, jumps and holes says where the pen must lift.
The far ends. End behavior says where each end goes; an asymptote is a line the graph nears as closely as you like.
Rates. Average rate of change is its own area: the slope \displaystyle \frac{f(b) - f(a)}{b - a} for a \ne b. Concavity asks whether that rate is itself increasing or decreasing.
A path through it
Intercepts come before increasing and decreasing intervals, which come before turning points and average rates; concavity follows the average rate. Symmetry comes before even and odd functions. Continuity, end behavior and asymptotes fit anywhere.
Next, graphing polynomial functions and degree and graph shape use them together.