Interpreting Function Graphs

A graph carries a lot of information if you know what to look for.

• x-intercepts: where y = 0. The zeros of the function.
• y-intercept: where x = 0. The starting value.
• Increasing: the graph rises left to right. Decreasing: it falls.
• Maximum or minimum: the highest or lowest turning point.

In a real situation these have meanings. On a graph of height against time for a thrown ball, the y-intercept is the release height, the maximum is the peak, and the positive x-intercept is when it lands.

Read intervals along the x-axis, not the y-axis: a function is increasing *on an interval of inputs*.

Key features are read on the axes they concern. Zeros are inputs where the output is 0 — the solutions of f(x) = 0 — and are stated as x-values. Intervals of increase and decrease are also stated in terms of x, while maxima and minima have both a location (x) and a value (y); a question asking "what is the maximum" wants the y-value, while "where does it occur" wants the x.

Extrema may be local or global. A local maximum is highest in a neighbourhood; a global maximum is highest overall. A quadratic has one extremum which is global; higher-degree polynomials can have several local ones.

Additional features that appear later include end behaviour (what happens as x → ±∞), asymptotes for rational and exponential functions, and symmetry: even functions satisfy f(−x) = f(x) and are symmetric about the y-axis, odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

The interpretation habit is the point. Modelling questions are graded on the meaning attached to a feature, not on locating it.