Zeros, signs, monotonicity and extrema from graphs

Zeros, signs, monotonicity and extrema from graphs

Interpret a function’s behavior directly from its graph. This precalculus set uses original coordinate diagrams with labelled unit scales to connect zeros, positive and negative outputs, increasing and decreasing intervals, constant segments and extreme values. No derivative rules are needed.

The questions distinguish an input where an extremum occurs from the extremum’s output value. They also examine flat intervals, several inputs attaining the same minimum, open endpoints and a local high point that is not an absolute maximum. A final reasoning task explains why matching outputs at two sampled inputs do not prove that a function is constant between them.

Prerequisites

Review coordinate reading, domains, ranges and piecewise graphs. Trace each graph from left to right. For sign questions compare height with the horizontal axis; for monotonicity compare how heights change as inputs increase.

Practice and feedback

Use exact graph coordinates where supplied. Interval answers describe horizontal inputs, and open or filled endpoints determine inclusion. Single-select and select-all items require checking each graphical claim separately. The explanations identify decisive points and intervals, including counterexamples to misleading shortcuts. When discussing local extrema here, look at nearby inputs on both sides of an interior point.