Sketch the graph of f by hand and use your sketch to find...

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 . )$ $f(x)=\left\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leq x<0} \\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 . )$
$f(x)=\left\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leq x<0} \\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$

Key Concepts

Critical Points and Endpoints

Critical points are points in the domain where the derivative of the function is zero or undefined; these are potential locations for local maxima or minima. Endpoints also must be examined, especially in closed or semi-closed domains, because absolute extrema can occur there even if the function behaves regularly in the interior.

Absolute and Local Extrema

Absolute extrema refer to the highest or lowest values a function attains over its entire domain, while local extrema are the highest or lowest values within a small neighborhood around a point. Finding these extrema often requires analyzing the behavior of the function at critical points as well as at the endpoints of its domain.

Piecewise Functions

Piecewise functions are defined by different formulas on different parts of their domain. Such functions require special care when sketching their graphs, as each segment may behave differently and the transitions between pieces—whether continuous or with jumps—must be clearly understood and represented.

Graph Sketching

Graph sketching involves drawing the function by hand as accurately as possible based on its mathematical definition and known transformations. This process includes plotting key points, intercepts, and understanding the behavior near boundaries, especially for piecewise functions.

Function Transformations

Function transformations involve shifting, scaling, reflecting, or otherwise modifying a basic function to obtain a new graph. Recognizing these transformations can make sketching the graph easier and more systematic, as they allow one to relate the current function to a known parent function.

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