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 $1.3 .$ ) $$ f(x)=\left\{\begin{array}{ll} x^{2} & \text { if }-1 \leqslant x \leqslant 0 \\ 2-3 x & \text { if } 0<x \leqslant 1 \end{array}\right. $$

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 $1.3 .$ )
$$
f(x)=\left\{\begin{array}{ll}
x^{2} & \text { if }-1 \leqslant x \leqslant 0 \\
2-3 x & \text { if } 0<x \leqslant 1
\end{array}\right.
$$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. Understanding how to work with piecewise functions involves treating each segment separately, graphing each function on its respective subinterval, and then combining them to form the overall graph. It is important to check the behavior at the points where the pieces meet to ensure continuity or identify any discontinuities.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting basic graphs to obtain the graph of a more complex function. In this context, familiar graphs such as parabolas for quadratic functions and straight lines for linear functions are used along with their transformations to help accurately sketch each segment of the piecewise function.

Extreme Value Theorem

The Extreme Value Theorem states that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum. This concept is crucial when finding the highest and lowest values of a function over a specific domain, as it implies that evaluating the function at critical points and endpoints will reveal these extreme values.

Local and Absolute Extremes

Local extremes are points where the function reaches a minimum or maximum relative to nearby values, whereas absolute extremes are the highest or lowest values of the function over the entire interval. Identifying these involves analyzing the function’s graph, checking critical points where the first derivative is zero or does not exist, and evaluating the function at the boundary of the interval.

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