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 $1.3 .$ )
$f(x)=1+(x+1)^{2}, \quad-2 \leqslant x<5$

Key Concepts

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting the graph of a basic function to obtain a new function. In general, knowing how to apply horizontal and vertical shifts helps to quickly sketch transformed functions by modifying key features such as the vertex or intercepts.

Quadratic Functions and Vertex Form

Quadratic functions are often written in vertex form, a(x - h)² + k, because this form clearly identifies the vertex (h, k) of the parabola. The vertex represents either a maximum or a minimum of the function depending on whether the parabola opens downward or upward, making it a critical point when analyzing extrema.

Absolute and Local Extrema

The concept of extrema involves finding the highest and lowest values (maximums and minimums) of a function. Absolute extrema refer to the highest or lowest value over the entire domain of the function, while local extrema are the maximums or minimums within a smaller interval or neighborhood. Techniques for identifying these include analyzing critical points and evaluating the function at endpoints when the domain is restricted.

Domain Restrictions and Endpoint Analysis

When a function is defined on a restricted interval, the endpoints play a key role in determining its absolute extrema. A closed endpoint means the value is included in the domain, and thus must be considered when finding maximums or minimums. Conversely, open endpoints are not included, which requires careful consideration of the limiting behavior of the function near those points.

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