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

Key Concepts

Graphical Analysis of Functions

This concept involves visually representing a function by sketching its graph. It helps in understanding the behavior of the function, identifying trends such as increasing or decreasing intervals, and spotting features like intercepts and symmetry. Using transformations of basic graph shapes is often employed to sketch more complex functions, making it a fundamental tool in understanding function behavior.

Absolute Extrema

Absolute extrema refer to the highest and lowest function values over a given interval. In problems involving a closed or partially closed interval, these extrema are determined by evaluating the function at critical points and at the endpoints of the interval. This concept is a key aspect of optimization and analysis in calculus.

Local Extrema

Local extrema are points in the function where it reaches a locally highest or lowest value relative to nearby points in the domain. They are typically found at critical points where the derivative is zero or undefined, and understanding them is central to analyzing the behavior of a function in a small neighborhood, even if these points are not the highest or lowest over the entire interval.

Domain Endpoints and Open/Closed Intervals

This concept emphasizes the importance of the type of interval (open or closed) in determining extrema. Endpoints of closed intervals are included in the function’s domain and must be evaluated to find absolute extrema, while points at the boundary of open intervals may create situations where an absolute extremum may not be attained if the endpoint is not included in the domain.

Quadratic Functions

Quadratic functions, often represented in the form f(x) = ax² + bx + c, create parabolic graphs that are symmetric about the vertex. Understanding the properties of quadratic functions, such as the direction of the opening and the location of the vertex, is crucial when sketching the graph and determining the function's behavior, including where maximum or minimum values occur.

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