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 $1.3 .$ )
$$f(x)=\frac{1}{2}(3 x-1), \quad x \leq 3$$

Key Concepts

Graphing Linear Functions

Linear functions have the form f(x) = mx + b, where m is the slope and b is the y-intercept. Graphing such functions involves plotting a straight line with a constant rate of change. This concept is key because it allows one to understand the basic behavior of the function and serves as a foundation for analyzing more complex functions.

Function Transformations

Function transformations involve scaling, translating, or reflecting a basic function to obtain a new function. In the given function, the multiplication by 1/2 and the rearrangement of terms indicate vertical stretching/compression and horizontal shifts, which are essential skills in sketching graphs quickly by hand.

Domain Restrictions

Domain restrictions limit the set of input values for which the function is defined. In problems like this, where x is constrained to x ? 3, it is important to understand how the behavior of the function is affected at the boundaries of the domain and to identify any consequences on the existence of extrema.

Absolute and Local Extrema

The concepts of absolute and local extrema refer to the maximum or minimum values a function attains either over its entire domain or in a neighborhood around a point. When dealing with a function that has a restricted domain, one must examine the endpoints and the overall behavior of the function (increasing or decreasing) to determine these extreme values.

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