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Sketching the graph of a function provides a visual representation of its behavior. When tackling the absolute value function like \(f(x) = |x|\), it's important to grasp how the function behaves over the specified interval. Here, the interval is from \([-1, 2]\). To begin sketching, identify some key points where the function's behavior changes:

• At \(x = -1\), the value of the function is \(|-1| = 1\). Therefore, the point \((-1, 1)\) is part of the graph.
• At \(x = 0\), the absolute value is zero so the graph passes through the origin \((0, 0)\).
• At \(x = 2\), the function returns 2, giving us the point \((2, 2)\).

Once these points are plotted, connect them with straight lines to form the characteristic V-shape of the absolute value graph. This V-shape inherently shows the symmetric property of the absolute value function across the y-axis. Remember to shade the region above the x-axis, as it represents the area beneath the graph of \(f(x) = |x|\) and above the x-axis, forming a clear triangular region.

Understanding piecewise functions is crucial when dealing with absolute value functions. The function \(f(x) = |x|\) can be rewritten using piecewise notation. This helps us break down the function into parts that are easier to interpret.For \(f(x) = |x|\):

• When \(x < 0\), \(f(x) = -x\). This indicates the graph reflects over the x-axis, resulting in a linear segment with a negative slope.
• When \(x \geq 0\), \(f(x) = x\). This part follows the typical linear growth with a positive slope.

By rewriting the function in this way, it becomes a piecewise function:\[ f(x) = \begin{cases} -x, & \text{if}\ x < 0 \ x, & \text{if}\ x \geq 0 \end{cases}\]Such piecewise definitions allow us to clearly see how the graph is constructed from different linear segments for various intervals of \(x\). This is key in accurately sketching the graph over given intervals.

Linear segments are straight portions of a graph and are essential to recognizing graph patterns, especially in piecewise functions like the absolute value function.In \(f(x) = |x|\), over \([-1, 2]\):

• From \([-1, 0)\), the line segment follows \(f(x) = -x\). This segment has a negative slope as it descends from \((-1, 1)\) to \((0, 0)\).
• From \([0, 2]\), the segment moves with a positive slope due to \(f(x) = x\). It goes upwards from \((0, 0)\) to \((2, 2)\).

The concept of linear segments is fundamental in breaking down complex functions into simpler parts. Each segment is linear, meaning it has a constant rate of change or slope. For the absolute value function, this breaks down into two segments around the vertex at the origin, illustrating its symmetry and consistent V-shape.Recognizing these segments helps us efficiently sketch the graph and understand how the overall function behaves across different intervals.