For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4.
y = |x - 2|

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Understand the given equation: \(y = |x - 2|\) represents the absolute value function shifted 2 units to the right along the x-axis. The absolute value function outputs the distance of the input from zero, so it is always non-negative.

To create a table of ordered pairs, choose at least three values of \(x\) around the point where the expression inside the absolute value is zero (i.e., \(x = 2\)). For example, select \(x = 1\), \(x = 2\), and \(x = 3\).

Calculate the corresponding \(y\) values by substituting each chosen \(x\) into the equation \(y = |x - 2|\). For each \(x\), compute \(y\) as the absolute value of \((x - 2)\).

Organize the results into a table of ordered pairs \((x, y)\), where each pair satisfies the equation. This table will help visualize the shape of the graph.

To graph the equation, plot the ordered pairs on the coordinate plane. Since \(y = |x - 2|\) forms a 'V' shape, draw two rays: one going upward to the left from the vertex at \((2, 0)\) and one going upward to the right from the same vertex.

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Absolute Value Function

The absolute value function outputs the non-negative magnitude of a number, defined as |x| = x if x ≥ 0, and -x if x < 0. In the equation y = |x - 2|, the expression inside the absolute value shifts the graph horizontally, affecting the vertex location.

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Creating Ordered Pairs from an Equation

To find ordered pairs (x, y) that satisfy the equation, select values for x, substitute them into the equation, and compute the corresponding y values. This process helps in plotting points and understanding the shape of the graph.

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Graphing Piecewise Functions

Absolute value functions can be expressed as piecewise linear functions, with different expressions depending on the sign of the input. Graphing involves plotting points and connecting them with lines that form a 'V' shape, reflecting the function's definition.

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