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When you're faced with an equation involving absolute values, like in our exercise, the key is to understand that the absolute value of a number is the distance between that number and zero on a number line, without considering direction. In other words, it's the non-negative value of that number. For example, both \(3\) and \( -3\) have an absolute value of \(3\).

To solve absolute value equations, you generally need to split the equation into two cases: one where the value inside the absolute value is positive, and one where it's negative. This is because absolute value will generate the same outcome for \(x\) and \( -x\). However, in our given exercise, the equation \( y = 4 - |x - 2|\) does not require splitting because we are checking if given points satisfy the equation rather than solving for \(x\). By substituting the \(x\) and \(y\) values from the points into the equation and seeing if the equation holds true, you can determine if the points lie on the graph.

Graphing absolute value functions involves a bit more visualization. The absolute value function typically has a 'V' shape, where the point of the 'V' corresponds to the vertex of the function. For the given function \( y = 4 - |x - 2|\), the graph is a reflection of the basic \( V\) shape and is shifted according to the equation components.

The vertex of this graph would be at the point where the expression inside the absolute value, \(x - 2\), equals zero. So here, that's when \(x = 2\). From there, the graph opens downwards because of the negative sign before the absolute value, and the function shifts up 4 units due to the constant \(4\). Recognizing these transformations is crucial as they determine the function's shape and position on the coordinate plane.

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points, lines, and shapes using coordinates on a plane. The most commonly used coordinate plane has two axes: the \(x\)-axis which runs horizontally, and the \(y\)-axis which runs vertically. Together, they divide the plane into four quadrants.

In our textbook problem, each point given as \( (x, y)\) represents a location on this plane. To check if these points are part of the graph of an equation, like \( y = 4 - |x - 2|\), we substitute them into the equation and verify the validity of the equality. This process illustrates how coordinate geometry helps us understand the relationship between algebraic expressions and their geometric representations.

Precalculus is a course that prepares students for the rigors of calculus. It involves the study of functions, including linear, polynomial, rational, exponential, logarithmic, and, as in our exercise, absolute value functions. An important skill in precalculus is translating among various representations of functions: from graphs to equations, from equations to numerical patterns, and so on.

This analytical ability is essential, and problems like the one we're examining reinforce comprehension of core precalculus concepts by making clear the connections between a function's algebraic description and its geometric representation.