a) The equation of the absolute value graph can be expressed as:
\[ y = |x| \]
b) To determine the equation for the absolute value graph, I considered the characteristics of the absolute value function. The absolute value of a number is defined as its distance from zero on the number line, regardless of direction. Therefore, the function \( y = |x| \) takes any input \( x \) and transforms it to a non-negative output.
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Basic Understanding: The graph of \( y = |x| \) is V-shaped with the vertex at the origin (0,0). For \( x \geq 0 \), \( |x| = x \), and for \( x < 0 \), \( |x| = -x \). This reflection across the y-axis is a key feature of absolute value functions.
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Graph Characteristics: The graph will rise linearly in both the first and second quadrants, creating the V-shape, and intersects the y-axis at (0,0).
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More General Form: If there are transformations involved (like shifts or stretches), the general form of an absolute value equation can be represented as: \[ y = a |x - h| + k \] where \( (h, k) \) is the vertex, \( a \) indicates vertical stretch or compression, and the sign of \( a \) determines if the V opens upwards (positive) or downwards (negative).
Overall, the fundamental equation for an absolute value function is \( y = |x| \), which acts as the baseline from which variations can be constructed.