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\( \left. \begin{array} { l } { y = | x - 6 | \quad y = x ^ { 2 } + 5 \quad y = - x ^ { 2 } + 5 \quad y = | x | + 6 \quad y = - ( x + 5 ) ^ { 2 } } \\ { y = - | x + 6 | \quad y = - | x | + 6 \quad y = - 6 | x | \quad y = ( x - 5 ) ^ { 2 } \quad y = - 5 x ^ { 2 } } \\ { y = 5 x ^ { 2 } \quad y = 6 | x | } \end{array} \right. \)

Ask by Li Greene. in the United States
Mar 17,2025

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Each function is described by its type (absolute value, parabola), vertex, domain, range, and general shape.

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Beyond the Answer

Did you know that absolute value functions and quadratics can be traced back to their mathematical origins? The absolute value function \(y=|x|\) arose from the need to express distances in algebraic terms, while quadratic equations, stemming from ancient civilizations like the Babylonians, demonstrate the beautiful intersection of geometry and algebra. Connecting these concepts to real-world situations can be super fun! For example, absolute value can model situations such as profit and loss—where you want to find the magnitude of gains or losses regardless of their direction. Quadratics often appear in physics, like when calculating the trajectory of a ball or optimizing areas in construction projects! Who knew math could be this useful?

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