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THQuadratic transformations describe how the graph of a quadratic function changes when we apply operations such as shifting, reflecting, stretching, or compressing. Starting from the basic function f(x) = x^2, vertical shifts occur when a constant is added or subtracted, moving the parabola up or down, while horizontal shifts occur when the input is adjusted, moving it left or right. Reflections are produced by multiplying the function by -1, flipping the parabola across the x-axis. Vertical stretches and compressions depend on the coefficient a: values of |a| > 1 make the parabola narrower, and 0 < |a| < 1 make it wider. All of these operations can be expressed in the vertex form f(x) = a(x - h)^2 + k, where (h, k) represents the vertex, a determines the stretch and reflection, and h and k determine the horizontal and vertical shifts respectively. Quadratic transformations are essential in understanding how the algebraic form of a function corresponds to changes in its graph.
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