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Alterations of function graphics are usually conducted through geometric methods. When a function \(y = f(x)\) is given, to shift the graph horizontally in the direction of stretching, \(y = f(x - a)\) expression is used, which shifts the graph to the right by \(a\) unit. To shift the graph vertically, \(y = f(x) + b\) is utilized, raising the graph upward by \(b\) unit. Multiplication or division, for example \(y = k \cdot f(x)\), expands or compresses the graph vertically

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