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**Order of function transformations**

Horizontal shifts Horizontal stretch/compression Reflection over y-axis Vertical stretch/compression Reflection over x-axis Vertical shifts Important note: Horizontal and vertical dilations are applied only to the portion of the function which is shifted horizontally. You NEVER multiply the dilation factors with vertical shifts.

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**Order of function transformations**

Example: f(x) = x² Shift 2 units left, 4 units up, horizontal stretch by a factor of 6, vertical compression by a factor of ½ and reflect over x-axis. Start with Horizontal shift: (x + 2)² Horizontal stretch: ((1/6)(x + 2))² Vertical compression: ½((1/6)(x + 2))² Reflection over x-axis: - ½((1/6)(x + 2))² Finally, Vertical shift: - ½((1/6)(x + 2))² + 4

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**Order of function transformations**

Example: g(x) = x² + 3 Shift 3 units right, reflect across y-axis, stretch vertically by a factor of 2, 2 units up. Start with horizontal shift: (x – 3)² + 3 Reflect across y-axis: (- x – 3)² + 3 Stretch vertically: 2(-x – 3)² + 3 Finally, vertical shift: 2(-x – 3)² +3+2=2(-x -3)² +5

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**Order of function transformations**

Now your turn: Given: x + 2 Shift 5 units to the right, 3 up, reflect across y-axis, reflect across x-axis, vertical stretch by a factor of 2.

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