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Translating right uses $x-5$ because horizontal shifts modify the input variable inside the function argument. To shift the graph right by $h$ units, you must replace $x$ with $x-h$ so that the function evaluates the original input at a point $h$ units to the left, effectively moving the graph right.

Conditions: The transformation is a horizontal translation of the graph of $y=f(x)$.; The shift amount is a positive constant.

Translating right uses $x-5$ because horizontal shifts modify the input variable inside the function argument. To shift the graph right by $h$ units, you must replace $x$ with $x-h$ so that the function evaluates the original input at a point $h$ units to the left, effectively moving the graph right.

Conditions: The transformation is a horizontal translation of the graph of $y=f(x)$.; The shift amount is a positive constant.

To translate the graph of $y=f(x)$ four units left and three units down, replace $x$ with $x+4$ inside the function argument and subtract 3 outside the function. The final transformed equation is $y=f(x+4)-3$.

Conditions: The starting graph has equation $y=f(x)$.; The intended transformation is a horizontal translation 4 units left and a vertical translation 3 units down.

To translate the graph of $y=f(x)$ four units left and three units down, replace $x$ with $x+4$ inside the function argument and subtract 3 outside the function. The final transformed equation is $y=f(x+4)-3$.

Conditions: The starting graph has equation $y=f(x)$.; The intended transformation is a horizontal translation 4 units left and a vertical translation 3 units down.