# How to translate a function of form y = f(x)

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In a translation the graph of the function is moved in either the x or y direction.

To perform a translation by amount a in the positive x direction (to the right on a graph) we replace every x in the equation with (x-a), so the function y = f(x) becomes y = f(x-a). A translation of a in the negative x direction can be thought of as a translation of -a in the positive direction, so this time y = f(x) becomes y = f(x-(-a)) = f(x+a).

For translations in the y direction a similar rule applies but this time we substitute y. A translation of in the positive y direction (upwards) will change y = f(x) into (y-b) = f(x), which is often written as y = f(x) + b. Similarly a translation by b in the negative y direction would give (y+b) = f(x) which may be written y = f(x) - b.

Example: Translate the function y = x^2 + 3x + 5 by vector (4,-1).

Solution: The vector given means that we want to translate the function by 4 units in the positive x direction and 1 units in the negative y direction. This transforms the original equation into

(y+1) = (x-4)^2 + 3(x-4) + 5
which expands to
y +1 = x^2 - 8x + 16 + 3x - 12 + 5
and simplifies to
y = x^2 - 5x + 8

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