Sketch Translations of Functions

Sketch Translations of Functions


Graphs can be translated horizontally and vertically.

To translate the graph horizontally, add or subtract a value before the function is applied. The movement is in the opposite direction to the value of the addition or subtraction. If `y = f(x)` then `f(x + a)` will cause the graph to be translated by `((-a),(0))`.

To translate a graph vertically, add or subtract a value after the function is applied. That is, if `y=f(x)` then `y=f(x)+a` will result in a translation of `((0),(a))`.


1. Sketch the graph of `f(x)=x^2`. Then sketch the graph of `y=(x+3)^2 - 2.`

Answer: sketch of y = x squared and translated

The function `f(x)=x^2` passes through (0, 0).

The addition of +3 before the function is evaluated causes a translation of `((-3),(0))`. The subtraction of -2 after the function is evaluated results in translation of `((0),(-2))`. Adding the translations gives `((-3),(-2))`.

2. The function `y=f(x)` requires to be translated by `((0), (-2))`. What is the new function?

Answer: `y=f(x)-2`

To translate a function vertically, then apply the value after the the function has been applied.