Question

What is the effect on the graph of the function f(x) = x2 when f(x) is changed to f(x − 6)?

Answer 1

Horizontal translation in the
`+x` direction.

Explanation:

A function of the form
`f(x)` experiments an horizontal translation when the following substitution is applied:

`x \to x + a`, where
`a \in \mathbb{R}`

If
`a > 0`, the function is translated in the
`-x` direction, whereas
`a < 0` is the case for the function being translated in the
`+x` direction.

The effect on the graph can be defined by a composition between two function:

`f(x) = x^(2)`,
`g (x) = x-6`

`f\,\circ\,g\,(x) = (x-6)^(2)`

The resulting expression represents a horizontal translation in the
`+x` direction.

Finally, we plot
`f(x)` (red) and
`f\,\circ g\,(x)` (blue) by a graphing tool and proved the certainty of this theory.