Question

Describe each graph in relation to its basic function.

a. Compare f(x) = (x - 2)^3 - 1 to the basic function g(x) = x^3
Explain in full sentences please!

Answer 1

In a nutshell,
`f(x)` is obtained by applying an horizontal translation on
`g(x)`, followed by a vertical translation.

Explanation:

Let be
`g(x) = x^(3)` the parent function,
`f(x)` is obtained by applying an horizontal translation on
`g(x)`, followed by a vertical translation, that is:

`f(x) = g(x-a)-b,\,a,b\in\mathbb{R}` (1)

If
`a > 0`, then translation is rightwards.

If
`b > 0`, then translation in downwars.

If
`a = 2` and
`b = 1`, then the resulting function is:

`f(x) = g(x-2)-1`

`f(x) = (x-2)^(3)-1` (2)

In a nutshell,
`f(x)` is obtained by applying an horizontal translation on
`g(x)`, followed by a vertical translation.