describe how the graph of `g(x)=root(3)(x-6)` can be obtained from the graph of `root(3)(x)`  then graph the function g(x).

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Given a base function `f(x)` , there are a number of transformations that can be applied to get a new function. If `g(x)=Af(B(x-h))+k` then g(x) is a transformation of f(x). The transformations include:

A: A creates a vertical stretch/compression of size A. Also, if A<0 the function is reflected over the x-axis.

B: B creates a horizontal stretch/compression. Usually this is most noticeable in periodic functions such as the trigonometric functions. In typical algebraic functions, horizontal stretch/compressions can be seen as vertical stretch/compressions. If B<0 the function is reflected over the y-axis.

h: h is a horizontal translation.

k: k is a vertical translation.

There are other possible transformations, but these are the basics.

For `f(x)=root(3)(x)` and `g(x)=root(3)(x-6)` we can see that g(x) is a transformation of f(x), and that the transformation is of the "h" type. `g(x)=root(3)(x-h)` where h=6: this is a horizontal translation -- g(x) looks like f(x) shifted 6 units to the right.

The graph of f(x) in red; the graph of g(x) in green:

Sources:

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