Question

What is the transformations needed to obtain `h(x)= (-1/2x)^3 + 4` from the graph of `x^2`?

Answer 1

Reverse x-axis, apply the power transformation `x to x^(3/2)`, apply the scalar multiple `1/8` and shift the origin to (0, 4). In brief, the transformation is `(x, y) to (-1/8x^(3/2), y-4)`

Explanation:

`y = h(x)=-1/8x^3`.

graph{x^2 [-10, 10, -5, 5]}

graph[ ((y-(--x)^2)=0[-40, 40, -20, 20]}

graph{(-x)^3 [-40, 40, -20, 20]}

graph{1/8(-x)^3 [-40, 40, -20, 20]}

graph[(y-1/8(-x)^3-4)=0 [-40, 40, -20, 20]}

The sequence is as follows.

Reverse x-axis.

Apply the power transformation `x to x^(3/2)`.

Apply the scalar multiple `1/8`.

Shift the origin to (0, 4).

In brief, the transformation is

`(x, y) to (-1/8x^(3/2), y-4)`.

Note that the graphs 1st and 2nd in the sequence are identical, and

so, given as one only. Also, referred to the transformed axes, the

new equation is `y =1/8x^3` (, with the reversed x--axis, and the origin shifted to (0, 3) ).