The function f(x) = x^3 has been transformed, resulting in function h. h(x) = -(x+2)^3-4

To create function h, function f was translated 2 units (down, to the left, up to the right) , translated 4 units (to the right, down to the left, up) , and reflected across the (y-axis, x-axis) .

a: to the left

b: up

c: x-axis

Step-by-step explanation:

First, let's write how the transformations work in a general way.

Horizontal shift.

A horizontal shift of A units is written as:

g(x) = f(x + A)

if A is positive, the shift is to the left.

If A is negative, the shift is to the right.

Vertical shift:

A vertical shift of A units is written as:

g(x) = f(x) + A

If A is positive, the shift is up

If A is negative, the shift is down.

A reflection over the x-axis changes the sign of the y-component, then:

g(x) = -f(x)

a reflection over the y-axis, changes the sign of the x-component, then:

g(x) = f(-x).

Ok, now let's solve this:

f(x) = x^3

g(x) = -(x + 2)^3 - 4

Then:

First, f(x) was translated 2 units to the left, to get:

g(x) = (x + 2)^3

Then, it was translated 4 units up, to get:

g(x) = (x + 2)^3 + 4.

Then, we have a reflenction over the x-axis, that changes the sign in the left side:

g(x) = -( (x + 2)^3 + 4) = -(x + 2)^3 - 4

Then the correct options are:

a: to the left

b: up

c: x-axis

part a: 1.08

part b: 27.59