Question

Write the equation of G(x)

Answer 1

The equation is

`G(x)=-(1)/(2)(x-3)^3 +2`

Explanation:

If the graph of the function
`G(x)=cf(x+h) +b` represents the transformations made to the graph of
`y= f(x)` then, by definition:

If
`0 <|c| <1` then the graph is compressed vertically by a factor c.

If
`|c| > 1` then the graph is stretched vertically by a factor c

If
`c <0` then the graph is reflected on the x axis.

If
`b> 0` the graph moves vertically upwards.

If
`b <0` the graph moves vertically down

If
`h <0` the graph moves horizontally h units to the right

If
`h >0` the graph moves horizontally h units to the left

In this problem we have the function
`G(x)` and our parent function is
`f(x) = x^3`

We know that G(x) is equal to f(x) but reflected on the x-axis (
`c <0`), compressed vertically by a multiple of 1/2 (
`0 <|c| <1` and
`c = -(1)/(2)`), displaced 2 units upwards (
`b = 2>0`) and moved to the right 3 units (
`h = -3<0`)

Then:

`G(x)=-(1)/(2)f(x-3) +2`

`G(x)=-(1)/(2)(x-3)^3 +2`

Answer 2

`G(x)=(1)/(2)(x-3)^3+2`

Explanation:

The given function is

`F(x)=x^(3)`

The transformations to this graph are in the form;

`G(x)=a(x-b)^3+c`

where
`a=(1)/(2)` is the vertical compression by a factor of
`(1)/(2)`

b=3 is a shift to the right by 3 units.

c=2 is an upward shift by 2 units.

Therefore
`G(x)=(1)/(2)(x-3)^3+2`