Answer:
The equation is
[tex]G(x)=-\frac{1}{2}(x-3)^3 +2[/tex]
Step-by-step explanation:
If the graph of the function [tex]G(x)=cf(x+h) +b[/tex] represents the transformations made to the graph of [tex]y= f(x)[/tex] then, by definition:
If [tex]0 <|c| <1[/tex] then the graph is compressed vertically by a factor c.
If [tex]|c| > 1[/tex] then the graph is stretched vertically by a factor c
If [tex]c <0[/tex] then the graph is reflected on the x axis.
If [tex]b> 0[/tex] the graph moves vertically upwards.
If [tex]b <0[/tex] the graph moves vertically down
If [tex]h <0[/tex] the graph moves horizontally h units to the right
If [tex]h >0[/tex] the graph moves horizontally h units to the left
In this problem we have the function [tex]G(x)[/tex] and our parent function is [tex]f(x) = x^3[/tex]
We know that G(x) is equal to f(x) but reflected on the x-axis ([tex]c <0[/tex]), compressed vertically by a multiple of 1/2 ([tex]0 <|c| <1[/tex] and [tex]c = -\frac{1}{2}[/tex]), displaced 2 units upwards ([tex]b = 2>0[/tex]) and moved to the right 3 units ([tex]h = -3<0[/tex])
Then:
[tex]G(x)=-\frac{1}{2}f(x-3) +2[/tex]
[tex]G(x)=-\frac{1}{2}(x-3)^3 +2[/tex]
