Given:
The given functions are:
[tex]f(x)=x^2[/tex]
[tex]g(x)=3(x+5)^2-2[/tex]
To find:
The transformations performed of f(x) to create g(x).
Solution:
The translation is defined as
[tex]g(x)=kf(x+a)+b[/tex] .... (i)
Where, k is stretch factor, a is horizontal shift and b is vertical shift.
If 0<k<1, then the graph compressed vertically by factor k and if k>1, then the graph stretch vertically by factor k.
If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.
If b>0, then the graph shifts b units up and if b<0, then the graph shifts b units down.
We have,
[tex]f(x)=x^2[/tex]
[tex]g(x)=3(x+5)^2-2[/tex]
Using these two function, we get
[tex]g(x)=3f(x+5)-2[/tex] ...(ii)
On comparing (i) and (ii), we get
[tex]k=3,a=5,b=-2[/tex]
It means the graph of f(x) is vertically stretched with a scale factor of 3, shifts 5 units left and 2 units down to get g(x).
Therefore, the correct options are A and C.