Given:
The function is:
[tex]y=-3(x+1)^2-1[/tex]
The parent function is:
[tex]y=x^2[/tex]
To find:
The transformation of the given function from the parent function.
Solution:
The transformation of a function is given by:
[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 k<0, then the graph of f(x) is reflected across the x-axis.
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.
The given function can be written as is:
[tex]g(x)=-3(x+1)^2-1[/tex]
Let the parent function be [tex]f(x)=x^2[/tex], then
[tex]g(x)=-3f(x+1)-1[/tex] ...(ii)
On comparing (i) and (ii), we get
[tex]k=-3,a=1,b=-1[/tex]
Therefore, the graph of parent function is stretched vertically by by factor 3 and reflected across the x-axis because [tex]k=-3[/tex], after that the graph of the function shifts 1 unit left and 1 unit down to get the graph of given function.
