Given:
The parent function is
[tex]f(x)=x^2[/tex]
The graphs of f(x) and g(x) are given.
To find:
The function g(x).
Solution:
From the given graph it is clear that the graph of f(x) is reflected over the x-axis and shifted 3 units down to get the graph of g(x).
The translation is defined as
[tex]g(x)=kf(x+a)+b[/tex] .... (1)
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) reflected over 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.
From the given graph it is clear that the graph of f(x) is reflected over the x-axis and shifted 3 units down to get the graph of g(x). So, k=-1, a=0 and b=-3.
[tex]g(x)=(-1)f(x+0)+(-3)[/tex]
[tex]g(x)=-f(x)-3[/tex]
[tex]g(x)=-x^2-3[/tex] [tex][\because f(x)=x^2][/tex]
Therefore, the required function is [tex]g(x)=-x^2-3[/tex]. So, the correct option is B.
