Given:
The parent sine function is [tex]f(x)=x^2[/tex].
It shifts three units to the left and four units down.
To find:
The new function.
Solution:
The translation is defined as
[tex]f_1(x)=f(x+a)+b[/tex] .... (1)
Where, a is horizontal shift and b is vertical shift.
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.
Graph shifts 3 units to the left. So, a=3.
Graph shifts 4 units down. So, b=-4.
Putting a=3 and b=-4, we get
[tex]f_1(x)=f(x+3)-4[/tex]
[tex]f_1(x)=(x+3)^2-4[/tex] [tex][\because f(x)=x^2][/tex]
Here, new function is also defined by f(x). So, [tex]f(x)=(x+3)^2-4[/tex].
Therefore, the correct option is d.
