[tex]f(x) = -(x-3)^{2} -2[/tex] is the required function.
Step-by-step explanation:
We have to find the transformation after the horizontal and vertical shifts and then reflection over the x-axis for the given function.
[tex]f(x) = x^{2}[/tex]
If there is a horizontal shift of 3, then the function becomes, [tex]f(x) = (x-3)^{2}[/tex]
If there is a vertical shift of -2, then the function becomes, [tex]f(x) = (x-3)^{2} -2[/tex]
The function is reflected over the x- axis then the function becomes,
[tex]f(x) = -(x-3)^{2} -2[/tex]
