Given:
The function if given as,
[tex]f(x)=x^2\text{ . . . .(1)}[/tex]
The objective is to write the formula for the transformations.
Explanation:
Reflection over the x-axis:
For the first transformation of reflection over the x-axis,
[tex]\begin{gathered} f(x)=-f(x) \\ f(x)=-(x^2)\text{. . . . .(2)} \end{gathered}[/tex]
Shifting 13 units down:
Then, for shifting down 13 units down the equation will be,
[tex]f(x)=f(x)-13[/tex]
Then, the equation (2) can be written as,
[tex]f(x)=-(x)^2-13\text{ . . . . (3)}[/tex]
Shifting 4 units right:
For, shifting 4 units to the right the equation will be,
[tex]f(x)=f(x-4)[/tex]
Then, the equation (3) can be written as,
[tex]f(x)=-(x-4)^2-13\text{ }[/tex]
Hence, the equation of the transformed graph is f(x) = -(x-4)²-13.
