Let's consider a simple example: color(red)(f(x) = x^2) is the parent function, represented by the red curve.
The graph of the transformed function color(blue)(g(x)) is colored in blue. Our goal is to find out the equation of the function color(blue)(g)
We notice that the transformed function color(blue)(g) is the outcome of several successive transformations applied to the parent function color(red)(f):
1. Reflection of f across the Ox axis : this gives us the first transformed function color(green)(f_1(x)=-x^2)
2. Translation of f_1 by 5 units to the right across the Ox axis: this gives us the second transformed function color(indigo)(f_2(x) = -(x-5)^2)
3. Translation of f_2 by 4 units up across the Oy axis : this gives us the transformed function g(x) = -(x-5)^2 + 4=x^2+10x-25 + 4 -> color(blue)(g(x)=-x^2+10x-21), that is the formula we were looking for.
Please find below the graphs of the successive transformations, represented in the corresponding colors (red, green, indigo, blue):
