
Transformation
Suppose for the moment that b^24ac >0, So that f(x) has two distinct real roots , and a>0.
A) Consider the transformation f(x)> f(x)+k. What is the value of K such that f(x)+k has two identical real roots: Hint, you will need to complete squares.
B) What is the condition on f(x)=ax^2+bx+c such that f(x) > f(x) leaves the function unchanged? Hint: draw a picture of a quadratic function that does not change under this transformation. What kind of transformation is this?

For A:
If $\displaystyle f(x)=ax^2 + bx + c$, then $\displaystyle f(x)+k = ax^2 + bx + (c+k)$.
Two identical roots: $\displaystyle \dfrac{\sqrt{b^2  4a(c+k)}}{2a} = 0$.
For B:
I don't know precisely what do you mean in "leaving the function unchanged", but I guess you are looking for a condition that makes f(x)=f(x). So what do we need for $\displaystyle ax^2+bx+c=a(x)^2 + b(x) + c$?

Thank You very much, that really helps :)