Translate on the horizontal axis the graph of a second degree polynomial

Question

I'm trying to find the parametric second degree polynomial that would allow me to translate the its graph on the horizontal axis. Basically the equivalent of increasing or decreasing c in ax^2 + bx + c (doing so will translate the graph in the vertical axis). Can somebody point me in the right direction?

Answer 1

If you have a graph of the form $$ y= a(x-x_0)^2 + h $$ then the lowest point (if $a>0$) of the parabola is located at $(x_0, h)$. So basically changing $h$ moves it up and down, and changing $x_0$ moves it left and right.

Answer 2

If you want to translate $f$ with $\alpha$ along the horizontal axis, you want to find what the value $f(x)$ at the point $x + \alpha$, so your translation $\tilde{f}$ must be $\tilde{f}(x) = f(x - \alpha)$.