What is the difference between shifting functions and shifting parabolas?

Question

For shifting a parabola to the right, do we write a "$+$" sign or "$-$" sign in the equation? Is this the same way for shifting a function, as well?

Here is the equation of parabola just for the reference:

$y=(x-0)^2+c $

(Where $c$ is a variable; Y-intercept.)

(Correct me if I'm wrong.)

Another question, is there a "$-$" sign already there in an ideal parabola equation?

P.S. Apologies for the simplicity of the question.

Answer 1

Let $f_0(x)$ be the function in hand with roots $\{r_i\}_{i=1}^{i=n}$ and let the roots of the shifted function $f_1(x)$ be $\{r_i'\}_{i=1}^{n}$. Clearly when we shift a function its roots must change in correspondence with the roots of the original function.

1. To the right: Since we're to shift to the right that means the corresponding new roots will be greater than the original roots i.e. $r_i'\gt r_i$. So for a shift by $r_i'-r_i$ to the right the new function would be $f(x-(r_i'-r_i))$.
2. To the left: Since we're to shift to the left that means the corresponding new roots will be smaller than the original roots i.e. $r_i'\lt r_i$. So for a shift by $r_i'-r_i$ to the left the new function would be $f(x+(r_i'-r_i))$.

Notice that Parabolas are functions described by Quadratic Equations, so these rules apply to them as well. You may also consider another approach by figuring out the vertex from the function.

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Take the example of $f(x)=(x+5)^2$. Zeroes are $\{-5, -5\}$. Let's say you want to shift it by $5$ to the right. Then the function becomes $g(x)=f(x-5)=\left((x-5)+5\right)^2=x^2$ which is your required function.