Can we write an expression for the constant in the quadratic equation such that changing $b$ results in a pure horizontal shift?

Question

I know this sounds completely strange but I've been pondering this question I've created and I'm not sure how to approach it. With the standard quadratic form $ax^2+bx+c$, modifying only $b$ value results in a horizontal and vertical shift. Is there a way to write the constant $c$ in terms of $a$ and $b$ such that modifying the $b$ coefficient in $ax^2+bx+c$ results in a pure horizontal shift?

Answer 1

If you complete the square of the quadratic, you get $$ax^2+bx+c=a\bigg(x+\frac{b}{2a}\bigg)^2+c-\frac{b^2}{4a}$$ Thus if you let $$c=\frac{b^2}{4a}$$ you have $$ax^2+bx+c=a\bigg(x+\frac{b}{2a}\bigg)^2$$ and in this case, a change in $b$ results in a horizontal shift only.

Answer 2

A quadratic polynomial

$$p(x) = a x^2 + bx + c$$

can be shifted $k$ units to the right as $p(x+k)$, i.e. we get the new polynomial

$$p_k(x) = a (x+k)^2 + b(x+k) + c$$

We see that

$$p_k(x) = ax^2+ ak^2+ 2akx + bx + bk + c = ax^2 + (2ak+b)x + (ak^2+bk+c)$$