multiple parabolas along the same line of symmetry with the same x-axis intercepts.

Question

Given the equation of the parabola: $$ f(x) = (x-78)(x+218 ). $$ Is it possible to have more than one parabola with the same axis of symmetry: (-70), and the same x-axis intersects: ( 78 , -218 ) ?

Answer 1

Absolutely. Any parabola of the form $$f(x) = a(x-78)(x+218),\quad a\ne 0$$ will be have $x-$intercepts at $78$ and $-218$ and will have an axis of symmetry $x = -70$. This is because any parabola of the form $f(x) = a(x-78)(x+218)$ will have $x$-intercepts at $78$ and $-218$, and the axis of symmetry is always halfway between the $x-$ intercepts, that is $$x = \frac{78 - 218}{2} = -70.$$

Answer 2

Since the equation for the axis of symmetry of a parabola is given by $\ x = \frac{-b}{2a}$, then by adjusting the two parameters a and b (while preserving the roots of the equation ), you can find infinitely many parabolas with the same axis of symmetry.