Question

What is the relationship between the
x-intercepts of a parabola and its vertex's x-coordinate?

Answer 1

if you average the x-intercepts , you get the x coordinate of the vertex.

y = x^2 - 2x - 8
= (x-4)(x+2)

so the x intercepts are 4 and -2

average of those = (-2 + 4)/2 = 1

so the vertex is (1, ...)

Answer 2

If I were to write down:
"When you find the median between the two intercepts, it is the x-intercept of the vertex."
do you think that's a true relationship still ?

Answer 3

I would use the word "mean"

"Mean" is what is known as the average in everyday language, "median" is the middle value of a set of data.
Of course if you only have two data values, as the x-intercepts, it actually does not matter.