What is the relationship between the

x-intercepts of a parabola and its vertex's x-coordinate?

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, ...)

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 ?

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.

The relationship between the x-intercepts of a parabola and its vertex's x-coordinate can be understood by studying the equation of the parabola, which is given in the form of y = ax^2 + bx + c.

The x-intercepts of the parabola represent the values of x for which the y-coordinate is equal to zero. These are the points where the parabola intersects the x-axis. Mathematically, the x-intercepts can be found by setting y equal to zero in the equation and solving for x.

On the other hand, the x-coordinate of the vertex of the parabola is given by the formula x = -b / (2a). This equation calculates the x-coordinate of the vertex using the coefficients a and b from the quadratic equation.

So, what is the relationship between the x-intercepts and the vertex's x-coordinate?

1. If the parabola opens upwards (a > 0), it means the graph of the parabola is U-shaped, and the vertex lies at the bottom of the curve. In this case, the x-intercepts will exist on either side of the vertex, equidistant from it. The x-coordinate of the vertex will be exactly in the middle of the two x-intercepts.

2. If the parabola opens downwards (a < 0), it means the graph of the parabola is ∩-shaped, and the vertex lies at the top of the curve. In this case, the x-intercepts will still exist on either side of the vertex, equidistant from it. The x-coordinate of the vertex will again be in the middle of the two x-intercepts.

In summary, the x-coordinate of the vertex of a parabola is located exactly in the middle of its x-intercepts when the parabola is symmetric.