What must be the other point on the graph?

Question

The axis of symmetery of a parabola iis the line x=4
the x-intercepts are 10 and -2
Suppose the point (12,4) is on the graph; what other point also must be on the graph?

Answer 1

You actually don't even need to know the intercepts. Given that the parabola is symmetric about x = 4, you can just travel from (12, 4), through the line x = 4, and as far to the other side. You need to travel 8 to the left from (12, 4) to get to x = 4, and if you travel 8 more left, you get to (-4, 4).

Answer 2

You dont just know one other point with the information you gave, YOU KNOW ALL THE POINTS, because you can derive the equation for the parabola.

If the x-intercepts (the zeros of the function) are 10 and -2, and the axis of symmetry is parallel to the y axis, then the function must be of form:
f(x) = a(x-10)(x+2). { notice if we put 10 or -2 in for x, f(x) = 0 }
If (12,4) is on the parabola, we know that
4 = a(12-10)(12+2)
4 = a * 2 * 14
4 = a * 28
a = 1/7

so f(x) = (x-10)(x+2)/7

This is more than you asked for, but why not?