How can I find the vertex of a parabola using only $x$ intercepts. My teacher gave me this problem where I did a long jump and recorded the distance I went. He then asked us the height. My distance was 80 inches so the x-intercepts are $0,0$ and $80,0$. My question Is how can I make a quadratic Equation using only the $x$ intercepts to find height.
You need additional data to find the height.

For example, the parabolas $y=-x^2+80x$ and $y=-2x^2+160x$ both have the $x$-intercepts you specify, but vertices at $(40,1600)$ and $(40,3200)$, respectively.
Basically, you need to know three points on the parabola to determine it uniquely. The general form of a parabola is $ax^2 + bx + c$, which is three unknowns, which means we need three linear relations (given by three points on the curve) to determine it. Just knowing the x-intercepts does not give enough information to determine all three coefficients.