How to find the coefficient a of a $y=ax^2$ parabola?

Question

If you have a parabola mirror with a $y=ax^2$ graph in real life, how could you calculate the coefficient $a$? The only measurement that should be performed in real life is some kind of straight line. With just these measurements, what's the best way to calculate $a$? $$$$ Edit: About the measurement part, we can measure the height of the parabola's axis symmetry for example. Again, any straight line would just be measurable, so if we need to construct a straight line and use its measure to calculate $a$ that would answer my question.

Answer 1

Hold your parabolic mirror in a vertical position, with some distant object in front of it (the Sun, a tall tree with the sky as background, and so on) and a vertical white screen where to project the image. Move your mirror until the image formed on the screen is clear and detailed. Measure then the distance between the image on the screen and the center of your mirror: that is the focal distance $f$ of your screen. And $a=1/(4f)$.

Answer 2

As a first qualitative step we know that if the parabola opens upward $a>0$ otherwise if it opens downward $a<0$.

For a quantitative evaluation we need at least to know the coordinates of a point $(x_0,y_0)$ on the parabola with $x_0\neq 0$ and then

$$y_0=ax_0^2 \implies a=\frac{y_0}{x_0^2}$$