First example. If the points are 3 I get the parabola that passes exactly through them.
x:   −3, −1, 5
y:   5, −2, 4

I get:  y = 0.5625*x^2 − 1.25*x − 3.8125.

 

How to graphically represent what was obtained:  see here.

Second example.
x:   -3, -1, 0, 4, 5, 7
y:   -9, -7, 1, 6, -2, -5

I get:  y = -0.4162*x^2 + 2.257*x - 0.2438.

 

How to graphically represent what was obtained:  see here.

Third example.
I have the multiflash photograph of a ball thrown in the air. The time intervals between two successive images are the same. With a graduated line I detect the distances, in centimeters, of the positions of the center of the ball from the left margin and from the low margin of the photo. Here, in order, the "abscissas" and the "ordinates" of the points thus determined:
x:   1.2, 1.5, 1.8, 2.2, 2.5, 2.8, 3.1, 3.4, 3.7, 4.1, 4.4, 4.7, 5.0, 5.3, 5.6, 6.0, 6.3, 6.6, 6.9, 7.2, 7.6, 7.9, 8.2, 8.5, 8.8, 9.1
y:  0.8, 2.6, 4.1, 6.0, 7.3, 8.4, 9.4, 10.3, 11.0, 11.8, 12.2, 12.5, 12.7, 12.7, 12.6, 12.3, 11.9, 11.4, 10.7, 9.9, 8.7, 7.6, 6.4, 5.0, 3.5, 1.9
I get:   y = −0.726*x^2 + 7.61*x - 7.20

 

How to graphically represent what was obtained:  see here.

Fourth example.
I have the multiflash photograph (one shot every 30th of a second) of a billiard ball dropped from a certain position, next to a graduated meter with notches one millimeter wide.

x: time (s);  y: position (m)
x:   0.03333333333, 0.06666666667, 0.1, 0.13333333333, 0 .16666666667, 0.2, 0 .23333333333, 0 .26666666667, 0.3, 0.3333333333, 0.36666666667, 0.4, 0 .43333333333, 0.46666666667
y:  0.095, 0.15, 0.215, 0.295, 0.385, 0.485, 0.595, 0.715, 0.85, 0.99, 1.14, 1.305, 1.48, 1.67
I get:   y = 4.837*x^2 + 1.213*x + 0.048

 

How to graphically represent what was obtained:  see here.