Quadratics in Swimming

Learn how quadratics can be applied in swimming.

The Example of quadratics

Notice how the swimmer dives from the block into the water and once they enter the water and they pass the minimum value and vertex ,which is the lowest depth they reach in the water, to do the breakout stroke before coming back to the surface of the water (note that the roots are the points where the swimmer enters and comes back to the surface of the water).


A swimmer dives from the block which is 1 foot above the deck (0,1) and once they enter the water, the lowest depth they achieve is 4 feet below the water (0,-4). The distance the swimmer swims till they resurface is 10 feet across (10,0) from the start.


With the information given the vertex would be (5,-4) , the axis of symmetry would be x=5, the minimum value would be -4, the y-intercept would be (0,1).

The equation in standard form would be y=.18x^2+-1.9x+1. To convert into Vertex form using the method of completing the square would greatly help. () would go around the ax^2+bx making it (.18x^2+-1.9x)factor out the .18, then half the b value and using that new number subtract it from the original y value and square it then plug it in giving you the vertex form of

y= .18(x-5)^2+ 1


Plug in the equation to discover the solutions of {.6,10} meaning the swimmer will enter the water at ~.6 feet and resurface at ~10 feet from the starting position.