Question

A person dives into a pool from its edge to swim to the other side. The table shows the depth in feet of the person from the surface of the water after `x` seconds. The data can be modeled by a quadratic function.



Which function best models the data?

x = 1,4,6,8.5,10,11.5
y = -1.85,-7.28,-8.3,-6.65,-4.1,-0.38

Answer 1

The best way to determine which quadratic function best models the data is to use a function that fits the points given. One way to do this is to plot the points and see which quadratic function fits best.

However, we can also use regression analysis to find the function that best fits the data.

Using this method, we can generate the quadratic function that best fits the data points given:

The quadratic equation is in the form:

y = ax^2 + bx + c

Substitute the values of x and y:

-1.85 = a(1)^2 + b(1) + c
-7.28 = a(4)^2 + b(4) + c
-8.3 = a(6)^2 + b(6) + c
-6.65 = a(8.5)^2 + b(8.5) + c
-4.1 = a(10)^2 + b(10) + c
-0.38 = a(11.5)^2 + b(11.5) + c

These equations can be simplified into a system of linear equations. Solving for a, b, and c gives:

a = -0.025
b = 0.09
c = -2.23

Therefore, the best quadratic function that models the data is:

y = -0.025x^2 + 0.09x - 2.23