Explain the real world meaning of the parameters a b and c o

Explain the real world meaning of the parameters a, b, and c of the quadratic function model.

Based on data from 1980 to 2004, the number of Medicare enrollees (in millions) can be modeled by M(t) = -0.00472t^2 + 0.663? + 28.4 where t is the number of years since 1980. (Source: Modeled from Statistical Abstract of the United States, 2006, Table 132)

Solution

M(t) = - 0.00472t^2 + 0.663t + 28.4

Its a quadratic function where similar to ax^2 +bx +c where a decides the shape of graph

a <0 , parabola is inverted downwards

Also, x = -b/2a gives the maxima if a < 0

So, at t = -( 0.663/2*-0.00472) = 70.23 years

So, In year 2051 no. of medicare enrolles is given M(70.23) = -0.00472*70.23^2 + 0.663*70.23 + 28.4 = 51.682 million enrollees