Finding the quadratic function in vertex form that models th

Finding the quadratic function in vertex form that models the data

The table to the right shows a country\'s gross domestic expenditures on research and development (in billions of U.S. dollars). Let x = 4 correspond to the year 2004. Use (4,51.3) as the vertex and the data from 2009 to find a quadratic function f(x) = a(x - h)^2 + k that models these data. Use the model to estimate the expenditures in the year 2012.

Solution

The vertex form of the equation of a parabola is y = a (x - h)² + k , where, (h, k) is the vertex. Here, h = 4 and k = 51.3. Therefore, the equation of the quadratic function changes to y = a (x - 4)² + 51.3 . In the year 2009, x = 9 and y = 154 8. Therefore, 154.8 = a (9 - 4)² + 51.3 = 25a + 51.3 or. 25 a = 154.8 - 51.3 = 103.5. Therefore a = 1/25(103.5) = 4.14. Thuen, the quadrtic function\'s equation becomes y = 4.14(x - 4)² + 51.3.

In the year 2012, x = 12, so that y = 4.14 (12 - 4)² + 51.3 or, y = 4.14 (64) + 51.3 = 264.96 + 51.3 = 316.26