Recall that the vertex form of a quadratic function (or parabolic function) is equal to
[tex]f(x) = a(x - h)^{2} + k[/tex]
Now, given that we have f(x) = x² + 14x + 40, to express f into its vertex form, we must first fill in the expression to form a perfect square.
One concept that we must remember when completing the square is that
(a + b)² = a² + 2ab + b²
So, to complete the square for (x² + 14x + ____), we have 2ab = 14 where a = 1. Thus, b = 14/2 = 7. Hence, the last term of the perfect square must equal to 7² = 49.
So, going back to the function, we have
[tex]f(x) = (x^{2} + 14x + 40 + 0) [/tex]
[tex]f(x) = (x^{2} + 14x + 40 + 49 - 49)[/tex]
[tex]f(x) = (x^{2} + 14x + 49) + 40 - 49[/tex]
[tex]f(x) = (x + 7)^{2} - 9[/tex]
Thus, we have derived the vertex form of the function.
Answer: f(x) = (x + 7)² - 9
