For this case we have an equation of the form:
[tex]y = ax ^ 2 + bx + c
[/tex]
This equation in vertex form is:
[tex]f (x) = a (x - h) ^ 2 + k
[/tex]
where (h, k) is the vertex of the parabola.
We have the following function:
[tex]f (x) = x ^ 2 + 14x + 40
[/tex]
We look for the vertice.
For this, we derive the equation:
[tex]f '(x) = 2x + 14
[/tex]
We equal zero and clear the value of x:
[tex]2x + 14 = 0
2x = -14
x = -14/2
x = -7[/tex]
Substitute the value of x = -7 in the function:
[tex]f (-7) = (- 7) ^ 2 + 14 * (- 7) +40
f (-7) = -9[/tex]
Then, the vertice is:
[tex](h, k) = (-7, -9)
[/tex]
Substituting values we have:
[tex]f (x) = (x + 7) ^ 2 - 9[/tex]
Answer:
The quadratic function in vertex form is:
[tex]f (x) = (x + 7) ^ 2 - 9[/tex]
