F(x) = 2x2 + 12x + 1
Determine the vertex !

(-3, -17)

Step-by-step explanation:

The best way to find the vertex is to rewrite the given quadratic in vertex form:  y - k = a(x - h)^2.

The given quadratic is f(x) = 2x2 + 12x + 1.  We will focus on the first two terms and "complete the square."  f(x) = 2x2 + 12x + 1 can be rewritten as

f(x) = 2(x^2 + 6x) + 1, which, after completing the square, is

f(x) = 2(x^2 + 6x + 9 - 9) + 1, or

f(x) = 2( (x + 3)^2  - 9 ) + 1, or

f(x) = 2(x + 3)^2 - 17

Comparing this to

f(x) = a(x - h)^2 + k, we see that the vertex is located at (h, k):  (-3, -17)

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step-by-step explanation: