Question

what is the vertex of the function ( f(x) = x^2 + 12x )?
( (-6, -36) )
( (-6, 0) )
( (6, 0) )
( (6, -36) )

Explanation:

Step1: Complete the square for the quadratic function

The function is \( f(x) = x^2 + 12x \). To complete the square, we take half of the coefficient of \( x \), square it, and add and subtract it. The coefficient of \( x \) is 12, half of it is 6, and squaring it gives \( 6^2 = 36 \). So we can rewrite the function as:

$$
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$$

Step2: Identify the vertex form of a parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. In our case, \( a = 1 \), \( h = -6 \) (because it's \( x - (-6) = x + 6 \)), and \( k = -36 \).

Answer:

\((-6, -36)\) (corresponding to the first option: \((-6, -36)\))