Question

$$ y = \frac { 1 } { 2 } x ^ { 2 } - 6 x $$

Explanation:

Step1: Find the vertex of the parabola

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
In the function \(y=\frac{1}{2}x^{2}-6x\), \(a = \frac{1}{2}\), \(b=-6\).

$$x=-\frac{-6}{2\times\frac{1}{2}}=\frac{6}{1} = 6$$

Substitute \(x = 6\) into the function \(y=\frac{1}{2}x^{2}-6x\):

$$y=\frac{1}{2}\times6^{2}-6\times6=\frac{1}{2}\times36 - 36=18 - 36=- 18$$

Step2: Find the \(y\) - intercept

Set \(x = 0\) in the function \(y=\frac{1}{2}x^{2}-6x\).
When \(x = 0\), \(y=0\).

Step3: Analyze the options

Option B has the vertex at \(x = 6\) (approximate position in the grid considering the scale) and passes through the origin \((0,0)\) (when \(x = 0\), \(y = 0\) for the function \(y=\frac{1}{2}x^{2}-6x\)).

Answer:

B.