Find the vertex of the parabola by applying the vertex formula. \[ f(x)=3 x^{2}-54 x-94 \]

To find the vertex of the parabola given by the quadratic function \( f(x) = 3x^2 - 54x - 94 \), we can use the vertex formula, which states that the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = -54 \). Calculating the x-coordinate: \[ x = -\frac{-54}{2 \cdot 3} = \frac{54}{6} = 9. \] Now, to find the y-coordinate, substitute \( x = 9 \) back into the function: \[ f(9) = 3(9)^2 - 54(9) - 94 = 3(81) - 486 - 94 = 243 - 486 - 94 = -337. \] Thus, the vertex is at the point \( (9, -337) \). Now, let's dig deeper into some context and useful insights! The vertex of a parabola is not just a point; it represents the highest or lowest point of the function, depending on the orientation. In this case, because the coefficient of \( x^2 \) is positive (3), the parabola opens upwards, making the vertex the lowest point! If you’re trying to graph this function or understand its behavior better, remember that the vertex form, \( f(x) = a (x-h)^2 + k \), can be helpful. This form is particularly useful for identifying shifts, stretches, and whether the parabola opens upwards or downwards. It transforms the equation into a format that shows the vertex \((h, k)\) closely.