Find the vertex of the parabola by applying the vertex formula. \[ f(x)=3 x^{2}-18 x+30 \]

To find the vertex of the parabola described by the function \( f(x) = 3x^2 - 18x + 30 \), we can use the vertex formula. The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 3 \) and \( b = -18 \). Calculating the x-coordinate: \[ x = -\frac{-18}{2 \times 3} = \frac{18}{6} = 3 \] Next, substitute \( x = 3 \) back into the original function to find the y-coordinate: \[ f(3) = 3(3)^2 - 18(3) + 30 = 3(9) - 54 + 30 = 27 - 54 + 30 = 3 \] So, the vertex of the parabola is \( (3, 3) \). Now, let’s add some fun facts! The parabola opens upwards because the coefficient of \( x^2 \) (which is 3) is positive. This means the vertex represents the lowest point on the graph, known as the minimum! In real-world applications, parabolas can model anything from the trajectory of a thrown ball to the shape of certain bridges. So knowing how to find that vertex can aid in understanding those phenomena more clearly!