Question Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an \( (x, y) \) point. \[ y=3 x^{2}+12 x-6 \] Answer

To find the vertex of the parabola given by the equation \( y = 3x^2 + 12x - 6 \) algebraically, we use the vertex formula. The x-coordinate of the vertex can be found using \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = 12 \). Calculating: \[ x = -\frac{12}{2 \cdot 3} = -\frac{12}{6} = -2. \] Next, substitute \( x = -2 \) back into the equation to find the y-coordinate: \[ y = 3(-2)^2 + 12(-2) - 6 = 3(4) - 24 - 6 = 12 - 24 - 6 = -18. \] Therefore, the coordinates of the vertex are \( (-2, -18) \). To confirm the vertex, you can check the properties of the parabola. Since it opens upwards (the value of \( a \) is positive), the vertex represents the minimum point. Remembering that the vertex splits the parabola into two symmetric halves can help visualize it. Sketching a simple graph with the vertex at \( (-2, -18) \ can be a fun way to see the graph's shape emerge!