To find the vertex of the parabola given by the equation y = x² + 8x + 12, we can use the formula for the vertex of a quadratic equation in the standard form y = ax² + bx + c. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In our equation, a = 1 and b = 8. Plugging these values into the formula gives us:
x = -8 / (2 * 1) = -8 / 2 = -4
Now that we have the x-coordinate of the vertex, we can substitute it back into the original equation to find the y-coordinate:
y = (-4)² + 8(-4) + 12
Simplifying this, we calculate:
y = 16 – 32 + 12 = -4
Thus, the coordinates of the vertex are (-4, -4). This point represents the maximum or minimum point of the parabola. Since the coefficient of x² is positive, the parabola opens upwards, confirming that the vertex is a minimum point.