Final answer:
The minimum value of the quadratic function y = 0.75(X + 4)² + 3 is found at the vertex (-4, 3). Since the quadratic equation is in vertex form and the coefficient of the squared term is positive, the y-value of the vertex, which is 3, represents the minimum value of the function.
Step-by-step explanation:
The minimum value of a quadratic function is found at the vertex if the coefficient of the squared term is positive. In the given quadratic relation y = 0.75(X + 4)² + 3, the coefficient of the squared term (0.75) is positive, indicating that the parabola opens upward and the vertex of the parabola represents the minimum point.
To find the coordinates of the vertex, we can see that the quadratic is already in vertex form, which is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our equation, h = -4 and k = 3. Therefore, the vertex of our parabola is at (-4, 3).
Since the vertex represents the minimum point and k is the y-value of the vertex, the minimum value of the function is y = 3.