Final answer:
The student's question involves determining or verifying a quadratic function using given characteristics such as the vertex, y-intercept, axis of symmetry, roots, and the direction of its opening. Utilizing the provided information, one can derive the equation of the parabola in vertex form.
Step-by-step explanation:
The question provided details about the features of a quadratic equation in vertex form and asks for further analysis or construction of the function. Given the vertex, the y-intercept, the axis of symmetry (AOS), the roots, and the orientation of the parabola (opening up),
we can determine that the quadratic function takes the form of f(x) = a(x - h)² + k, where (h, k) represents the vertex and a is a positive coefficient because the parabola opens up. Since the vertex is at (1, -16) and the AOS is x = 1, that confirms our vertex h value.
The y-intercept being (0, -15) allows us to solve for a as it must satisfy f(0) = -15. Finally, having the roots (5,0) and (-3,0) means that those x-values must also satisfy the equation when f(x) = 0.