The table represents a quadratic function if and only if f(h - d) = f(h + d), where d is the distance of the x-value with respect to the x-coordinate of the vertex of the parabola, represented by the variable h.
Quadratic functions are polynomials of the form y = a · x² + b · x + c, which can be rewritten on vertex form: y - k = a · (x - h)², where (h, k) is the vertex of the function. Graphically speaking, quadratic functions are parabolae.
The axis of symmetry passes through the function and therefore we can infer that the table given represents a quadratic function if and only if the following condition is met:
f(h - d) = f(h + d) (1)
Where d is the distance of the x-value with respect to the x-coordinate of the vertex of the parabola, represented by the variable h. If all possible pairs observes (1), then we are in front of a quadratic equation.
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