Which inequality in vertex form represents the region less than or equal to the quadratic function with vertex open paren negative 6 comma 12 close paren and includes the point open paren negative 9 comma 15 close paren on the boundary?

Vertex form is y = a(x − h)² + k, so a vertex of (−6, 12) gives y = a(x + 6)² + 12. Substitute the boundary point (−9, 15): 15 = a(−9 + 6)² + 12, so 15 = 9a + 12 and a = 1/3. Thus the inequality is y ≤ (1/3)(x + 6)² + 12, choice C.

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