An AI programmer is optimizing a model's performance using a quadratic cost function \( f(x) = ax^2 + bx + c \). If the vertex of this parabola corresponds to the minimum point at \( x = 3 \) and the cost at the vertex is \(-5\), find the relationship between \( a \) and \( c \) given \( b = -6 \).
["Optimizing AI Models with Quadratic Cost Functions: Deriving the Relationship Between Model Parameters", "In AI model training, minimizing a cost function is essential for improving performance. A classic approach uses quadratic cost functions of the form ( f(x) = ax^2 + bx + c ), where optimizing the model involves locating the vertex of this parabola—this point represents the minimum (or maximum) of the function.", "This article explains how, using given conditions—specifically a vertex at ( x = 3 ), a minimum cost of (-5), and ( b = -6 )—we derive a precise relationship between the coefficients ( a ) and ( c ).", "---", "1. Vertex of a Quadratic Function", "For any quadratic ( f(x) = ax^2 + bx + c ), the x-coordinate of the vertex is given by:", "[\nx = -\frac{b}{2a}\n]", "We’re told the minimum occurs at ( x = 3 ), so substitute this into the vertex formula:", "[\n3 = -\frac{b}{2a}\n]", "With ( b = -6 ), substitute:", "[\n3 = -\frac{-6}{2a} = \frac{6}{2a} = \frac{3}{a}\n]", "Solving for ( a ):", "[\na = 1\n]", "So far, we know ( a = 1 ) and ( b = -6 ).", "---", "2. Using the Minimum Value of the Function", "The value at the vertex is ( f(3) = -5 ). Substitute ( x = 3 ), ( a = 1 ), ( b = -6 ) into the function:", "[\nf(3) = a(3)^2 + b(3) + c = -5\n]", "[\n1 \cdot 9 + (-6)(3) + c = -5\n]", "[\n9 - 18 + c = -5\n]", "[\n-9 + c = -5\n]", "[\nc = 4\n]", "---", "3. Deriving the Relationship Between ( a ) and ( c )", "So far, we have ( a = 1 ), ( b = -6 ), and ( c = 4 ). But the core goal is to find a general relationship between ( a ) and ( c ) under the given conditions—specifically, that the vertex is at ( x = 3 ) and ( f(3) = -5 ).", "From the vertex formula:", "[\n3 = -\frac{b}{2a} \Rightarrow b = -6a\n]", "But we’re told ( b = -6 ), so equating:", "[\n-6a = -6 \Rightarrow a = 1\n]", "Still, let’s eliminate ( b ) explicitly using both conditions.", "We already used:\n- Vertex at ( x = 3 \Rightarrow b = -6a )\n- ( f(3) = -5 \Rightarrow a(9) + b(3) + c = -5 )", "Substitute ( b = -6a ) into the cost equation:", "[\n9a + 3(-6a) + c = -5\n]\n[\n9a - 18a + c = -5\n]\n[\n-9a + c = -5\n]", "Now solve for ( c ):", "[\nc = 9a - 5\n]", "This expresses ( c ) directly in terms of ( a ).", "Thus, the relationship between ( a ) and ( c ) is:", "[\n\boxed{c = 9a - 5}\n]", "---", "Conclusion", "In optimizing AI models using quadratic cost functions, identifying the vertex’s location and value allows derivation of exact parameter relationships. Given the vertex at ( x = 3 ), ( b = -6 ), and minimum cost (-5), we’ve shown that ( c = 9a - 5 ). This formula helps practitioners compute and validate cost functions efficiently, ensuring models converge to optimal solutions.", "Understanding these parametric ties strengthens both theoretical insight and practical implementation in AI development."]
