A programmer is optimizing a neural network and models the loss function as \( L(x) = 2x^2 - 8x + k \). Find \( k \) if the minimum loss occurs at \( x = 2 \) and the loss there is 4.

["Title: How to Find the Constant ( k ) in a Neural Network Loss Function When the Minimum Loss and Its Value Are Known", "---", "Introduction\nIn neural network optimization, minimizing the loss function is essential to improve model performance. When modeling a loss function as a quadratic function ( L(x) = 2x^2 - 8x + k ), understanding how to determine the unknown constant ( k ) based on the location and value of the minimum is crucial. In this article, we’ll analyze how to solve for ( k ) when the function reaches its minimum at ( x = 2 ) and the loss there equals 4.", "---", "Understanding the Quadratic Loss Function\nThe given loss function is:\n[\nL(x) = 2x^2 - 8x + k\n]\nThis is a quadratic function in standard form ( ax^2 + bx + c ), where ( a = 2 ), ( b = -8 ), and ( c = k ). Since ( a > 0 ), the parabola opens upward and guarantees a global minimum.", "The vertex of a parabola ( ax^2 + bx + c ) occurs at:\n[\nx = -\frac{b}{2a}\n]\nSubstitute ( a = 2 ) and ( b = -8 ):\n[\nx = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2\n]\nThis confirms that the minimum occurs at ( x = 2 ), matching the given condition.", "---", "Finding ( k ) Using the Minimum Value\nWe are told that the minimum value of the function is ( L(2) = 4 ). Substitute ( x = 2 ) into the loss function:\n[\nL(2) = 2(2)^2 - 8(2) + k = 4\n]\nCompute the terms:\n[\n2(4) - 16 + k = 4\n]\n[\n8 - 16 + k = 4\n]\n[\n-8 + k = 4\n]\nSolve for ( k ):\n[\nk = 4 + 8 = 12\n]", "---", "Conclusion\nBy leveraging the vertex formula and substituting known values, we determined that the constant ( k ) must be 12. This ensures the minimum loss ( L(x) = 2x^2 - 8x + 12 ) reaches exactly 4 when ( x = 2 ), validating both the location and magnitude of the minimum. Properly tuning ( k ) helps align the model’s loss curve with desired optimization outcomes, improving neural network convergence and performance.", "---", "Key Takeaways:\n- The vertex of ( L(x) = 2x^2 - 8x + k ) occurs at ( x = 2 ).\n- Set ( L(2) = 4 ) to solve for ( k ).\n- Calculating ( k ) requires evaluating the function at the minimum point and solving the resulting linear equation.\n- Accurate knowledge of ( k ) enables precise control over the loss landscape in neural network training.", "---", "Keywords: neural network optimization, loss function minimum, quadratic loss model, find k in ( L(x) = 2x^2 - 8x + k ), small business blog, AI learning, machine learning optimization, parameter tuning", "---", "Meta Description:\nLearn how to determine the constant ( k ) in a quadratic loss function ( L(x) = 2x^2 - 8x + k ) when the minimum occurs at ( x = 2 ) and ( L(2) = 4 ). Step-by-step guide for neural network optimization."]