The function $ T(x) = x^2 - 6x + 13 $ is a quadratic. The minimum value of a quadratic $ ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $.
["# Understanding the Quadratic Function $ T(x) = x^2 - 6x + 13 $", "The function $ T(x) = x^2 - 6x + 13 $ is a classic example of a quadratic expression. Quadratics are fundamental in algebra and appear frequently in both theoretical mathematics and real-world applications. In this article, we explore the properties of this particular quadratic, especially focusing on how to find its minimum value using the vertex formula.", "## What is a Quadratic Function?", "A quadratic function has the general form $ T(x) = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants and $ a <br/>\neq 0 $. The graph of any quadratic function is a parabola, which opens upwards if $ a > 0 $ and downwards if $ a < 0 $. Since the coefficient of $ x^2 $ in $ T(x) $ is $ 1 $ (which is positive), the parabola opens upwards, meaning the function has a unique minimum point.", "## Finding the Vertex—the Minimum Point", "For any quadratic $ ax^2 + bx + c $, the $ x $-coordinate of the vertex—the point at which the function reaches its minimum (or maximum)—occurs at:", "$$\nx = -\frac{b}{2a}\n$$", "In our case, comparing $ T(x) = x^2 - 6x + 13 $ with the general form:", "- $ a = 1 $\n- $ b = -6 $\n- $ c = 13 $", "Substitute these into the vertex formula:", "$$\nx = -\frac{-6}{2(1)} = \frac{6}{2} = 3\n$$", "So, the minimum value occurs at $ x = 3 $.", "## Calculating the Minimum Value", "To find the minimum value of $ T(x) $, substitute $ x = 3 $ back into the function:", "$$\nT(3) = (3)^2 - 6(3) + 13 = 9 - 18 + 13 = 4\n$$", "Thus, the minimum value of the function is $ 4 $, and it occurs at $ x = 3 $.", "## Why This Matters", "Understanding where the minimum (or maximum) of a quadratic occurs helps in optimization problems. For instance, in physics, economics, or engineering, quadratic functions often model cost, height, or motion. Knowing how to compute and interpret the vertex allows for precise predictions and better decision-making.", "## Final Thoughts", "The function $ T(x) = x^2 - 6x + 13 $ elegantly demonstrates how vertex form and algebraic formulas simplify finding key features of quadratics. Remember: the vertex gives the minimum value when $ a > 0 $, located at $ x = -\frac{b}{2a} $. This insight is invaluable for mastering quadratic functions and their applications.", "---", "Keywords: $ T(x) = x^2 - 6x + 13 $, quadratic function, vertex formula, minimum value of a quadratic, algebra, mathematics education, optimization, functions."]
