Solution: The maximum height of a parabola $ y = ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $. Here, $ a = -1 $, $ b = 6 $, so $ x = -\frac{6}{2(-1)} = 3 $. Substituting $ x = 3 $ into the equation:

["Understanding the Maximum Height of a Parabola: A Step-by-Step Solution", "When analyzing quadratic functions, one key concept is the maximum or minimum point of the parabola—this determines the parabola’s peak or trough. For a quadratic equation in standard form, $ y = ax^2 + bx + c $, the vertex represents the highest or lowest point, depending on the direction the parabola opens. In this article, we’ll explore how to find this maximum height using a specific quadratic equation, with real-world context and clear mathematical reasoning.", "---", "### The Vertex Formula: Finding the Maximum Height", "The $ x $-coordinate of the vertex (and thus the vertex point’s height) is given by:", "$$\nx = -\frac{b}{2a}\n$$", "This formula works regardless of whether the parabola opens upward or downward. Notably:\n- If $ a > 0 $, the parabola opens upward and has a minimum at the vertex.\n- If $ a < 0 $, the parabola opens downward and has a maximum at the vertex.", "In our example, we are given:", "- $ a = -1 $ → the parabola opens downward\n- $ b = 6 $", "Calculating the $ x $-coordinate of the maximum point:", "$$\nx = -\frac{b}{2a} = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3\n$$", "So the maximum height occurs at $ x = 3 $.", "---", "### Finding the Maximum Value: Substituting into the Equation", "To determine the actual height (i.e., the $ y $-coordinate of the vertex), we substitute $ x = 3 $ back into the original quadratic equation:", "$$\ny = ax^2 + bx + c\n$$", "Using $ a = -1 $, $ b = 6 $, and $ x = 3 $:", "$$\ny = (-1)(3)^2 + 6(3) + c\n$$", "$$\ny = -9 + 18 + c = 9 + c\n$$", "Assuming $ c $ represents the y-intercept of the parabola, if $ c = 0 $ (i.e., the graph passes through the origin), then:", "$$\ny = 9 + 0 = 9\n$$", "If $ c <br/>\ne 0 $, the vertex height becomes $ y = 9 + c $, which reflects vertical shifts of the parabola.", "---", "### Visual and Real-World Interpretation", "Graphically, this means the parabola peaks at $ x = 3 $, reaching a maximum height of $ y = 9 $ (if $ c = 0 $). In practical applications—such as physics for projectile motion or optimization in business—the vertex represents the optimal outcome: maximum height, maximum profit, or minimum cost.", "---", "### Summary", "- The maximum height of a downward-opening parabola $ y = ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $.\n- With $ a = -1 $, $ b = 6 $, $ x = 3 $.\n- Substituting $ x = 3 $ into the equation gives the maximum $ y $-value: $ y = 9 + c $.\n- This formula applies broadly across mathematics, physics, and engineering to model peak performance.", "By mastering the vertex calculation and substitution, you gain deep insight into parabolic behavior—essential for solving equations, optimizing systems, and interpreting data.", "---", "Keywords: parabola maximum height, vertex formula $ x = -\frac{b}{2a} $, quadratic vertex calculation, find maximum of $ y = ax^2 + bx + c $, downward opening parabola, maximum value of quadratic function", "---", "Optimize your understanding today: use $ x = -\frac{b}{2a} $ to locate the vertex—and reveal the peak of any quadratic in seconds."]