Solution: The altitude is modeled by the quadratic function $ y = 3x^2 - 12x + 15 $. Since the coefficient of $ x^2 $ is positive, the parabola opens upward, and the minimum value occurs at the vertex. The x-coordinate of the vertex is:
["Understanding the Altitude Model: Analyzing the Quadratic Function $ y = 3x^2 - 12x + 15 $", "When modeling physical phenomena like altitude over horizontal distance, quadratic functions often provide accurate and insightful representations. The given equation, $ y = 3x^2 - 12x + 15 $, describes an altitude profile that follows this mathematical form.", "### The Shape of the Parabola", "Since the coefficient of $ x^2 $ is $ 3 $, which is positive, the parabola opens upward—indicating that the function has a minimum value but no maximum. This upward opening is key to locating the lowest point, which corresponds to the point of minimum altitude in practical terms.", "### Finding the Vertex—the Minimum Point", "The vertex of a parabola defined by $ y = ax^2 + bx + c $ occurs at the x-coordinate:", "$$\nx = -\frac{b}{2a}\n$$", "For our function:\n- $ a = 3 $\n- $ b = -12 $", "Substituting into the formula:", "$$\nx = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2\n$$", "So, the x-coordinate of the vertex—and thus the point where the altitude reaches its minimum—is $ x = 2 $.", "### Vertex Coordinates and Minimum Altitude", "To find the corresponding minimum altitude (y-coordinate), substitute $ x = 2 $ back into the original equation:", "$$\ny = 3(2)^2 - 12(2) + 15 = 3(4) - 24 + 15 = 12 - 24 + 15 = 3\n$$", "Therefore, the minimum altitude is 3 units, occurring at $ x = 2 $.", "### Why This Matters in Real-World Applications", "Modeling altitude as $ y = 3x^2 - 12x + 15 $ helps engineers, urban planners, and environmental scientists analyze elevation changes smoothly. The vertex provides critical insight into where the lowest or highest point lies, aiding in design, safety planning, and efficient resource use.", "### Conclusion", "Modeling altitude with a quadratic function like $ y = 3x^2 - 12x + 15 $ allows precise identification of key features. Using the vertex formula $ x = -\frac{b}{2a} $, we find the minimum altitude occurs at $ x = 2 $. Recognizing the parabola’s upward shape confirms this is a minimum point—essential for accurate analysis and prediction.", "Key takeaway: The x-coordinate of the vertex, $ x = 2 $, is found by $ x = -\frac{b}{2a} = -\frac{-12}{2 \cdot 3} = 2 $. This makes it a powerful tool in understanding the function’s behavior."]
