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b/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/plot.py @@ -0,0 +1,66 @@ +# /// script +# requires-python = ">=3.11" +# dependencies = ["numpy", "matplotlib"] +# /// + +import numpy as np +import matplotlib.pyplot as plt + +plt.rcParams["font.sans-serif"] = [ + "WenQuanYi Micro Hei", + "Noto Sans CJK SC", + "Microsoft YaHei", + "DejaVu Sans", +] +plt.rcParams["axes.unicode_minus"] = False + +fig, ax = plt.subplots(figsize=(10, 7), dpi=150) + +x = np.linspace(-1, 5, 500) +y = -2 * x**2 + 8 * x - 3 + +ax.plot(x, y, "b-", linewidth=2, label="$y = -2x^2 + 8x - 3$") + +vertex_x, vertex_y = 2, 5 +ax.scatter( + [vertex_x], + [vertex_y], + color="red", + s=150, + zorder=5, + label=f"顶点 ({vertex_x}, {vertex_y})", +) +ax.annotate( + f"顶点 ({vertex_x}, {vertex_y})\n最大值: y = {vertex_y}", + xy=(vertex_x, vertex_y), + xytext=(vertex_x + 0.8, vertex_y + 1.5), + fontsize=11, + ha="left", + arrowprops=dict(arrowstyle="->", color="red", lw=1.5), + bbox=dict(boxstyle="round,pad=0.3", facecolor="yellow", alpha=0.7), +) + +ax.axhline(y=0, color="gray", linestyle="--", linewidth=1, alpha=0.7) +ax.axvline(x=0, color="gray", linestyle="--", linewidth=1, alpha=0.7) +ax.axvline( + x=vertex_x, + color="green", + linestyle=":", + linewidth=1.5, + alpha=0.7, + label=f"对称轴 x = {vertex_x}", +) + +ax.set_xlabel("x", fontsize=12) +ax.set_ylabel("y", fontsize=12) +ax.set_title("二次函数 $y = -2x^2 + 8x - 3$ 的图像", fontsize=14, fontweight="bold") +ax.legend(loc="upper right", fontsize=10) +ax.grid(True, alpha=0.3) + +ax.set_xlim(-0.5, 5.5) +ax.set_ylim(-6, 8) + +plt.tight_layout() +plt.savefig("figure.png", bbox_inches="tight", dpi=150) +plt.close() +print("图像已保存: figure.png") diff --git a/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/report.md b/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/report.md new file mode 100644 index 0000000..c0b69b6 --- /dev/null +++ b/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/report.md @@ -0,0 +1,80 @@ +# 二次函数 $y=-2x^2+8x-3$ 求解报告 + +## 1. 🎯 问题描述 + +已知二次函数 $y = -2x^2 + 8x - 3$,求: +1. 函数的顶点坐标 +2. 函数的最大值 + +## 2. ✅ 最终结论 + +二次函数 $y = -2x^2 + 8x - 3$ 的顶点坐标为 $(2, 5)$,由于抛物线开口向下,该顶点即为函数的最大值点,因此函数的最大值为 $y = 5$。 + +## 3. 📈 可视化 + +![函数图像](figure.png) + +**图表说明**: + +- 蓝色曲线:函数 $y = -2x^2 + 8x - 3$ 的图像 +- 红色圆点:顶点位置 $(2, 5)$,也是函数的最大值点 +- 绿色虚线:对称轴 $x = 2$ +- 灰色虚线:坐标轴参考线 + +## 4. 🧠 数学建模与解题过程 + +
+点击展开 + +**问题分析**:这是一个标准的一元二次函数 $y = ax^2 + bx + c$ 的求极值问题。 + +**方法选择**:使用顶点公式法。对于二次函数 $y = ax^2 + bx + c$,其顶点横坐标为 $x = -\frac{b}{2a}$,顶点纵坐标为函数在该点的函数值。 + +**推导过程**: + +已知函数: +$$y = -2x^2 + 8x - 3$$ + +其中 $a = -2$,$b = 8$,$c = -3$。 + +**步骤 1:求顶点横坐标** + +$$x_{vertex} = -\frac{b}{2a} = -\frac{8}{2 \times (-2)} = -\frac{8}{-4} = 2$$ + +**步骤 2:求顶点纵坐标** + +将 $x = 2$ 代入原函数: +$$y_{vertex} = -2(2)^2 + 8(2) - 3 = -2 \times 4 + 16 - 3 = -8 + 16 - 3 = 5$$ + +**步骤 3:判断极值类型** + +由于二次项系数 $a = -2 < 0$,抛物线开口向下,因此顶点为最高点,即函数的最大值点。 + +**结论**: +- 顶点坐标:$(2, 5)$ +- 函数最大值:$y = 5$ + +
+ +## 5. 📊 运行结果 + +
+点击展开 + +``` +================================================== +二次函数 y = -2x² + 8x - 3 求解结果 +================================================== + +1. 顶点坐标: (2, 5) + 即: x = 2.0, y = 5.0 + +2. 函数最大值: y = 5.0 + 由于 a = -2 < 0,抛物线开口向下, + 顶点为最高点,即为最大值点。 + +3. 对称轴: x = 2.0 +================================================== +``` + +
diff --git a/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/solve.py b/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/solve.py new file mode 100644 index 0000000..632e30c --- /dev/null +++ b/20260109_161620_quadratic_vertex-工具openode模型MiniMax21/solve.py @@ -0,0 +1,26 @@ +# /// script +# requires-python = ">=3.11" +# dependencies = ["sympy"] +# /// + +import sympy as sp + +x = sp.symbols("x", real=True) + +a, b, c = -2, 8, -3 + +y = a * x**2 + b * x + c + +vertex_x = -b / (2 * a) +vertex_y = a * vertex_x**2 + b * vertex_x + c + +print("=" * 50) +print("二次函数 y = -2x² + 8x - 3 求解结果") +print("=" * 50) +print(f"\n1. 顶点坐标: ({sp.nsimplify(vertex_x)}, {sp.nsimplify(vertex_y)})") +print(f" 即: x = {vertex_x}, y = {vertex_y}") +print(f"\n2. 函数最大值: y = {vertex_y}") +print(f" 由于 a = -2 < 0,抛物线开口向下,") +print(f" 顶点为最高点,即为最大值点。") +print(f"\n3. 对称轴: x = {vertex_x}") +print("=" * 50)