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b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/plot.py new file mode 100644 index 0000000..c6a5f21 --- /dev/null +++ b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/plot.py @@ -0,0 +1,90 @@ +# /// script +# requires-python = ">=3.11" +# dependencies = ["numpy", "matplotlib"] +# /// + +import numpy as np +import matplotlib.pyplot as plt + +# 设置字体 +plt.rcParams['font.sans-serif'] = ['WenQuanYi Micro Hei', 'Noto Sans CJK SC', 'Microsoft YaHei', 'SimHei', 'SimSun', 'DejaVu Sans'] +plt.rcParams['axes.unicode_minus'] = False + +# 创建图形 +fig, ax = plt.subplots(figsize=(10, 8), dpi=150) + +# 定义函数 +def f(x): + return -2*x**2 + 8*x - 3 + +# 生成 x 数据(根据顶点位置设置合理范围) +x = np.linspace(-1, 5, 500) +y = f(x) + +# 绘制函数曲线 +ax.plot(x, y, 'b-', linewidth=2.5, label=r'$y = -2x^2 + 8x - 3$') + +# 标记顶点 +vertex_x, vertex_y = 2, 5 +ax.scatter([vertex_x], [vertex_y], color='red', s=120, zorder=5, edgecolors='darkred', linewidths=2) +ax.annotate(f'顶点 ({vertex_x}, {vertex_y})\n最大值', + xy=(vertex_x, vertex_y), + xytext=(vertex_x + 0.8, vertex_y + 0.5), + fontsize=12, + ha='left', + arrowprops=dict(arrowstyle='->', color='red', lw=1.5), + bbox=dict(boxstyle='round,pad=0.3', facecolor='lightyellow', edgecolor='orange')) + +# 标记与 x 轴的交点(求根) +# -2x² + 8x - 3 = 0 → x = (8 ± √(64-24))/(-4) = (8 ± √40)/(-4) +discriminant = 64 - 24 +x1 = (8 - np.sqrt(discriminant)) / 4 +x2 = (8 + np.sqrt(discriminant)) / 4 +ax.scatter([x1, x2], [0, 0], color='green', s=100, zorder=5, marker='s', edgecolors='darkgreen', linewidths=2) +ax.annotate(f'x ≈ {x1:.2f}', xy=(x1, 0), xytext=(x1 - 0.3, -1.5), fontsize=10, ha='center', + arrowprops=dict(arrowstyle='->', color='green', lw=1)) +ax.annotate(f'x ≈ {x2:.2f}', xy=(x2, 0), xytext=(x2 + 0.3, -1.5), fontsize=10, ha='center', + arrowprops=dict(arrowstyle='->', color='green', lw=1)) + +# 绘制对称轴 +ax.axvline(x=vertex_x, color='purple', linestyle='--', linewidth=1.5, alpha=0.7, label=f'对称轴 x = {vertex_x}') + +# 绘制参考线 +ax.axhline(y=0, color='gray', linestyle='-', linewidth=0.8, alpha=0.5) +ax.axhline(y=vertex_y, color='orange', linestyle=':', linewidth=1.5, alpha=0.7, label=f'最大值 y = {vertex_y}') + +# 填充顶点到 x 轴的区域(可视化最大值) +x_fill = np.linspace(x1, x2, 100) +y_fill = f(x_fill) +ax.fill_between(x_fill, y_fill, 0, alpha=0.15, color='blue', label='函数值 > 0 的区域') + +# 设置坐标轴 +ax.set_xlim(-1, 5) +ax.set_ylim(-4, 7) +ax.set_xlabel('x', fontsize=14) +ax.set_ylabel('y', fontsize=14) +ax.set_title(r'二次函数 $y = -2x^2 + 8x - 3$ 图像', fontsize=16, fontweight='bold') + +# 添加网格 +ax.grid(True, linestyle='--', alpha=0.4) + +# 添加图例 +ax.legend(loc='lower right', fontsize=11, framealpha=0.9) + +# 添加说明文字框 +textstr = '\n'.join([ + '关键信息:', + f'• 顶点坐标: ({vertex_x}, {vertex_y})', + f'• 最大值: {vertex_y}', + f'• 开口方向: 向下 (a < 0)', + f'• 对称轴: x = {vertex_x}' +]) +props = dict(boxstyle='round', facecolor='white', edgecolor='gray', alpha=0.9) +ax.text(0.02, 0.98, textstr, transform=ax.transAxes, fontsize=11, + verticalalignment='top', bbox=props) + +plt.tight_layout() +plt.savefig('figure.png', bbox_inches='tight', facecolor='white') +plt.close() + +print("图像已保存: figure.png") diff --git a/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/report.md b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/report.md new file mode 100644 index 0000000..bac0eef --- /dev/null +++ b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/report.md @@ -0,0 +1,116 @@ +# 二次函数 $y = -2x^2 + 8x - 3$ 顶点与最值 - 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 + +对于二次函数 $y = -2x^2 + 8x - 3$: + +**顶点坐标为 $(2, 5)$**。由于二次项系数 $a = -2 < 0$,抛物线开口向下,因此函数在顶点处取得**最大值 $y_{max} = 5$**,此时 $x = 2$。 + +换句话说,当 $x = 2$ 时,函数值达到最大,为 $5$;当 $x$ 偏离 $2$ 时(无论向左还是向右),函数值都会减小。 + +## 3. 📈 可视化 + +![函数图像](figure.png) + +**图表说明**: + +- **蓝色曲线**:二次函数 $y = -2x^2 + 8x - 3$ 的图像 +- **红色圆点**:顶点 $(2, 5)$,即函数的最高点 +- **绿色方块**:函数与 $x$ 轴的两个交点(零点) +- **紫色虚线**:对称轴 $x = 2$ +- **橙色点线**:最大值参考线 $y = 5$ +- **浅蓝色区域**:函数值大于零的区域 + +## 4. 🧠 数学建模与解题过程 + +
+点击展开 + +**问题分析**:这是一个关于二次函数顶点和最值的基本问题。对于一般形式的二次函数 $y = ax^2 + bx + c$,其图像是一条抛物线,顶点坐标和最值可以通过多种方法求解。 + +**方法选择**:本题采用三种方法相互验证: + +### 方法一:顶点公式 + +对于 $y = ax^2 + bx + c$,顶点坐标为: + +$$\left( -\frac{b}{2a}, \frac{4ac - b^2}{4a} \right)$$ + +本题中 $a = -2$,$b = 8$,$c = -3$,代入得: + +$$x_{顶点} = -\frac{8}{2 \times (-2)} = -\frac{8}{-4} = 2$$ + +$$y_{顶点} = \frac{4 \times (-2) \times (-3) - 8^2}{4 \times (-2)} = \frac{24 - 64}{-8} = \frac{-40}{-8} = 5$$ + +### 方法二:求导法 + +对函数求导: + +$$y' = \frac{d}{dx}(-2x^2 + 8x - 3) = -4x + 8$$ + +令 $y' = 0$,解得 $x = 2$。 + +将 $x = 2$ 代入原函数:$y = -2(2)^2 + 8(2) - 3 = -8 + 16 - 3 = 5$ + +### 方法三:配方法 + +$$y = -2x^2 + 8x - 3$$ + +$$= -2(x^2 - 4x) - 3$$ + +$$= -2(x^2 - 4x + 4 - 4) - 3$$ + +$$= -2(x - 2)^2 + 8 - 3$$ + +$$= -2(x - 2)^2 + 5$$ + +顶点式为 $y = -2(x - 2)^2 + 5$,直接读出顶点 $(2, 5)$。 + +**结论**:三种方法结果一致,顶点为 $(2, 5)$。由于 $a = -2 < 0$,抛物线开口向下,函数在 $x = 2$ 处取得最大值 $5$。 + +
+ +## 5. 📊 运行结果 + +
+点击展开 + +``` +================================================== +二次函数 y = -2x² + 8x - 3 求解 +================================================== + +【方法1:顶点公式】 +a = -2, b = 8, c = -3 +顶点横坐标 x = -b/(2a) = -8/(2×-2) = 2.0 +顶点纵坐标 y = (4ac-b²)/(4a) = 5.0 + +【方法2:SymPy 求导验证】 +y' = 8 - 4*x +令 y' = 0,解得 x = [2] +将 x = 2 代入原函数:y = 5 + +【方法3:配方法】 +y = -2x² + 8x - 3 + = -2(x² - 4x) - 3 + = -2(x² - 4x + 4 - 4) - 3 + = -2(x - 2)² + 8 - 3 + = -2(x - 2)² + 5 +顶点形式:y = -2(x - 2)² + 5 + +================================================== +【最终结果】 +================================================== +顶点坐标:(2, 5) +由于 a = -2 < 0,抛物线开口向下 +函数最大值:y_max = 5(在 x = 2 处取得) +``` + +
diff --git a/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/solve.py b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/solve.py new file mode 100644 index 0000000..86e220a --- /dev/null +++ b/20260109_160634_quadratic_vertex-ClaudeCode+Opus45/solve.py @@ -0,0 +1,61 @@ +# /// script +# requires-python = ">=3.11" +# dependencies = ["sympy"] +# /// + +import sympy as sp + +# 定义符号变量 +x = sp.symbols('x', real=True) + +# 定义二次函数 +y = -2*x**2 + 8*x - 3 + +print("=" * 50) +print("二次函数 y = -2x² + 8x - 3 求解") +print("=" * 50) + +# 方法1:使用配方法/顶点公式 +# 对于 y = ax² + bx + c,顶点为 (-b/(2a), (4ac-b²)/(4a)) +a, b, c = -2, 8, -3 + +# 顶点横坐标 +x_vertex = -b / (2*a) +# 顶点纵坐标(最值) +y_vertex = (4*a*c - b**2) / (4*a) + +print(f"\n【方法1:顶点公式】") +print(f"a = {a}, b = {b}, c = {c}") +print(f"顶点横坐标 x = -b/(2a) = -{b}/(2×{a}) = {x_vertex}") +print(f"顶点纵坐标 y = (4ac-b²)/(4a) = {y_vertex}") + +# 方法2:使用 SymPy 求导 +print(f"\n【方法2:SymPy 求导验证】") +dy = sp.diff(y, x) +print(f"y' = {dy}") + +# 令导数为0,求驻点 +critical_points = sp.solve(dy, x) +print(f"令 y' = 0,解得 x = {critical_points}") + +if critical_points: + x_val = critical_points[0] + y_val = y.subs(x, x_val) + print(f"将 x = {x_val} 代入原函数:y = {y_val}") + +# 方法3:配方法展示 +print(f"\n【方法3:配方法】") +print("y = -2x² + 8x - 3") +print(" = -2(x² - 4x) - 3") +print(" = -2(x² - 4x + 4 - 4) - 3") +print(" = -2(x - 2)² + 8 - 3") +print(" = -2(x - 2)² + 5") +print("顶点形式:y = -2(x - 2)² + 5") + +# 最终结果 +print("\n" + "=" * 50) +print("【最终结果】") +print("=" * 50) +print(f"顶点坐标:({int(x_vertex)}, {int(y_vertex)})") +print(f"由于 a = {a} < 0,抛物线开口向下") +print(f"函数最大值:y_max = {int(y_vertex)}(在 x = {int(x_vertex)} 处取得)")