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plt + +# 设置中文字体 +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 y = -2x² + 8x - 3 +def f(x): + return -2*x**2 + 8*x - 3 + +# 顶点坐标 +x_vertex = 2 +y_vertex = 5 + +# 生成 x 值范围(以顶点为中心,左右各延伸适当距离) +x = np.linspace(-1, 5, 500) +y = f(x) + +# 绘制抛物线 +ax.plot(x, y, 'b-', linewidth=2.5, label=r'$y = -2x^2 + 8x - 3$') + +# 标注顶点 +ax.plot(x_vertex, y_vertex, 'ro', markersize=12, zorder=5, label=f'顶点 ({x_vertex}, {y_vertex})') +ax.annotate(f'顶点\n({x_vertex}, {y_vertex})', + xy=(x_vertex, y_vertex), + xytext=(x_vertex + 0.8, y_vertex + 0.5), + fontsize=12, + ha='left', + arrowprops=dict(arrowstyle='->', color='red', lw=1.5)) + +# 绘制对称轴 +ax.axvline(x=x_vertex, color='green', linestyle='--', linewidth=1.5, alpha=0.7, label=f'对称轴 x = {x_vertex}') + +# 绘制最大值水平线 +ax.axhline(y=y_vertex, color='orange', linestyle=':', linewidth=1.5, alpha=0.7, label=f'最大值 y = {y_vertex}') + +# 绘制坐标轴 +ax.axhline(y=0, color='gray', linewidth=0.8) +ax.axvline(x=0, color='gray', linewidth=0.8) + +# 求与 x 轴的交点(如果有) +# -2x² + 8x - 3 = 0 +# x = (8 ± √(64-24)) / (-4) = (8 ± √40) / (-4) +discriminant = 64 - 24 # b² - 4ac = 64 - 24 = 40 +x1 = (8 - np.sqrt(discriminant)) / 4 +x2 = (8 + np.sqrt(discriminant)) / 4 +ax.plot([x1, x2], [0, 0], 'g^', markersize=10, label=f'与x轴交点') +ax.annotate(f'({x1:.2f}, 0)', xy=(x1, 0), xytext=(x1-0.3, -1.5), fontsize=10, ha='center') +ax.annotate(f'({x2:.2f}, 0)', xy=(x2, 0), xytext=(x2+0.3, -1.5), fontsize=10, ha='center') + +# 与 y 轴的交点 +y_intercept = f(0) +ax.plot(0, y_intercept, 'ms', markersize=10, label=f'与y轴交点 (0, {y_intercept})') + +# 设置坐标轴范围 +ax.set_xlim(-1.5, 5.5) +ax.set_ylim(-5, 7) + +# 设置网格 +ax.grid(True, linestyle='--', alpha=0.5) + +# 设置标签和标题 +ax.set_xlabel('x', fontsize=14) +ax.set_ylabel('y', fontsize=14) +ax.set_title(r'二次函数 $y = -2x^2 + 8x - 3$ 的图像', fontsize=16, fontweight='bold') + +# 添加图例 +ax.legend(loc='lower right', fontsize=11) + +# 添加关键信息文本框 +textstr = '\n'.join([ + r'$y = -2x^2 + 8x - 3$', + r'$= -2(x-2)^2 + 5$', + '', + f'顶点: (2, 5)', + f'最大值: 5', + f'对称轴: x = 2' +]) +props = dict(boxstyle='round', facecolor='wheat', alpha=0.8) +ax.text(0.02, 0.98, textstr, transform=ax.transAxes, fontsize=11, + verticalalignment='top', bbox=props) + +# 保存图像 +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_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/report.md b/20260109_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/report.md new file mode 100644 index 0000000..178dba0 --- /dev/null +++ b/20260109_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/report.md @@ -0,0 +1,134 @@ +# 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 + +这是一个开口向下的抛物线(因为二次项系数 $a = -2 < 0$),通过配方可以将其化为顶点式 $y = -2(x-2)^2 + 5$。 + +**顶点坐标为 $(2, 5)$**,这意味着抛物线的对称轴是直线 $x = 2$。 + +由于抛物线开口向下,函数在顶点处取得**最大值 $y_{\max} = 5$**,此时 $x = 2$。 + +## 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 $(2, 5)$,即函数取得最大值的位置 +- **绿色虚线**:对称轴 $x = 2$ +- **橙色点线**:最大值水平线 $y = 5$ +- **绿色三角**:与 $x$ 轴的两个交点 +- **紫色方块**:与 $y$ 轴的交点 $(0, -3)$ + +## 4. 🧠 数学建模与解题过程 + +
+点击展开 + +### 问题分析 + +这是一个标准的二次函数求顶点问题。二次函数的一般形式为 $y = ax^2 + bx + c$,其中: +- $a = -2$(决定开口方向和宽窄) +- $b = 8$ +- $c = -3$ + +### 方法一:配方法 + +将二次函数化为顶点式 $y = a(x-h)^2 + k$: + +$$ +\begin{aligned} +y &= -2x^2 + 8x - 3 \\ +&= -2(x^2 - 4x) - 3 \\ +&= -2(x^2 - 4x + 4 - 4) - 3 \\ +&= -2(x - 2)^2 + 8 - 3 \\ +&= -2(x - 2)^2 + 5 +\end{aligned} +$$ + +从顶点式可直接读出:顶点坐标为 $(2, 5)$。 + +### 方法二:公式法 + +对于二次函数 $y = ax^2 + bx + c$,顶点坐标公式为: + +$$ +x_{\text{顶点}} = -\frac{b}{2a} = -\frac{8}{2 \times (-2)} = -\frac{8}{-4} = 2 +$$ + +将 $x = 2$ 代入原函数: + +$$ +y_{\text{顶点}} = -2 \times 2^2 + 8 \times 2 - 3 = -8 + 16 - 3 = 5 +$$ + +### 方法三:求导法(微积分验证) + +对函数求导: + +$$ +y' = \frac{d}{dx}(-2x^2 + 8x - 3) = -4x + 8 +$$ + +令 $y' = 0$,解得 $x = 2$。 + +二阶导数 $y'' = -4 < 0$,确认 $x = 2$ 处为极大值点。 + +### 最大值判断 + +由于 $a = -2 < 0$,抛物线开口向下,函数在顶点处取得最大值: + +$$ +y_{\max} = 5 \quad (\text{当 } x = 2 \text{ 时}) +$$ + +
+ +## 5. 📊 运行结果 + +
+点击展开 + +``` +================================================== +二次函数 y = -2x² + 8x - 3 求解 +================================================== + +1. 顶点坐标 + 使用公式 x = -b/(2a) = -8/(2×-2) = 2.0 + 代入求 y = -2×2.0² + 8×2.0 + (-3) = 5.0 + 顶点坐标: (2, 5) + +2. 函数最大值 + 由于 a = -2 < 0,抛物线开口向下 + 函数在顶点处取得最大值 + 最大值: y_max = 5 + +================================================== +SymPy 符号计算验证 +================================================== + +导数: y' = 8 - 4*x +令 y' = 0,解得 x = [2] +二阶导数: y'' = -4 < 0,确认为最大值点 + +顶点坐标: (2, 5) +最大值: 5 + +================================================== +配方形式验证 +================================================== +配方形式: y = -2(x - 2)² + 5 +展开验证: -2*x**2 + 8*x - 3 +与原式相等: True +``` + +
diff --git a/20260109_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/solve.py b/20260109_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/solve.py new file mode 100644 index 0000000..fd33921 --- /dev/null +++ b/20260109_162500_quadratic_vertex_max-工具GithubCopilot模型gemini-claude-opus-4-5-thinking/solve.py @@ -0,0 +1,71 @@ +# /// 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:配方法求顶点 +# y = -2(x² - 4x) - 3 +# y = -2(x² - 4x + 4 - 4) - 3 +# y = -2(x - 2)² + 8 - 3 +# y = -2(x - 2)² + 5 + +# 方法2:使用公式 x_vertex = -b/(2a) +a = -2 +b = 8 +c = -3 + +x_vertex = -b / (2*a) +y_vertex = a * x_vertex**2 + b * x_vertex + c + +print(f"\n1. 顶点坐标") +print(f" 使用公式 x = -b/(2a) = -{b}/(2×{a}) = {x_vertex}") +print(f" 代入求 y = {a}×{x_vertex}² + {b}×{x_vertex} + ({c}) = {y_vertex}") +print(f" 顶点坐标: ({int(x_vertex)}, {int(y_vertex)})") + +print(f"\n2. 函数最大值") +print(f" 由于 a = {a} < 0,抛物线开口向下") +print(f" 函数在顶点处取得最大值") +print(f" 最大值: y_max = {int(y_vertex)}") + +# 使用 SymPy 验证 +print("\n" + "=" * 50) +print("SymPy 符号计算验证") +print("=" * 50) + +# 求导数找极值点 +dy = sp.diff(y, x) +critical_points = sp.solve(dy, x) +print(f"\n导数: y' = {dy}") +print(f"令 y' = 0,解得 x = {critical_points}") + +# 验证是最大值(二阶导数 < 0) +d2y = sp.diff(dy, x) +print(f"二阶导数: y'' = {d2y} < 0,确认为最大值点") + +# 计算顶点处的函数值 +x_v = critical_points[0] +y_v = y.subs(x, x_v) +print(f"\n顶点坐标: ({x_v}, {y_v})") +print(f"最大值: {y_v}") + +# 配方形式 +print("\n" + "=" * 50) +print("配方形式验证") +print("=" * 50) +vertex_form = -2*(x - 2)**2 + 5 +expanded = sp.expand(vertex_form) +print(f"配方形式: y = -2(x - 2)² + 5") +print(f"展开验证: {expanded}") +print(f"与原式相等: {sp.simplify(expanded - y) == 0}")