91 lines
3.2 KiB
Python
91 lines
3.2 KiB
Python
# /// script
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# requires-python = ">=3.11"
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# dependencies = ["numpy", "matplotlib"]
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# ///
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import numpy as np
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import matplotlib.pyplot as plt
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# 设置字体
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plt.rcParams['font.sans-serif'] = ['WenQuanYi Micro Hei', 'Noto Sans CJK SC', 'Microsoft YaHei', 'SimHei', 'SimSun', 'DejaVu Sans']
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plt.rcParams['axes.unicode_minus'] = False
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# 创建图形
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fig, ax = plt.subplots(figsize=(10, 8), dpi=150)
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# 定义函数
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def f(x):
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return -2*x**2 + 8*x - 3
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# 生成 x 数据（根据顶点位置设置合理范围）
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x = np.linspace(-1, 5, 500)
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y = f(x)
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# 绘制函数曲线
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ax.plot(x, y, 'b-', linewidth=2.5, label=r'$y = -2x^2 + 8x - 3$')
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# 标记顶点
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vertex_x, vertex_y = 2, 5
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ax.scatter([vertex_x], [vertex_y], color='red', s=120, zorder=5, edgecolors='darkred', linewidths=2)
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ax.annotate(f'顶点 ({vertex_x}, {vertex_y})\n最大值',
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xy=(vertex_x, vertex_y),
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xytext=(vertex_x + 0.8, vertex_y + 0.5),
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fontsize=12,
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ha='left',
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arrowprops=dict(arrowstyle='->', color='red', lw=1.5),
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bbox=dict(boxstyle='round,pad=0.3', facecolor='lightyellow', edgecolor='orange'))
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# 标记与 x 轴的交点（求根）
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# -2x² + 8x - 3 = 0 → x = (8 ± √(64-24))/(-4) = (8 ± √40)/(-4)
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discriminant = 64 - 24
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x1 = (8 - np.sqrt(discriminant)) / 4
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x2 = (8 + np.sqrt(discriminant)) / 4
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ax.scatter([x1, x2], [0, 0], color='green', s=100, zorder=5, marker='s', edgecolors='darkgreen', linewidths=2)
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ax.annotate(f'x ≈ {x1:.2f}', xy=(x1, 0), xytext=(x1 - 0.3, -1.5), fontsize=10, ha='center',
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arrowprops=dict(arrowstyle='->', color='green', lw=1))
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ax.annotate(f'x ≈ {x2:.2f}', xy=(x2, 0), xytext=(x2 + 0.3, -1.5), fontsize=10, ha='center',
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arrowprops=dict(arrowstyle='->', color='green', lw=1))
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# 绘制对称轴
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ax.axvline(x=vertex_x, color='purple', linestyle='--', linewidth=1.5, alpha=0.7, label=f'对称轴 x = {vertex_x}')
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# 绘制参考线
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ax.axhline(y=0, color='gray', linestyle='-', linewidth=0.8, alpha=0.5)
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ax.axhline(y=vertex_y, color='orange', linestyle=':', linewidth=1.5, alpha=0.7, label=f'最大值 y = {vertex_y}')
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# 填充顶点到 x 轴的区域（可视化最大值）
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x_fill = np.linspace(x1, x2, 100)
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y_fill = f(x_fill)
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ax.fill_between(x_fill, y_fill, 0, alpha=0.15, color='blue', label='函数值 > 0 的区域')
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# 设置坐标轴
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ax.set_xlim(-1, 5)
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ax.set_ylim(-4, 7)
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ax.set_xlabel('x', fontsize=14)
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ax.set_ylabel('y', fontsize=14)
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ax.set_title(r'二次函数 $y = -2x^2 + 8x - 3$ 图像', fontsize=16, fontweight='bold')
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# 添加网格
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ax.grid(True, linestyle='--', alpha=0.4)
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# 添加图例
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ax.legend(loc='lower right', fontsize=11, framealpha=0.9)
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# 添加说明文字框
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textstr = '\n'.join([
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'关键信息:',
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f'• 顶点坐标: ({vertex_x}, {vertex_y})',
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f'• 最大值: {vertex_y}',
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f'• 开口方向: 向下 (a < 0)',
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f'• 对称轴: x = {vertex_x}'
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])
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props = dict(boxstyle='round', facecolor='white', edgecolor='gray', alpha=0.9)
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ax.text(0.02, 0.98, textstr, transform=ax.transAxes, fontsize=11,
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verticalalignment='top', bbox=props)
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plt.tight_layout()
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plt.savefig('figure.png', bbox_inches='tight', facecolor='white')
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plt.close()
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print("图像已保存: figure.png")
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