98 lines
2.9 KiB
Python
98 lines
2.9 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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# 设置中文字体
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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 y = -2x² + 8x - 3
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def f(x):
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return -2*x**2 + 8*x - 3
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# 顶点坐标
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x_vertex = 2
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y_vertex = 5
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# 生成 x 值范围（以顶点为中心，左右各延伸适当距离）
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x = np.linspace(-1, 5, 500)
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y = f(x)
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# 绘制抛物线
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ax.plot(x, y, 'b-', linewidth=2.5, label=r'$y = -2x^2 + 8x - 3$')
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# 标注顶点
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ax.plot(x_vertex, y_vertex, 'ro', markersize=12, zorder=5, label=f'顶点 ({x_vertex}, {y_vertex})')
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ax.annotate(f'顶点\n({x_vertex}, {y_vertex})',
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xy=(x_vertex, y_vertex),
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xytext=(x_vertex + 0.8, y_vertex + 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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# 绘制对称轴
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ax.axvline(x=x_vertex, color='green', linestyle='--', linewidth=1.5, alpha=0.7, label=f'对称轴 x = {x_vertex}')
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# 绘制最大值水平线
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ax.axhline(y=y_vertex, color='orange', linestyle=':', linewidth=1.5, alpha=0.7, label=f'最大值 y = {y_vertex}')
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# 绘制坐标轴
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ax.axhline(y=0, color='gray', linewidth=0.8)
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ax.axvline(x=0, color='gray', linewidth=0.8)
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# 求与 x 轴的交点（如果有）
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# -2x² + 8x - 3 = 0
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# x = (8 ± √(64-24)) / (-4) = (8 ± √40) / (-4)
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discriminant = 64 - 24 # b² - 4ac = 64 - 24 = 40
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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.plot([x1, x2], [0, 0], 'g^', markersize=10, label=f'与x轴交点')
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ax.annotate(f'({x1:.2f}, 0)', xy=(x1, 0), xytext=(x1-0.3, -1.5), fontsize=10, ha='center')
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ax.annotate(f'({x2:.2f}, 0)', xy=(x2, 0), xytext=(x2+0.3, -1.5), fontsize=10, ha='center')
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# 与 y 轴的交点
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y_intercept = f(0)
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ax.plot(0, y_intercept, 'ms', markersize=10, label=f'与y轴交点 (0, {y_intercept})')
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# 设置坐标轴范围
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ax.set_xlim(-1.5, 5.5)
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ax.set_ylim(-5, 7)
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# 设置网格
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ax.grid(True, linestyle='--', alpha=0.5)
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# 设置标签和标题
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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 $y = -2x^2 + 8x - 3$ 的图像', fontsize=16, fontweight='bold')
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# 添加图例
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ax.legend(loc='lower right', fontsize=11)
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# 添加关键信息文本框
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textstr = '\n'.join([
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r'$y = -2x^2 + 8x - 3$',
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r'$= -2(x-2)^2 + 5$',
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'',
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f'顶点: (2, 5)',
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f'最大值: 5',
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f'对称轴: x = 2'
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])
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props = dict(boxstyle='round', facecolor='wheat', alpha=0.8)
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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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# 保存图像
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plt.tight_layout()
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plt.savefig('figure.png', bbox_inches='tight', dpi=150)
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plt.close()
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print("图像已保存: figure.png")
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