数学 - Python - 代数学 - 1次関数、2次関数 - 通る点、頂点、直線、x軸、接点、判別式

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第6章(1次方程式、2次方程式)、練習問題6の解答を求めてみる。

1. 頂点座標が（3，-1）の2次関数は
  
  $y = a {(x - 3)}^{2} - 1$
  
  これが点（4，2）を通ればいいのでもある放物線は
  
  $\begin{array}{l} 2 = a {(4 - 3)}^{2} - 1 \\ a = 3 \\ y = 3 {(x - 3)}^{2} - 1 \\ = 3 x^{2} - 18 x + 26 \end{array}$
2. $y = a x^{2} + b x + c$
  
  ${\begin{cases} b^{2} - 4 a c = 0 \\ a + b + c = 1 \\ 16 a + 4 b + c = 4 \end{cases}$
  
  $\begin{array}{l} c = 1 - a - b \\ 16 a + 4 b + 1 - a - b = 4 \\ 15 a + 3 b = 3 \\ b = \frac{3 - 15 a}{3} = 1 - 5 a \\ c = 1 - a - 1 + 5 a = 4 a \end{array}$
  
  $\begin{array}{l} {(1 - 5 a)}^{2} - 16 a^{2} = 0 \\ 1 - 10 a + 25 a^{2} - 16 a^{2} = 0 \\ 9 a^{2} - 10 a + 1 = 0 \\ a = \frac{5 \pm \sqrt{25 - 9}}{9} \\ = \frac{5 \pm 4}{9} \\ = 1, \frac{1}{9} \end{array}$
  
  $\begin{array}{l} b = - 4, \frac{4}{9} \\ c = 4, \frac{4}{9} \end{array}$
  
  $\begin{array}{l} y = x^{2} - 4 x + 4 \\ y = \frac{1}{9} x^{2} + \frac{4}{9} x + \frac{4}{9} \end{array}$
3. $\begin{array}{l} y = a x^{2} + b x + c \\ = a {(x + \frac{b}{2 a})}^{2} - \frac{b^{2} - 4 a c}{4 a} \end{array}$
  
  ${\begin{cases} 4 = a - b + c \\ - 8 = 4 a + 2 b + c \\ 2 (- \frac{b}{2 a}) - 4 (- \frac{b^{2} - 4 a c}{4 a}) + 27 = 0 \end{cases}$
  
  $\begin{array}{l} 12 = - 3 a - 3 b \\ a = \frac{- 12 - 3 b}{3} = - b - 4 \end{array}$
  
  $\begin{array}{l} 4 = - b - 4 - b + c \\ c = 2 b + 8 \end{array}$
  
  $- b + b^{2} - 4 a c + 27 a = 0$
  
  $- b + b^{2} - 4 (- b - 4) (2 b + 8) + 27 (- b - 4) = 0$
  
  $\begin{array}{l} - b + b^{2} + 8 (b^{2} + 8 b + 16) - 27 b - 108 = 0 \\ 9 b^{2} + 36 b + 20 = 0 \\ b = \frac{- 18 \pm 12}{9} = - \frac{10}{3}, - \frac{2}{3} \end{array}$
  
  $\begin{array}{l} a = - \frac{2}{3}, - \frac{10}{3} \\ c = \frac{4}{3}, \frac{20}{3} \end{array}$
  
  $\begin{array}{l} y = - \frac{2}{3} x^{2} - \frac{10}{3} x + \frac{4}{3} \\ y = - \frac{10}{3} x^{2} - \frac{2}{3} x + \frac{20}{3} \end{array}$

#!/usr/bin/env python3 from sympy import symbols, plot, Rational from sympy.abc import x print('6.') p = plot(3 * x ** 2 - 18 * x + 26, -1, 2, x ** 2 - 4 * x + 4, x ** 2 / 9 + 4 * x / 9 + Rational(4, 9), 1, 4, -2 * x ** 2 / 3 - 10 * x / 3 + Rational(4, 3), -10 * x ** 2 / 3 - 2 * x / 3 + Rational(20, 3), (2 * x + 27) / 4, -8, (x, -10, 10), ylim=(-10, 10), legend=True, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color p.save('sample6.png') p.show()

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2020年6月15日月曜日

数学 - Python - 代数学 - 1次関数、2次関数 - 通る点、頂点、直線、x軸、接点、判別式

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