三次函數的特徵
Author: 藍邦偉  老師

$ O $

$ 三次函數的特徵 $

$ 三次函數f(x)=ax^3+bx^2+cx+d\;的圖$
$形對稱於反曲點\left( -\frac{b}{3a},\;f(-\frac{b}{3a})  \right)  $
$Pf：$

$ f(x)=(ax³+bx²)+cx+d $

$ =a(x+\frac{b}{3a})³+cx+d-\frac{b²}{3a}x-\frac{b³}{27a²} $

$ =a(x+\frac{b}{3a})³+(c-\frac{b²}{3a})(x+\frac{b}{3a}) $
$\;\;\;\;+d-\frac{b³}{27a²}+\frac{b³}{9a²}-\frac{bc}{3a}$

$ =a(x+\frac{b}{3a})³+p(x+\frac{b}{3a})+q $
$where\;p=c-\frac{b²}{3a}=f'(-\frac{b}{3a}),$
$q=d-\frac{b³}{27a²}+\frac{b³}{9a²}-\frac{bc}{3a}=f(-\frac{b}{3a})$

$設\;r=-\frac{b}{3a} 時，\;(r,f(r))是函數f\;圖形上$
$唯一的反曲點，\;即：$

$f(x)=a(x-r)³+p(x-r)+f(r) $

$ \left\lbrace \begin{array}{ll} f(r+h)=ah³+ph+f(r)  \ f(r-h)=a(-h)³+p(-h)+f(r) \ \end{array}  \right.\  $
$,∀\;h∈\mathbb{R}$

$ ⇒f(r+h)+f(r-h)=2f(r) 即圖形上$
$的點(r-h,f(r-h))與(r+h,f(r+h))$
$都對稱於(r,f(r))$

$ 三次函數\;y=f(x)\;的圖形上有極大點$
$A(α,f(α))與極小點B(β,f(β))時，\;其連線$
$段的中點即為反曲點(\frac{α+β}{2},f(\frac{α+β}{2})) $
$Pf：$

$ 設f(x)=ax^3+bx^2+cx+d ，且：$
$f'(x)=3ax^2+2bx+c$
$\;\;\;\;\;\;\;\;\;=3a(x-α)(x-β),α<β$

$ 則：\;α=\frac{-b-\sqrt{b²-3ac}}{3a}=r-h $
$\;\;\;\;\;\;\;β=\frac{-b+\sqrt{b²-3ac}}{3a}=r+h$
$\;\;\;\;\;\;\;\;r=-\frac{b}{3a},h=\frac{\sqrt{b²-3ac}}{3a}$

$ 又反曲點(r,f(r)) 是函數f圖形上的對稱中$
$心故A與B對稱於反曲點(\frac{α+β}{2},f(\frac{α+β}{2}))$

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.6 * Corner[1] + 0.4 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
n = 0 (slider from 0 to 12, step 1)
A_1 = (-4, -2)
B_1 = (3, 3)
C_1 = Midpoint(A_1, B_1)
D_1 = (4.424940004944452, 2.1425476713903437)
f_1 = Polynomial({A_1, B_1, C_1, D_1})
A = TurningPoint(f_1)
B = Point(f_1)
B' = Mirror(B, A)
a = Segment(B, B')
C, D = Extremum(f_1)
b = Segment(C, D)