x = 0,y = 3/2
x = 2,y = 3/2 = 4(t + 1) + 4(t + 2) + 3/2
t = -3/2
y = -x²/2 + x + 3/2 = (-1/2)(x + 1)(x - 3)
B(-1,0),C(3,0)
二次函数的图像在B,C间的部分含B,C点向左平移n个单位,解析式变为:
y = -(1/2)(x + n + 1)(x + n - 3)
与x轴交于B'(-1-n,0),C'(3-n,0)
y = -x²/2 + x + 3/2 向上平移n个单位,解析式变为y = -x²/2 + x + 3/2 + n = 0
x₁,₂ = 1 ± √(2n + 4)
与x轴交于B"(1-√(2n + 4),0),C"(1 - √(2n + 4),0)
要使二者有公共点,只需B"在B'C'(含B',C')上:
-1 - n ≤ 1-√(2n + 4) ≤ 3 - n
n - 2 ≤ √(2n + 4) ≤ n + 2
(a) √(2n + 4) ≤ n + 2
2n + 4 ≤ n² + 4n + 4
n² + 2n ≥ 0
n ≥ 0 (舍去n ≤ -2)
(b) n - 2 ≤ √(2n + 4)
(i) 0 < n < 2,此时不等式总成立
(ii) n ≥ 2:
n² - 4n + 4 ≤ 2n + 4
n² - 6n ≤ 0
0 ≤ n ≤ 6
结合前提n ≥ 2,2 ≤ n ≤ 6
结合(i)(ii):0 < n ≤ 6
结合(a)(b):0 < n ≤ 6