将抛物线y=x^2向下平移后,设它与x轴的两个交点分别为,A,B.且抛物线的顶点为c

1.设抛物线y=x²向下平移a个单位(a > 0),新抛物线的方程为y = x² - a = (x + √a)(x - √a)
顶点为C(0,-a)
y轴为对称轴,AC = BC
A(-√a,0),B(√a,0)
AB = 2√a
BC = AC = √[(√a - 0)² + (0 + a)²] = √(a + a²)
三角形ABC为等边三角形,AB = BC,2√a = √(a + a²),a² = 3a
a = 0 (舍去)或a = 3
抛物线解析式:y = x² - 3
(2)y轴为对称轴,AC = BC,若三角形ABC为等腰直角三角形,只需AC⊥BC即可.
AC的斜率m = (-a - 0)/(0 + √a) = -√a
BC的斜率n = (-a - 0)/(0 - √a) = √a
AC⊥BC,mn = -√a*√a = -1
抛物线解析式:y = x² - 1