抛物线y=kx²+4x+k²-1(k为正整数)经过坐标原点,与x轴的另一个交点为A,

抛物线y=kx²+4x+k²-1(k为正整数)经过坐标原点,与x轴的另一个交点为A,
抛物线y=kx²+4x+k²-1(k为正整数)经过坐标原点,与x轴的另一个交点为A,

抛物线y=kx²+4x+k²-1(k为正整数)经过坐标原点,与x轴的另一个交点为A,
(1)
过原点,x = 0,y = k² - 1 = 0
k = 1 (舍去k = -1 < 0)
y = x² + 4x = x(x + 4) = (x + 2)² - 4
A(-4,0),B(-2,-4)
面积为原来的1/2,则A,B的坐标均变为原来的1/√2
A -> C(-2√2,0)
B -> D(-√2,-2√2)
过O,C,则可表达为y = ax(x + 2√2)
x = -√2,y = -2a = -2√2
a = √2
F:y = √2x(x + 2√2) = √2x² + 4x
(2)
令M(m,√2m² + 4m),m < -2√2
N与M关于对称轴x = -√2对称,N的横坐标n = -2√2-m,纵坐标与M的相同
周长l = 2(PM + PQ)
= 2(√2m² + 4m -2√2-m - m)
= 2√2(m + √2/2)² - 5√2
此为以m = -√2/2为对称轴的抛物线(开口向上),x趋近于负无穷时,l为无穷大,即无最大值．