如图,已知抛物线y=ax2+bx+c经过O(0,0)

1个回答

  • (1)

    经过O,A(4,0),可表达为y = ax(x - 4)

    经过B(3,√3):-3a = √3

    a = -√3/3,b = 4√3/3

    抛物线的函数解析式:y = -√3/3(x² - 4x)

    (2)

    t秒时:P(t,0)

    (i) Q在AB上

    AB的解析式:(y - 0)/(√3 - 0) = (x - 4)/(3 - 4),y = -√3(x - 4)

    AB = √[(3 - 4)² + (√3 - 0)²] = 2

    0 < t < 2

    设Q(q,-√3(q - 4)),显然0 < q < 4

    AQ² = OP² = t² = (q - 4)² + [-√3(q - 4) - 0]² = 4(q - 4)²

    q = 4 - t/2 (舍去q = 4 + t/2)

    Q(4 - t/2,√3t/2)

    S = (1/2)PA*Q的纵坐标 = (1/2)(4 - t)*√3t/2 = √3(4 - t)t/4

    (ii)Q在BC上

    2 < t < 4

    QB = t - AB = t - 2,Q的横坐标 = B的横坐标 - QB = 3 - (t - 2) = 5 - t

    Q(5 - t,√3)

    S = (1/2)PA*Q的纵坐标 = (1/2)(4 - t)√3 = √3(4 - t)/2