以抛物线为载体的一题多问综合题1VIP免费

xy12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCo已知抛物线，与x轴交于A（－3，0）、B（1，0），与y轴交于点C，（1）求抛物线的解析式及顶点坐标；（2）在平面直角坐标系中，是否存在点D，是以A、B、C、D为顶点的四边形为平行四边形，若存在，请直接写出点D的坐标，若不存在，请说明理由．（3）在抛物线的对称轴上是否存在点M，使△BCM为直角三角形，若存在，请求出点M的坐标，若不存在，请说明理由．xy(-2,-3)(-4,3)(4,3)D3D1D212345–1–2–3–4–512345–1–2–3–4–5ABCo，（－1，4）xyHGM12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCo，xyHGM12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCoxyHM12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCoxy12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCo（4）在抛物线的对称轴上是否存在点P，使△BCP为等腰三角形，若存在，请求出点P的坐标，若不存在，请说明理由．xyP1P212345–1–2–3–412345–1–2–3–4–5ABCoxyP312345–1–2–3–4–5–612345–1–2–3CBAoxyP5舍去P412345–1–2–3–4–5–612345–1–2–3–4–5CBAoxy12345–1–2–3–4–5–612345–1–2–3–4–5–6ABCo（5）在抛物线的对称轴上是否存在点Q，使△ACQ为直角三角形，若存在，请求出点Q的坐标，若不存在，请说明理由．xyQ112345–1–2–3–4–512345–1–2–3–4–5ABCoxyIQ212345–1–2–3–4–512345–1–2–3–4–5ABCoxyO1Q3Q412345–1–2–3–4–512345–1–2–3–4–5ABCo,xyFE12345–1–2–3–412345–1–2–3–4–5–6ABCo（6）若点E从点C出发，沿线段CB向点B运动，F从点B出发，沿线段BA向点A运动，两点同时出发，速度分别为每秒个单位长度和每秒3个单位长度，点F运动到A即停止，设运动时间为t秒．①设△BEF的面积为S，求S与t之间的函数关系式，并求t为何值时，△BEF的面积最大，最大值是多少？②当△BEF为等腰三角形时，求运动时间t的值．BF=BE，，EF=EB，，FE=FB，