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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">如图，已知抛物线经过两点<i>A</i>（﹣3，0），<i>B</i>（0，3），且其对称轴为直线<i>x</i>＝﹣1．</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（1）求此抛物线的解析式；</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（2）若点<i>P</i>是抛物线上点<i>A</i>与点<i>B</i>之间的动点（不包括点<i>A</i>，点<i>B</i>），求△<i>PAB</i>的面积的最大值，并求出此时点<i>P</i>的坐标．</span></span></span></span></span><br />
                                                                                                                                                    <img height="157" src="/uploads/meth/20201226/fc664c3c9caf698ceb4b26f886b7389f.png" width="142" />

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<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">解：（1）∵抛物线对称轴是直线<i>x</i>＝﹣1且经过点<i>A</i>（﹣3，0）</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">由抛物线的对称性可知：抛物线还经过点（1，0）</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">设抛物线的解析式为<i>y</i>＝<i>a</i>（<i>x</i>﹣<i>x</i><sub>1</sub>）（<i>x</i>﹣<i>x</i><sub>2</sub>）（<i>a</i>≠0）</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">即：<i>y</i>＝<i>a</i>（<i>x</i>﹣1）（<i>x</i>+3）</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">把<i>B</i>（0，3）代入得：3＝﹣3<i>a</i></span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>a</i>＝﹣1</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴抛物线的解析式为：<i>y</i>＝﹣<i>x</i><sup>2</sup>﹣2<i>x</i>+3．</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（2）设直线<i>AB</i>的解析式为<i>y</i>＝<i>kx</i>+<i>b</i>，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∵<i>A</i>（﹣3，0），<i>B</i>（0，3），</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴</span></span></span></span></span><img height="38" src="/uploads/meth/20201226/1efa25c5265708514d75ee5b837958b4.png" width="68" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，                                            </span></span></span></span></span><img height="157" src="/uploads/meth/20201226/aeb512180fb7f78aed6a9aaf97abe5a3.png" width="142" /><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴直线<i>AB</i>为<i>y</i>＝<i>x</i>+3，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">作<i>PQ</i>⊥<i>x</i>轴于<i>Q</i>，交直线<i>AB</i>于<i>M</i>，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">设<i>P</i>（<i>x</i>，﹣<i>x</i><sup>2</sup>﹣2<i>x</i>+3），则<i>M</i>（<i>x</i>，<i>x</i>+3），</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>PM</i>＝﹣<i>x</i><sup>2</sup>﹣2<i>x</i>+3﹣（<i>x</i>+3）＝﹣<i>x</i><sup>2</sup>﹣3<i>x</i>，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴<i>S</i>＝</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/81ce6ccd05f3dd6ca71d52fd3739ec55.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（﹣<i>x</i><sup>2</sup>﹣3<i>x</i>）×3＝﹣</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/10575ef0686d549683b8757946937e82.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">（<i>x</i>+</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/6a374d7c27bad127a22fd68db7e07be8.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">）<sup>2</sup>+</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/e6162469f5b6b56b41ee05f4c4173147.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">．</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">当<i>x</i>＝﹣</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/71173c2b3cec859e82d6dad0a104396b.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">时，<i>S</i><sub>最大</sub>＝</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/df217956b05ee2537d3a69e74ef76d84.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，<i>y</i>＝﹣（﹣</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/b2f38c20572ec5e9ee1dfcd553066192.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">）<sup>2</sup>﹣2×（﹣</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/f21abffa8f7199a0813dbc896da7fba5.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">）+3＝</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/1c80d6d65a1ad838cdd9ba4fd1c4682a.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><br />
<span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">∴△<i>PAB</i>的面积的最大值为</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/c7eae7eb17f86e4aa404ab51703a7449.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，此时点<i>P</i>的坐标为（﹣</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/9cd383ef5470719339c95fbfbada62db.png" width="13" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">，</span></span></span></span></span><img height="35" src="/uploads/meth/20201226/3a5e01da9b15535b79e9b5913ceecfab.png" width="21" /><span style="font-size:10.5pt"><span style="line-height:150%"><span style="font-family:Calibri"><span style="line-height:150%"><span style="font-family:宋体">）</span></span></span></span></span>

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