【答案】
分析:(1)點(diǎn)C在直線AB:y=-2x+42上,將C點(diǎn)的橫坐標(biāo)代入即可求出C點(diǎn)的縱坐標(biāo),同理可知:D點(diǎn)在直線OB:y=x上,將D點(diǎn)的橫坐標(biāo)代入解析式即可求出D點(diǎn)的縱坐標(biāo);
(2)拋物線y=ax
2-2x+c經(jīng)過C、D兩點(diǎn),列出關(guān)于a和c二元一次方程組,解出a和c即可;
(3)根據(jù)Q為線段OB上一點(diǎn),P、Q兩點(diǎn)的縱坐標(biāo)都為5,則可以求出Q點(diǎn)的坐標(biāo),又知P點(diǎn)在拋物線上,求出P點(diǎn)的坐標(biāo),P、Q兩點(diǎn)的橫坐標(biāo)的差的絕對值即為線段PQ的長;
(4)根據(jù)PQ⊥x軸,可知P和Q兩點(diǎn)的橫坐標(biāo)相同,都為m,用含m的代數(shù)式分別表示P、Q兩點(diǎn)的坐標(biāo),求出B點(diǎn)的坐標(biāo),分兩種情況討論:①Q(mào)是線段OB上的一點(diǎn);②Q是線段AB上的一點(diǎn).分別求出d與m之間的函數(shù)解析式,根據(jù)二次函數(shù)的性質(zhì),即可求出d隨m的增大而減小時(shí)m的取值范圍.
解答:解:(1)∵點(diǎn)C在直線AB:y=-2x+42上,且C點(diǎn)的橫坐標(biāo)為16,
∴y=-2×16+42=10,即點(diǎn)C的縱坐標(biāo)為10;
∵D點(diǎn)在直線OB:y=x上,且D點(diǎn)的橫坐標(biāo)為4,
∴點(diǎn)D的縱坐標(biāo)為4;
(2)由(1)知點(diǎn)C的坐標(biāo)為(16,10),點(diǎn)D的坐標(biāo)為(4,4),
∵拋物線y=ax
2-2x+c經(jīng)過C、D兩點(diǎn),
∴
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,
解得:
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.
∴拋物線的解析式為y=
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x
2-2x+10;
(3)∵Q為線段OB上一點(diǎn),縱坐標(biāo)為5,
∴Q點(diǎn)的橫坐標(biāo)也為5,
∵點(diǎn)P在拋物線上,縱坐標(biāo)為5,
∴
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x
2-2x+10=5,
解得x
1=8+2
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,x
2=8-2
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.
當(dāng)點(diǎn)P的坐標(biāo)為(8+2
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,5),點(diǎn)Q的坐標(biāo)為(5,5),線段PQ的長為2
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+3;
當(dāng)點(diǎn)P的坐標(biāo)為(8-2
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,5),點(diǎn)Q的坐標(biāo)為(5,5),線段PQ的長為2
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-3.
所以線段PQ的長為2
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+3或2
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-3;
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(4)∵PQ⊥x軸,
∴P、Q兩點(diǎn)的橫坐標(biāo)相同,都為m,
∴P(m,
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m
2-2m+10),Q(m,m)(此時(shí)Q在線段OB上)或Q(m,-2m+42)(此時(shí)Q在線段AB上).
由
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,
解得
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.
∴點(diǎn)B的坐標(biāo)為(14,14).
①當(dāng)點(diǎn)Q為線段OB上時(shí),如圖所示,
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在OD段,即當(dāng)0≤m<4時(shí),d=(
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m
2-2m+10)-m=
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m
2-3m+10=
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(m-12)
2-8,d隨m的增大而減�。�
在BD段,即當(dāng)4≤m≤14時(shí),d=m-(
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m
2-2m+10)=-
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m
2+3m-10=-
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(m-12)
2+8,
在對稱軸右側(cè),d隨m的增大而減小,即當(dāng)12<m≤14時(shí),d隨m的增大而減小.
則當(dāng)0≤m<4或12≤m≤14時(shí),d隨m的增大而減��;
②當(dāng)點(diǎn)Q為線段AB上時(shí),如圖所示,
在BC段,即當(dāng)14≤m<16時(shí),d=(-2m+42)-(
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m
2-2m+10)=-
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m
2+32,
在對稱軸右側(cè),d隨m的增大而減小,即當(dāng)14≤m<16時(shí),d隨m的增大而減小;
在CA段,即當(dāng)16≤m≤21時(shí),d=(
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m
2-2m+10)-(-2m+42)=
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m
2-32,
在對稱軸左側(cè),d隨m的增大而減小,m不滿足條件.
綜上所述,當(dāng)0≤m<4或12≤m<16時(shí),d隨m的增大而減小.
點(diǎn)評(píng):本題考查了二次函數(shù)的綜合題型,其中涉及到的知識(shí)點(diǎn)有運(yùn)用待定系數(shù)法求二次函數(shù)的解析式,函數(shù)圖象上點(diǎn)的坐標(biāo)特征,平行于坐標(biāo)軸上的兩點(diǎn)之間的距離,二次函數(shù)的增減性,難度中等,解題關(guān)鍵是運(yùn)用數(shù)形結(jié)合及分類討論的思想.