问题 解答题
已知函数f(x)=2cos2x+2asinxcosx-1的图象关于直线x=
π
8
对称.
(Ⅰ)求a的值;
(Ⅱ)把函数y=f(x)的图象按向量
a
平移后与函数g(x)=
2
sin2x
-1的图象重合,求:
a
的坐标.
答案

解(1):f(x)=cos2x+asin2x…(2分)

=

1+a2
sin(2x+ϕ)…(4分)

f(

π
8
)=±
1+a2
=
2
2
(a+1)…(6分)

a=1…(8分)

另f(0)=f(

π
4
)⇒2=1+a∴a=1

(2)f(x)=

2
sin(2x+
π
4
)=
2
sin2(x+
π
8
)-------g(x)=
2
sin2x+1

f(x)向右移动

π
8
个单位向上移动1个单位即可得g(x)图象

b
=(
π
8
,1)….(14分)

完形填空

Taking part in charity is not just for grown- up people. Children can do it too! Melvin Sheppard, 14, an American student, has done charity work for eight years. He has also encouraged his classmates to join him.

Melvin studies at William Allen Middle School in Philadelphia. Recently, he collected $1,500 by himself and $487 from his classmates to donate to Cooper University Hospital.

Melvin was born nine weeks premature (早产) in Cooper University Hospital, weighing 2.6kg. His father told him how Cooper saved his life, so he felt thankful to the hospital.

Melvin's classmates joined him as soon as they found out about his kind behavior. Their teacher, Michael Berner, also held a class with them about helping others. “It wasn’t really about the money,” Berner said. “It was about the face that they were doing something good.”

“It is great, helping someone that I know,” said Jordan, Melvin's classmate. Melvin's neighbors also helped out.

Since Melvin was 8, he and his parents have given money to Cooper University Hospital every year. This year, Gary E. Stahl, the head of Cooper's division of neonatology (新生儿科), went to Melvin's school to accept the money.

The money goes to a part of the hospital that helps about 500 young patients every year. “When they leave, we call them graduates,” Stahl said, “We are pleased when our graduates and their families do well, like Melvin and his family.”  

小题1:How old was Melvin when he started to do charity work?

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小题2:What caused Melvin to raise money for Cooper University Hospital? 

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小题3:Did Melvin's teacher and classmates think his behavior was kind?

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小题4: How many young patients get help every year?

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小题5:What can we learn from Melvin?

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