已知函數(shù)f（x）=sin（wx+Q）（w>0,絕對(duì)值Q>π/2）在同一周期內(nèi),當(dāng)x=pai／4時(shí),y取最大值1,當(dāng)7pai／12時(shí),y取最小值-1
⑴求函數(shù)y=f（x）
⑵函數(shù)y=sinx的圖像經(jīng)過怎樣的變換可得到y(tǒng)=f（x）
⑶若函數(shù)f（x）滿足方程f（x）=（0＜a＜1）,求在[0,2pai]內(nèi)所有實(shí)數(shù)根之和
(1)解析：∵函數(shù)f（x）=sin（wx+Q）（w>0,絕對(duì)值Q>π/2）在同一周期內(nèi),當(dāng)x=pai／4時(shí),y取最大值1,當(dāng)7pai／12時(shí),y取最小值-1
∴T/2=7π/12-π/4=π/3==>T=2π/3==>w=2π/T=3
∴f(x)=sin(3x+Q)==> f(π/4)=sin(3π/4+Q)=1==>3π/4+Q=π/2==>Q=-π/4
∴f(x)=sin(3x-π/4)
(2)解析：函數(shù)y=sinx
A:將函數(shù)y水平右移π/4單位得y=sin(x-π/4)
B:將X軸坐標(biāo)單位水平壓縮到原單位的1/3,得y=sin(3x-π/4)
或
A:將X軸坐標(biāo)單位水平壓縮到原單位的1/3,得y=sin(3x)
B:將函數(shù)y水平右移π/12單位得y=sin(3(x-π/12))= sin(3x-π/4)
(3)解析：方程f（x）=a（0＜a＜1）
sin(3x-π/4)= √2/2==>3x-π/4=2kπ+π/4==>x=2kπ/3+π/6；3x-π/4=2kπ+3π/4==>x=2kπ/3+π/3
sin(3x-π/4)=1/2==>3x-π/4=2kπ+π/6==>x=2kπ/3+5π/36；3x-π/4=2kπ+5π/6==>x=2kπ/3+13π/36
sin(3x-π/4)= √3/2==>3x-π/4=2kπ+π/3==>x=2kπ/3+7π/36；3x-π/4=2kπ+4π/6==>x=2kπ/3+11π/36
在[0,2pai]內(nèi)所有實(shí)數(shù)根之和
π/6+2π/6+5π/6+6π/6+9π/6+10π/6=11π/2
5π/36+13π/36+29π/36+37π/36+53π/36+61π/36=11π/2
7π/36+11π/36+31π/36+35π/36+55π/36+59π/36=11π/2可見,無論a∈(0,1)取何值,在[0,2pai]內(nèi)所有實(shí)數(shù)根之和均為11π/2