Question

把函數(shù)的圖象y=cos(x+

4π

3

)沿x軸平移|φ|個(gè)單位，所得圖象關(guān)于原點(diǎn)對(duì)稱，則|φ|的最小值是

 

．

Answer 1

分析：先根據(jù)圖象平移的法則推斷出圖象平移后函數(shù)的方程，進(jìn)而根據(jù)原點(diǎn)對(duì)稱可推斷出函數(shù)為奇函數(shù)，進(jìn)而求得φ的集合，則|φ|的最小值可得．

解答：解：將函數(shù)的圖象y=cos(x+

4π

3

)沿x軸平移|φ|個(gè)單位后得到y=cos(x+φ+

4π

3

)
∵圖象關(guān)于原點(diǎn)對(duì)稱
∴cos(-x+φ+

4π

3

)=-cos(x+φ+

4π

3

)
∴cosxcos(φ+

4π

3

)+sinxsin(φ+

4π

3

)=-cosxcos(φ+

4π

3

)+sinxsin(φ+

4π

3

)
∴cos(φ+

4π

3

)=0∴φ+

4π

3

=

π

2

+kπ
∴φ=-

5π

6

+kπ，∴當(dāng)k=1時(shí)，|φ|=

π

6

最小
故答案為：

π

6

點(diǎn)評(píng)：本題主要考查了函數(shù)y=Asin（ωx+φ）的圖象變換，三角函數(shù)的奇偶性的應(yīng)用．考查了學(xué)生綜合分析問(wèn)題和解決問(wèn)題的能力．