解:∵
![](http://thumb.1010pic.com/pic5/latex/44.png)
=(
![](http://thumb.1010pic.com/pic5/latex/21.png)
cosωx,sinωx),
![](http://thumb.1010pic.com/pic5/latex/45.png)
=(sinωx,0),
∴
![](http://thumb.1010pic.com/pic5/latex/44.png)
+
![](http://thumb.1010pic.com/pic5/latex/45.png)
=(
![](http://thumb.1010pic.com/pic5/latex/21.png)
cosωx+sinωx,sinωx).
故f(x)=(
![](http://thumb.1010pic.com/pic5/latex/44.png)
+
![](http://thumb.1010pic.com/pic5/latex/45.png)
)•
![](http://thumb.1010pic.com/pic5/latex/45.png)
+k=
![](http://thumb.1010pic.com/pic5/latex/21.png)
sinωxcosωx+sin
2ωx+k
=
![](http://thumb.1010pic.com/pic5/latex/376.png)
sin2ωx+
![](http://thumb.1010pic.com/pic5/latex/118585.png)
+k=
![](http://thumb.1010pic.com/pic5/latex/376.png)
sin2ωx-
![](http://thumb.1010pic.com/pic5/latex/13.png)
cos2ωx+
![](http://thumb.1010pic.com/pic5/latex/13.png)
+k
=sin(2ωx-
![](http://thumb.1010pic.com/pic5/latex/198.png)
)+k+
![](http://thumb.1010pic.com/pic5/latex/13.png)
.
(1)由题意可知
![](http://thumb.1010pic.com/pic5/latex/4356.png)
=
![](http://thumb.1010pic.com/pic5/latex/68115.png)
≥
![](http://thumb.1010pic.com/pic5/latex/73.png)
,∴ω≤1.
又ω>0,∴0<ω≤1.
(2)∵T=
![](http://thumb.1010pic.com/pic5/latex/18462.png)
=π,∴ω=1.
∴f(x)=sin(2x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
)+k+
![](http://thumb.1010pic.com/pic5/latex/13.png)
.
∵x∈[-
![](http://thumb.1010pic.com/pic5/latex/198.png)
,
![](http://thumb.1010pic.com/pic5/latex/198.png)
],∴2x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
∈[-
![](http://thumb.1010pic.com/pic5/latex/73.png)
,
![](http://thumb.1010pic.com/pic5/latex/198.png)
].
从而当2x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
=
![](http://thumb.1010pic.com/pic5/latex/198.png)
,即x=
![](http://thumb.1010pic.com/pic5/latex/198.png)
时,f(x)
max=f(
![](http://thumb.1010pic.com/pic5/latex/198.png)
)=sin
![](http://thumb.1010pic.com/pic5/latex/198.png)
+k+
![](http://thumb.1010pic.com/pic5/latex/13.png)
=k+1=
![](http://thumb.1010pic.com/pic5/latex/13.png)
,
∴k=-
![](http://thumb.1010pic.com/pic5/latex/13.png)
.故f(x)=sin(2x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
).
由函数y=sinx的图象向右平移
![](http://thumb.1010pic.com/pic5/latex/198.png)
个单位长度,得到函数y=sin(x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
)的图象,再将得到的函数图象上所有点的横坐标变为原来的
![](http://thumb.1010pic.com/pic5/latex/13.png)
倍(纵坐标不变),得到函数y=sin(2x-
![](http://thumb.1010pic.com/pic5/latex/198.png)
)的图象.
分析:利用向量的数量积,化简函数的表达式,通过二倍角、两角差的正弦函数化简函数为一个角的一个三角函数的形式,
(1)利用周期与函数f(x)的图象中相邻两条对称轴间的距离不小于
![](http://thumb.1010pic.com/pic5/latex/73.png)
,得到关系式,求出ω的取值范围;
(2)通过周期求出ω,通过函数的最大值,求出x的值,然后确定k的值.利用函数图象平移的原则:左加右减,上加下减由函数y=sinx的图象变换得到函数y=f(x)的图象.
点评:本题是中档题,考查三角函数的化简、公式的应用、周期的求法、最值的应用及函数图象的变换,还考查发现问题解决问题的能力、计算能力,是常考题型.