Question

已知函數(shù)y=cos²x+sinxcosx+1,x∈R.

(1)求它的振幅、周期和初相；

(2)用五點法作出它的簡圖；

(3)該函數(shù)的圖象是由y=sinx(x∈R)的圖象經(jīng)過怎樣的平移和伸縮變換得到的？

Answer 1

解：y=cos²x+sinxcosx+1=cos2x+sin2x+

=sin(2x+)+.

(1)y=cos²x+sinxcosx+1的振幅為A=，周期為T==π，初相為φ=.

(2)令x₁=2x+，則y=sin(2x+)+=sinx₁+，列出下表，并描出如下圖象：

x
x1	0		π		2π
y=sinx1	0	1	0	-1	0
y=sin(2x+)+

(3)解法一：將函數(shù)圖象依次作如下變換：

函數(shù)y=sinx的圖象函數(shù)y=sin(x+)的圖象

函數(shù)y=sin(2x+)的圖象

函數(shù)y=sin(2x+)的圖象

函數(shù)y=sin(2x+)+的圖象.

即得函數(shù)y=cos²x+sinxcosx+1的圖象.

解法二：函數(shù)y=sinx的圖象

函數(shù)y=sin2x的圖象函數(shù)y=sin(2x+)的圖象

函數(shù)y=sin(2x+)+的圖象

函數(shù)y=sin(2x+)+的圖象.

即得函數(shù)y=cos²x+sinxcosx+1的圖象.