Question

已知函數(shù)y=cos²x+sinx·cosx+1(x∈R).

(1)當(dāng)函數(shù)y取得最大值時(shí),求自變量x的集合.

(2)該函數(shù)的圖象可由y=sinx(x∈R)的圖象經(jīng)過(guò)怎樣的平移和伸縮變換得到?

Answer 1

解:(1)y=cos²x+sinxcosx+1

=(2cos²x-1)++(2sinxcosx)+1

=cos2x+sin2x+

=(cos2x·sin+sin2x·cos)+

=sin(2x+)+,

y取得最大值必須且只需

2x+=2kπ+,k∈Z,

即x=kπ+,k∈Z,

∴當(dāng)函數(shù)y取最大值時(shí),自變量x的集合為

{x|x=kπ+,k∈Z}.

(2)將函數(shù)y=sinx依次進(jìn)行如下變換:

①把函數(shù)y=sinx的圖象向左平移,得到函數(shù)y=sin(x+)的圖象;

②把得到的圖象上各點(diǎn)橫坐標(biāo)縮短到原來(lái)的(縱坐標(biāo)不變),得到函數(shù)y=sin(2x+)的圖象;

③把得到的圖象上各點(diǎn)縱坐標(biāo)縮短到原來(lái)的(橫坐標(biāo)不變),得到函數(shù)y=sin(2x+)的圖象;

④把得到的圖象向上平移個(gè)單位長(zhǎng)度,得到函數(shù)y=sin(2x+)+的圖象.