已知函数f(x)=2cos(ωx+φ)(ω>0,-π<φ<0)的最小正周期为π,其图象的一条对称轴是直线x=π8.

1个回答

  • (Ⅰ)由题意可得 [2π/ω]=π,∴ω=2.

    ∵图象的一条对称轴是直线x=

    π

    8,

    ∴2cos(2×[π/8]+φ)=±2.

    再由,-π<φ<0可得 φ=-[π/4],

    ∴函数f(x)=2cos(2x-[π/4]).

    (Ⅱ)令 2kπ≤2x-[π/4]≤2kπ+π,k∈z,解得 kπ+[π/8]≤x≤kπ+[5π/8],k∈z,故函数的减区间为[kπ+[π/8],kπ+[5π/8]],k∈z.

    (Ⅲ)列表:

    x 0 [π/8] [2π/8] [3π/8] [4π/8] [5π/8] [6π/8] [7π/8] π

    2x-[π/4] -[π/4] 0 [π/4] [π/2] [3π/4] π [5π/4] [3π/2] [π/4]

    f(x)