I can't do math problems in grade one. Please explain and point out why we can find the function half period from the distance between two adjacent symmetry axes.

Note: Two adjacent symmetry axes must pass through the highest point and the lowest point, so the distance between them is half a period.

f(x)=√3 sin(ωx+φ)-cos(ωx+φ)= 2 sin(ωx+φ-π/6)

(1) function y=f(x) The distance between two adjacent symmetry axes of an image is π/2, so the function period is π, and ω = 2 is obtained.

And the function is odd function, so f(0)=0, 2sin (φ-π/6) = 0, and the solution is φ = π/6.

So the analytic expression of the function is f (x) = 2sin (2x).

(2) The increasing interval of the function is (kπ-π/4, kπ+π/4), (k∈Z).

The subtraction interval of the function is (kπ-π/4, kπ+π/4), (k∈Z).