Question

Graph the function in the interval from 0 to 2pi. y = sin(θ − 2)

Answer 1

Answer: It is the second graph. See the attached figure.

Explanation:

1) You can consider the function y = sin(θ − 2) as a transformation of the parent function y = sin θ

2) Subtracting a constant from the argument of a function leads to shifting the graph as many units to the right as the constant.

3) So, subtracting 2 from the argument of y = sin θ shifts its graph 2 units to the right.

4) Since the x-intercepts of y = sin θ in the interval 0 to 2π are 0, π, and 2π, the x-intercepts of y = sin(θ − 2) will be 2, π + 2, and 2π + 2.

5) The x-intercepts are not enough to differentiate between some graphs, so take into account the local maxima or minima.

The local maxima of the function y = sin θ in the interval 0 to 2π are at x = π/2, and 3π/2,  then the local maximum of y = sin(θ − 2) is at π/2 + 2.

That is the second graph.

I attach the graph for avoiding confussions.

Answer 2

The answer is the second graphSOlution: y = sin(θ − 2) =>y = sin θ 
Subtract y = sin θ in the interval 0 to 2π are 0, π, and 2π,
the x-intercepts of y = sin(θ − 2) will be 2, π + 2, and 2π + 2.
then the local maximum of y = sin(θ − 2) is at π/2 + 2.
SO it is the second graph.