Which of the following is the graph of f(x) = 5 cos (2x)?

a.) graph with points 0, 5 and pi over 4, 0 and pi over 2, negative 5 and 3 pi over 4, 0 and pi, 5
b.) graph with points at 0, 0 and pi over 4, negative 3 and pi over 2, 0 and 3 pi over 4, 3 and pi, 0
c.) graph with points 0, negative 5 and pi over 4, 0 and pi over 2, 5 and 3 pi over 4, 0 and pi, negative 5
d.) graph with points at 0, 5 and pi over 4, 2 and pi over 2, negative 1 and 3 pi over 4, 2 and pi, 5

A) graph with points (0, 5) and (π/4, 0) and (π/2, -5) and (3π/4, 0) and (π, 5).

First of all, consider that the function f(x) is a transformation of the cosine function, generically given by the formula:

y = A·cos(ωx + Ф) + B

In our case, Ф and B are zero, which means that there is no vertical or horizontal shift. What changes is the amplitude A and the period given by T = 2π/ω.

The cosine function has a period of 2π, a range [-1, 1] starts at (0, 1) where it has a maximum, intersects the x-axis on x = π/2 and 3π/2, has a minimum at (π, -1) and another maximum at 2π.

Similarly, the function f(x) has a period of T = 2π/2 = π, which means that it completes a cycle in half the space of the cosine, and an amplitude A = 5, which means that it is stretched in a range [-5, 5]. Therefore, we can say that it starts at (0, 5) where it has a maximum, intersects the x-axis on x = π/4 and 3π/4, has a minimum at (π/2, -5) and another maximum at π.

The option describing this behavior is option A.