`f(x) = cos(x), g(x) = cos(2x)` Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts.
Based on your query, I am considering the three criteria that you have asked for:
- Amplitude: Their amplitude will be exactly the same, i.e. they have equal amplitude.
- Period: The period of g(x) is 1/2 of that of f(x). Thus, g(x) completes two cycles as f(x) completes one.
- Shift: There is no shift. Because the graphs originate form the same point as you can see and neither function has moved to the left or right compared to the other.
The graph of g(x) has the same amplitude as the graph of f(x). The period of g(x) is pi units, where the period of f(x) is 2pi units. This means that g(x) completes two cycles for every 1 cycle of f(x). The graph of g(x) is compressed horizontally compared to the graph of f(x). The graphs coincide every 2pi units.
The graph of f(x) in black and g(x) in red: