True or False: The graphs of the functions f(x)= cos x and f(x)= -cos x have the same period, amplitude, domain and range, and x-intercepts. Explain.
True.
The period of a cosine function is 2π. This means that for any value of x, cos(x) will repeat its values every 2π units. The graph of f(x) = cos(x) will complete one full cycle from 0 to 2π.
The negative sign in front of the cosine function -cos(x) does not change the period. It only reflects the graph across the x-axis. The graph of f(x) = -cos(x) will also complete one full cycle from 0 to 2π.
Therefore, both functions have the same period.
The amplitude of a cosine function is 1, which represents how much the graph oscillates between its maximum and minimum values. Both f(x) = cos(x) and f(x) = -cos(x) have an amplitude of 1.
The domain and range of both functions are all real numbers, as they can take any value of x.
Since cosine functions have x-intercepts at intervals of π, both f(x) = cos(x) and f(x) = -cos(x) will have x-intercepts at intervals of π.
In conclusion, both functions have the same period, amplitude, domain and range, and x-intercepts.