D. The graph of g (x) is the graph of f (x) translated 4 units down.
Explanation:
To show a translation left or right, we add or subtract a value to x before the function is applied. That is not what has happened to this function, therefore this was not a horizontal translation.
To show a translation up or down, we add or subtract a value at the end of a function. The function g(x) = f(x) - 4 subtracts 4 from the end of the function f(x); this means that the function has been translated down 4 units.
The frequency of sine function is given by the number of periods in a given range. For example:
Frequency for is 1 in the interval .
This means that, if we want another sine function with frequency 20% higher, we need that function to have a frequency of 1.2 in the interval .
To be easier to see we will consider interval instead of . In this interval has 5 periods, therefore our new sine function should have 6 periods.
Finally, as we can see in the graph, the function (in blue) has a frequency 20% higher than (in red). This can be easily seen counting the number of periods between 0 and for both functions. 5 for and 6 for .