Understanding the Function f(x) = 3 tan(4x)
To analyze the function f(x) = 3 tan(4x), we need to identify its period and phase shift. This function is a transformation of the basic tangent function.
1. Period of the Function
The period of the basic tangent function, tan(x), is π. However, when the tangent function is altered by a coefficient of the variable, as in tan(kx), the period changes. The new period can be calculated using the formula:
Period = π / |k|
In our case, k = 4. Thus, we can find the period:
Period = π / |4| = π / 4
2. Phase Shift of the Function
Next, we consider the phase shift. Typically, the phase shift is determined from functions of the form f(x) = a tan(b(x – c)) + d, where:
- a is the vertical stretch factor,
- b affects the period,
- c indicates a horizontal shift,
- d provides vertical shifting.
In our function f(x) = 3 tan(4x), we do not have any horizontal shifts represented (the term in parentheses is (4x), with no subtraction). This means:
Phase Shift = 0
Summary
In conclusion, for the function f(x) = 3 tan(4x), we find:
- Period = π / 4
- Phase Shift = 0
Understanding these characteristics helps us plot the graph of this function accurately and comprehend its behavior over intervals.