What are the period and phase shift of the function f(x) = 5 tan(2x + p)?

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The function given is f(x) = 5 tan(2x + p). Let’s break down the components of this function to determine the period and phase shift.

1. Understanding the Period

The standard period of the tangent function, tan(x), is π. However, when the argument of the tangent function is altered, the period changes based on the coefficient of x.

For the general form tan(kx), the period is calculated using the formula:

Period = π / |k|

In our case, k = 2, so we can substitute this value into the formula:

Period = π / |2| = π / 2

Thus, the period of the function f(x) = 5 tan(2x + p) is π / 2.

2. Determining the Phase Shift

The phase shift of a trigonometric function is determined by the horizontal transformation of the function inside the argument. For the function tan(2x + p), the phase shift can be calculated using the formula:

Phase Shift = - (C / k)

where C is the constant added or subtracted inside the argument (which is p here) and k is the coefficient of x (which is 2 in this case).

Substituting these values into the formula gives us:

Phase Shift = - (p / 2)

This indicates that if p is positive, the phase shift is to the left, while if p is negative, the phase shift is to the right.

Summary

  • The period of the function f(x) = 5 tan(2x + p) is π / 2.
  • The phase shift is – (p / 2).

Overall, the transformation of the original tangent function by both vertical scaling (due to the 5) and the manipulations of 2x + p provides us with critical shifts in the graphical representation of this function.

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