Question

the function \\(f\\) is given by \\(f(\theta) = 2 \tan\left(\frac{\pi}{3}(\theta + 4)\
ight) - 6\\).
identify the period and phase shift of \\(f\\).

• the function \\(f\\) has a period of 3 units and a phase shift of -4 units.
• the function \\(f\\) has a period of 3 units and a phase shift of 4 units.
• the function \\(f\\) has a period of 4 units and a phase shift of -3 units.
• the function \\(f\\) has a period of 4 units and a phase shift of 3 units.

Explanation:

Identify the general form of the tangent function

Using the Periodic Functions knowledge point

$$ f(\theta) = a \tan(b(\theta - c)) + d $$

Calculate the period of the function

Using the Periodic Functions knowledge point

$$ \text{Period} = \frac{\pi}{|b|} = \frac{\pi}{\frac{\pi}{3}} = 3 $$

Determine the phase shift of the function

Using the Phase Shifts knowledge point

$$
LATEXBLOCK0
$$

Answer:

• (A) The function f has a period of 3 units and a phase shift of -4 units. (Correct answer)
• (B) The function f has a period of 3 units and a phase shift of 4 units.
• (C) The function f has a period of 4 units and a phase shift of -3 units.
• (D) The function f has a period of 4 units and a phase shift of 3 units.