Find the period and phase shift (horizontal shift) of the function. Answer exactly. $$ f(x)=3 an (2 x+8) $$ Period: Phase shift:

Question

Find the period and phase shift (horizontal shift) of the function. Answer exactly. $$ f(x)=3 	an (2 x+8) $$ Period: Phase shift:

UpStudy AI · Accepted Answer

To determine the **period** and **phase shift** of the function $$ f(x) = 3 \tan(2x + 8) $$, we'll use the standard form of the tangent function:

$$
f(x) = A \tan(Bx + C)
$$

where:
- $$ A $$ is the amplitude (not relevant for period and phase shift).
- $$ B $$ affects the period.
- $$ C $$ affects the phase shift.

### 1. **Period**

The period of the tangent function is given by:

$$
\text{Period} = \frac{\pi}{|B|}
$$

For $$ f(x) = 3 \tan(2x + 8) $$:
- $$ B = 2 $$

Thus,

$$
\text{Period} = \frac{\pi}{2}
$$

### 2. **Phase Shift**

The phase shift (horizontal shift) is calculated by:

$$
\text{Phase Shift} = -\frac{C}{B}
$$

For $$ f(x) = 3 \tan(2x + 8) $$:
- $$ C = 8 $$
- $$ B = 2 $$

Thus,

$$
\text{Phase Shift} = -\frac{8}{2} = -4
$$

This means the graph is shifted **4 units to the left**.

---

**Final Answers:**

- **Period:** $$ \dfrac{\pi}{2} $$
- **Phase shift:** $$ -4 $$
The period is $$ \frac{\pi}{2} $$ and the phase shift is 4 units to the left.

Find the period and phase shift (horizontal shift) of the function. Answer exactly. \( f(x)=3 \tan (2 x+8) \) Period: Phase shift: