### Trigonometry Graph Sketching Activity**4. Sketch the graph of the function $ f(x) $. Include two periods.**\[ f(x) = 2 \sin \left( \frac{x}{3} \right) + 4 \]a) **Period:** b) **Amplitude:** c) **Key Points:** d) **Graph:**The attached graph grid helps you to plot $ f(x) $ accurately over two periods. Use the function properties, such as its amplitude, period, and phase shift, to mark critical points and sketch the sinusoidal curve accordingly.#### Detailed Explanation of the Function Components:1. **Period Calculation:** The period of the sine function $\sin \left( \frac{x}{3} \right)$ is determined by the coefficient of $x$. The standard period of $\sin(x)$ is $2\pi$. For $\sin \left( \frac{x}{3} \right)$, it scales by the factor of $\frac{1}{3}$, making the period: \[ \text{Period} = 2\pi \cdot 3 = 6\pi \]2. **Amplitude:** The amplitude is the coefficient in front of the sine function, which is 2 in this case. This means the function oscillates 2 units above and below its central value.3. **Vertical Shift:** The function $ f(x) = 2 \sin \left( \frac{x}{3} \right) + 4 $ is vertically shifted up by 4 units.4. **Graphing Key Points:** - **Maximum points:** Occur at $ 4 + 2 = 6 $ - **Minimum points:** Occur at $ 4 - 2 = 2 $ - **Zero points:** Occur at baseline shifted to $ y = 4 $#### Graphing on Provided Grid:- **Step 1:** Identify and mark the range on the x-axis over which you will plot the function. Since the period is $6\pi$, you need to plot from $0$ to $12\pi$ to cover two periods.- **Step 2:** Mark key points (maximum, minimum, and zero crossings) using the amplitude and vertical shift.- **

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Answered: Sketch the graph of the function f (x).…

Math

Trigonometry

Sketch the graph of the function f (x). Include two periods. :2 sin +4 Period: Amplitude Key Points

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

### Trigonometry Graph Sketching Activity

**4. Sketch the graph of the function $ f(x) $. Include two periods.**

\[ f(x) = 2 \sin \left( \frac{x}{3} \right) + 4 \]

a) **Period:**

b) **Amplitude:**

c) **Key Points:**

d) **Graph:**

The attached graph grid helps you to plot $ f(x) $ accurately over two periods. Use the function properties, such as its amplitude, period, and phase shift, to mark critical points and sketch the sinusoidal curve accordingly.

#### Detailed Explanation of the Function Components:
1. **Period Calculation:**
The period of the sine function $\sin \left( \frac{x}{3} \right)$ is determined by the coefficient of $x$. The standard period of $\sin(x)$ is $2\pi$. For $\sin \left( \frac{x}{3} \right)$, it scales by the factor of $\frac{1}{3}$, making the period:
\[
\text{Period} = 2\pi \cdot 3 = 6\pi
\]

2. **Amplitude:**
The amplitude is the coefficient in front of the sine function, which is 2 in this case. This means the function oscillates 2 units above and below its central value.

3. **Vertical Shift:**
The function $ f(x) = 2 \sin \left( \frac{x}{3} \right) + 4 $ is vertically shifted up by 4 units.

4. **Graphing Key Points:**
- **Maximum points:** Occur at $ 4 + 2 = 6 $
- **Minimum points:** Occur at $ 4 - 2 = 2 $
- **Zero points:** Occur at baseline shifted to $ y = 4 $

#### Graphing on Provided Grid:
- **Step 1:** Identify and mark the range on the x-axis over which you will plot the function. Since the period is $6\pi$, you need to plot from $0$ to $12\pi$ to cover two periods.
- **Step 2:** Mark key points (maximum, minimum, and zero crossings) using the amplitude and vertical shift.
- **

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