Sketch the graph of the function f (x). Include two periods. :2 sin +4 Period: Amplitude Key Points
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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![### 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.
- **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b69bab9-9204-435f-8698-dc9cd67a06b8%2F4d0daf09-aa7c-4d76-a68e-24a101b1546d%2Fiyoh53k.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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