Sketch two periods of the graph of the function h(x) = 3 sec ( 7 (x+3)). Identify the stretching factor, period, and asymptotes. Enter the exact answers. Stretching factor = Number Period: P =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Sketch two periods of the graph of the function \( h(x) = 3 \sec \left( \frac{\pi}{4} (x + 3) \right) \). Identify the stretching factor, period, and asymptotes.**

Enter the exact answers.

- **Stretching factor** = [Number]

- **Period: \( P \) =** [Input Box]

Enter the asymptotes of the function on the domain \([-P, P]\).

To enter \(\pi\), type Pi.

**The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2; 4; 6 or \( x + 1; x - 1 \)). The order of the list does not matter.**
Transcribed Image Text:**Sketch two periods of the graph of the function \( h(x) = 3 \sec \left( \frac{\pi}{4} (x + 3) \right) \). Identify the stretching factor, period, and asymptotes.** Enter the exact answers. - **Stretching factor** = [Number] - **Period: \( P \) =** [Input Box] Enter the asymptotes of the function on the domain \([-P, P]\). To enter \(\pi\), type Pi. **The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2; 4; 6 or \( x + 1; x - 1 \)). The order of the list does not matter.**
## Understanding Asymptotes and Graphs of Trigonometric Functions

### Asymptotes

- **Function**: \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \)

**Identify the vertical asymptotes** of the function. These are the values of \( x \) where the function is undefined.

### Graph Explanation

The graph presented is of the function \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \).

#### Key Features:

1. **Vertical Asymptotes**: 
   - Identified by dashed red lines.
   - Positions at \( x = -5, -3, -1, 1, 3 \).

2. **Secant Function**: 
   - The graph of the secant function is characterized by repeating U-shaped curves above and below the x-axis.
   - There are sections of the graph close to the asymptotes that approach positive or negative infinity.

3. **Periodic Behavior**:
   - The function is periodic, repeating every \( \frac{8}{\pi} \) horizontally.

4. **Amplitude**: 
   - The vertical stretch factor is \( 3 \), which multiplies the standard secant function, making the peaks and troughs reach values of approximately \( 3 \) and \( -3 \).

5. **Coordinate Plane**:
   - The x-axis is marked at intervals of 1 unit.
   - The y-axis ranges from -10 to 10.

### Conclusion

Understanding the graph of a secant function involves recognizing the behavior near asymptotes, the amplitude, and the periodicity. For \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \), practice finding the vertical asymptotes and noting the stretching effect of the coefficient \( 3 \).
Transcribed Image Text:## Understanding Asymptotes and Graphs of Trigonometric Functions ### Asymptotes - **Function**: \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \) **Identify the vertical asymptotes** of the function. These are the values of \( x \) where the function is undefined. ### Graph Explanation The graph presented is of the function \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \). #### Key Features: 1. **Vertical Asymptotes**: - Identified by dashed red lines. - Positions at \( x = -5, -3, -1, 1, 3 \). 2. **Secant Function**: - The graph of the secant function is characterized by repeating U-shaped curves above and below the x-axis. - There are sections of the graph close to the asymptotes that approach positive or negative infinity. 3. **Periodic Behavior**: - The function is periodic, repeating every \( \frac{8}{\pi} \) horizontally. 4. **Amplitude**: - The vertical stretch factor is \( 3 \), which multiplies the standard secant function, making the peaks and troughs reach values of approximately \( 3 \) and \( -3 \). 5. **Coordinate Plane**: - The x-axis is marked at intervals of 1 unit. - The y-axis ranges from -10 to 10. ### Conclusion Understanding the graph of a secant function involves recognizing the behavior near asymptotes, the amplitude, and the periodicity. For \( h(x) = 3 \sec\left( \frac{\pi}{4}(x + 3) \right) \), practice finding the vertical asymptotes and noting the stretching effect of the coefficient \( 3 \).
Expert Solution
Step 1

Given that

h(x)=3secπ4(x+3)compare toy=A sec B(x+C)+D

Here

A is Stretching factor

Period is 2πB

A=3

B=π4

C=3

D=0

steps

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