**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 ExplanationThe 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.### ConclusionUnderstanding 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 $.

Answered: Sketch two periods of the graph of the…

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

Section: Chapter Questions

Problem 1RQ

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Question

**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 $.

Expert Solution

Step 1

Given that

$h (x) = 3 s e c (\frac{π}{4} (x + 3)) c o m p a r e t o y = A s e c (B (x + C)) + D$

Here

A is Stretching factor

Period is $\frac{2 π}{B}$

A=3

B= $\frac{π}{4}$

C=3

D=0

Step by step

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