Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
F: #8.
![## Graphing a Tangent Function
### Problem Statement:
Graph the function:
\[ y = \tan\left(\frac{1}{2}x + \frac{\pi}{8}\right) \]
### Explanation:
The given equation is a transformation of the tangent function. The base function for tangent is:
\[ y = \tan(x) \]
#### Transformations:
1. **Horizontal Stretch/Compression:**
- The factor \(\frac{1}{2}\) inside the tangent function causes a horizontal stretch by a factor of 2. This means the period of the function is now \(2\pi\) instead of \(\pi\).
2. **Horizontal Shift:**
- The term \(\frac{\pi}{8}\) inside the argument of the tangent function results in a horizontal shift. Specifically, the function is shifted to the left by \(\frac{\pi}{8}\).
### Steps to Graph:
1. **Determine the period** of the new function. For the function \(y = \tan(Bx)\), the period is \(\frac{\pi}{|B|}\). In this case, \(B = \frac{1}{2}\), so the period is:
\[
\text{Period} = \frac{\pi}{|\frac{1}{2}|} = 2\pi
\]
2. **Identify the asymptotes** of the tangent function. For \(\tan(Bx + C)\):
\[
Bx + C = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{2B} - \frac{C}{B} + \frac{k\pi}{B}
\]
Substituting \(B = \frac{1}{2}\) and \(C = \frac{\pi}{8}\):
\[
x = \frac{\pi}{2} - \frac{\pi}{8} + k(2\pi) \implies x = \frac{3\pi}{8} + 2k\pi
\]
Therefore, the vertical asymptotes occur at \(x = \frac{3\pi}{8} + 2k\pi\), where \(k\) is an integer.
3. **Plot Key Points:**
- At \(x = -\frac{\pi}{8](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c34de7e-5935-4303-bb64-ce025f7422a2%2Fb6ae9fb2-ba48-41d9-bb8d-029b36734908%2F7hgd59y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Graphing a Tangent Function
### Problem Statement:
Graph the function:
\[ y = \tan\left(\frac{1}{2}x + \frac{\pi}{8}\right) \]
### Explanation:
The given equation is a transformation of the tangent function. The base function for tangent is:
\[ y = \tan(x) \]
#### Transformations:
1. **Horizontal Stretch/Compression:**
- The factor \(\frac{1}{2}\) inside the tangent function causes a horizontal stretch by a factor of 2. This means the period of the function is now \(2\pi\) instead of \(\pi\).
2. **Horizontal Shift:**
- The term \(\frac{\pi}{8}\) inside the argument of the tangent function results in a horizontal shift. Specifically, the function is shifted to the left by \(\frac{\pi}{8}\).
### Steps to Graph:
1. **Determine the period** of the new function. For the function \(y = \tan(Bx)\), the period is \(\frac{\pi}{|B|}\). In this case, \(B = \frac{1}{2}\), so the period is:
\[
\text{Period} = \frac{\pi}{|\frac{1}{2}|} = 2\pi
\]
2. **Identify the asymptotes** of the tangent function. For \(\tan(Bx + C)\):
\[
Bx + C = \frac{\pi}{2} + k\pi \implies x = \frac{\pi}{2B} - \frac{C}{B} + \frac{k\pi}{B}
\]
Substituting \(B = \frac{1}{2}\) and \(C = \frac{\pi}{8}\):
\[
x = \frac{\pi}{2} - \frac{\pi}{8} + k(2\pi) \implies x = \frac{3\pi}{8} + 2k\pi
\]
Therefore, the vertical asymptotes occur at \(x = \frac{3\pi}{8} + 2k\pi\), where \(k\) is an integer.
3. **Plot Key Points:**
- At \(x = -\frac{\pi}{8
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