Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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![### Function Transformations
#### Problem Statement
Suppose \( f(x) \) is a known function with a known graph.
**Question:**
Give a formula for a function, \( g(x) \), whose graph looks like the graph of \( f(x) \) but shifted up 4 units and reflected about the y-axis.
#### Solution
To transform the graph of \( f(x) \) to \( g(x) \) with the given modifications, follow these steps:
1. **Reflection About the Y-axis:**
- Reflecting a function \( f(x) \) about the y-axis involves replacing \( x \) with \( -x \) in the function. This gives us \( f(-x) \).
2. **Vertical Shift:**
- To shift a function up by 4 units, you add 4 to the function. Therefore, if the function is \( f(-x) \), it becomes \( f(-x) + 4 \).
Combining these two steps, the new function \( g(x) \) is obtained by reflecting \( f(x) \) about the y-axis and shifting it up by 4 units:
\[ g(x) = f(-x) + 4 \]
Thus, the desired formula for \( g(x) \) is:
\[ g(x) = f(-x) + 4 \]
This completes the transformation of the graph of \( f(x) \) as described.
### Visual Explanation
Since the problem does not provide a specific graph, we can imagine how the transformations would appear:
- **Reflection:** The graph of \( f(x) \) reflected across the y-axis.
- **Vertical Shift:** After reflecting, the whole graph is moved 4 units up along the y-axis.
Understanding these operations helps to visualize and handle transformations of functions effectively.](https://content.bartleby.com/qna-images/question/21d92c2f-6485-4b0b-a513-40194c3c2abd/663cb9cd-9c38-474a-9297-5120f72a6304/vkoxnn7_processed.jpeg)
Transcribed Image Text:### Function Transformations
#### Problem Statement
Suppose \( f(x) \) is a known function with a known graph.
**Question:**
Give a formula for a function, \( g(x) \), whose graph looks like the graph of \( f(x) \) but shifted up 4 units and reflected about the y-axis.
#### Solution
To transform the graph of \( f(x) \) to \( g(x) \) with the given modifications, follow these steps:
1. **Reflection About the Y-axis:**
- Reflecting a function \( f(x) \) about the y-axis involves replacing \( x \) with \( -x \) in the function. This gives us \( f(-x) \).
2. **Vertical Shift:**
- To shift a function up by 4 units, you add 4 to the function. Therefore, if the function is \( f(-x) \), it becomes \( f(-x) + 4 \).
Combining these two steps, the new function \( g(x) \) is obtained by reflecting \( f(x) \) about the y-axis and shifting it up by 4 units:
\[ g(x) = f(-x) + 4 \]
Thus, the desired formula for \( g(x) \) is:
\[ g(x) = f(-x) + 4 \]
This completes the transformation of the graph of \( f(x) \) as described.
### Visual Explanation
Since the problem does not provide a specific graph, we can imagine how the transformations would appear:
- **Reflection:** The graph of \( f(x) \) reflected across the y-axis.
- **Vertical Shift:** After reflecting, the whole graph is moved 4 units up along the y-axis.
Understanding these operations helps to visualize and handle transformations of functions effectively.
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