Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Transformations of Graphs
**Problem [5 points]:**
**Question:**
1. Describe how to obtain the graph of \( g \) from the graph of \( f(x) = \sqrt{x} \).
**Given Function:**
\[ f(x) = \sqrt{x - 3} + 2 \]
**Answer/Response:**
\[ 3 \text{ to the right}, 2 \text{ up} \]
### Explanation:
To transform the graph of \( f(x) = \sqrt{x} \) into \( g(x) = \sqrt{x - 3} + 2 \), perform the following steps:
1. **Horizontal Shift:**
- The term \( x - 3 \) inside the square root function indicates a horizontal shift.
- Specifically, \( x - 3 \) shifts the graph 3 units to the right.
2. **Vertical Shift:**
- The constant \( +2 \) outside the square root shifts the graph vertically.
- Add 2 to every y-coordinate, shifting the entire graph 2 units up.
### Visualization:
- Start with the base function \( f(x) = \sqrt{x} \).
- Move the entire graph 3 units to the right to account for \( -3 \) inside the square root.
- Then, move the graph 2 units up to account for the \( +2 \) outside the square root.
By following these transformations, the graph of the original function is shifted to obtain the new function \( g(x) = \sqrt{x - 3} + 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76555388-6a8d-4d01-aabe-a551666898f4%2Fca672252-63fa-40ad-b10a-5a0c2e3abfb7%2Fnbin0tf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transformations of Graphs
**Problem [5 points]:**
**Question:**
1. Describe how to obtain the graph of \( g \) from the graph of \( f(x) = \sqrt{x} \).
**Given Function:**
\[ f(x) = \sqrt{x - 3} + 2 \]
**Answer/Response:**
\[ 3 \text{ to the right}, 2 \text{ up} \]
### Explanation:
To transform the graph of \( f(x) = \sqrt{x} \) into \( g(x) = \sqrt{x - 3} + 2 \), perform the following steps:
1. **Horizontal Shift:**
- The term \( x - 3 \) inside the square root function indicates a horizontal shift.
- Specifically, \( x - 3 \) shifts the graph 3 units to the right.
2. **Vertical Shift:**
- The constant \( +2 \) outside the square root shifts the graph vertically.
- Add 2 to every y-coordinate, shifting the entire graph 2 units up.
### Visualization:
- Start with the base function \( f(x) = \sqrt{x} \).
- Move the entire graph 3 units to the right to account for \( -3 \) inside the square root.
- Then, move the graph 2 units up to account for the \( +2 \) outside the square root.
By following these transformations, the graph of the original function is shifted to obtain the new function \( g(x) = \sqrt{x - 3} + 2 \).
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