Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
#### Options for the Correct Graph:
Choose the correct graph below.
**Option A:**
**Option B:**
**Option C:**
**Option D:**
### Detailed Explanation of Graphs:
1. **Original Graph \( f(x) = 5x \)**
- The given graph shows a linear function \( f(x) = 5x \).
- The line passes through the origin (0,0) with a slope of 5, indicating that for every unit increase in \( x \), \( y \) increases by 5 units.
2. **Transforming to \( f(x) = 5x + 3 \)**
- To determine the graph of \( f(x) = 5x + 3 \), we need to apply a vertical shift of the original line.
- Adding 3 to the function causes the entire graph to shift up by 3 units.
- The slope remains the same (5), but the line will now intersect the y-axis at (0, 3) instead of (0, 0).
### Graph Options Analysis:
- **Option A**: Shows a graph passing through (0, 0), not shifted up.
- **Option B**: This graph intersects the y-axis at (0, 3) and maintains the same slope as the original.
- **Option C**: This graph shows an incorrect slope alteration.
- **Option D**: This graph shows a vertical shift but not by the correct amount.
**Correct Answer: Option B**
Graph **Option B** accurately represents the solution \( f(x) = 5x + 3 \), which is the vertical shift of the original function \( f(x) = 5x \) by 3 units upward.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45e6926f-54da-4187-acd9-fbbebfb1651c%2F0a6eee70-9fb9-46d1-9856-f834ebec9e52%2Fdzpiw88_processed.png&w=3840&q=75)
Transcribed Image Text:### Graph Equations and Transformations
#### Problem Statement:
The graph of \( f(x) = 5x \) is shown to the right. Determine the graph of \( f(x) = 5x + 3 \).
#### Options for the Correct Graph:
Choose the correct graph below.
**Option A:**
**Option B:**
**Option C:**
**Option D:**
### Detailed Explanation of Graphs:
1. **Original Graph \( f(x) = 5x \)**
- The given graph shows a linear function \( f(x) = 5x \).
- The line passes through the origin (0,0) with a slope of 5, indicating that for every unit increase in \( x \), \( y \) increases by 5 units.
2. **Transforming to \( f(x) = 5x + 3 \)**
- To determine the graph of \( f(x) = 5x + 3 \), we need to apply a vertical shift of the original line.
- Adding 3 to the function causes the entire graph to shift up by 3 units.
- The slope remains the same (5), but the line will now intersect the y-axis at (0, 3) instead of (0, 0).
### Graph Options Analysis:
- **Option A**: Shows a graph passing through (0, 0), not shifted up.
- **Option B**: This graph intersects the y-axis at (0, 3) and maintains the same slope as the original.
- **Option C**: This graph shows an incorrect slope alteration.
- **Option D**: This graph shows a vertical shift but not by the correct amount.
**Correct Answer: Option B**
Graph **Option B** accurately represents the solution \( f(x) = 5x + 3 \), which is the vertical shift of the original function \( f(x) = 5x \) by 3 units upward.
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